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==CENTRO INTERNACIONAL DE METODOS NUMERICOS EN INGENIERIA==
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==Acknowledgments==
 
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<div style="text-align: right; direction: ltr; margin-left: 1em;">
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<big>'''Monografías de Ingeniería Sísmica'''</big></div>
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<div style="text-align: right; direction: ltr; margin-left: 1em;">
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Editor A.H. Barbat</div>
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<div style="text-align: right; direction: ltr; margin-left: 1em;">
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<big>'''Seismic Protection of Cable-Stayed Bridges Applying Fluid Viscous Dampers'''</big></div>
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<div style="text-align: right; direction: ltr; margin-left: 1em;">
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<big>Galo E. Valdebenito </big></div>
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<div style="text-align: right; direction: ltr; margin-left: 1em;">
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<big>Ángel C. Aparicio</big></div>
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<div style="text-align: right; direction: ltr; margin-left: 1em;">
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<big>'''Acknowledgments'''</big></div>
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The authors wish to thank to the Department of Geotechnical Engineering and Geosciences and the Department of Construction Engineering at Technical University of Catalonia for their help and support during the doctorate years of Mr. Galo Valdebenito. This work is inspired in the basic result of that investigation. Likewise, thank to the Faculty of Engineering Sciences and the Department of Research and Development ''(DID)'' at Universidad Austral de Chile for their help and support in this publication.
 
The authors wish to thank to the Department of Geotechnical Engineering and Geosciences and the Department of Construction Engineering at Technical University of Catalonia for their help and support during the doctorate years of Mr. Galo Valdebenito. This work is inspired in the basic result of that investigation. Likewise, thank to the Faculty of Engineering Sciences and the Department of Research and Development ''(DID)'' at Universidad Austral de Chile for their help and support in this publication.
  
<div style="text-align: right; direction: ltr; margin-left: 1em;">
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==Preface==
<big>'''Preface'''</big></div>
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Earthquakes can be really destructive. There is no doubt. Recent seismic events have demonstrated the important effects on structures, and especially on bridges. In this sense, cable-stayed bridges are not an exception, although their seismic performance during recent events has been satisfactory. Their inherent condition as part of life-lines makes the seismic design and retrofitting of such structures be seriously considered.
 
Earthquakes can be really destructive. There is no doubt. Recent seismic events have demonstrated the important effects on structures, and especially on bridges. In this sense, cable-stayed bridges are not an exception, although their seismic performance during recent events has been satisfactory. Their inherent condition as part of life-lines makes the seismic design and retrofitting of such structures be seriously considered.
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<big>Llavaneras (Barcelona), October 2009.</big></div>
 
<big>Llavaneras (Barcelona), October 2009.</big></div>
  
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<div style="text-align: right; direction: ltr; margin-left: 1em;">
 
<div style="text-align: right; direction: ltr; margin-left: 1em;">
 
<big>'''Contents'''</big></div>
 
<big>'''Contents'''</big></div>
  
  
[[Image:draft_Samper_432909089-picture-Lienzo 2.svg|center|600px]]
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[[Image:draft_Samper_432909089-monograph-picture-Lienzo 2.svg|center|600px]]
  
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|  style="vertical-align: top;"|<big>'''Chapter 1. Introduction'''</big>
 
|  style="vertical-align: top;"|<big>'''Chapter 1. Introduction'''</big>
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==Chapter 1. Introduction==
<big>Chapter 1 </big></div>
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<div style="text-align: right; direction: ltr; margin-left: 1em;">
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<big>Introduction</big></div>
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:<big>1.1 Cable-Stayed Bridges and Seismic Protection </big>
 
:<big>1.1 Cable-Stayed Bridges and Seismic Protection </big>
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image1-c.png|516px]] </div>
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  [[Image:draft_Samper_432909089-monograph-image1-c.png|516px]] </div>
  
 
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image2-c.png|516px]] </div>
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  [[Image:draft_Samper_432909089-monograph-image2-c.png|516px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
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|  style="vertical-align: top;width: 64%;"| [[Image:draft_Samper_432909089-image3-c.jpeg|384px]]
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|  style="vertical-align: top;width: 63%;"|[[Image:draft_Samper_432909089-monograph-image3-c.jpeg|384px]]
  
 
'''Fig. 1.2''' (a) Energy Dissipation of a Traditional Bridge, (b) Energy Dissipation of a Seismic Isolated Bridge [Adapted from Jara and Casas, 2002]
 
'''Fig. 1.2''' (a) Energy Dissipation of a Traditional Bridge, (b) Energy Dissipation of a Seismic Isolated Bridge [Adapted from Jara and Casas, 2002]
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
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|-
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image4.png|312px]]
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|  style="text-align: center;vertical-align: top;width: 52%;"|[[Image:draft_Samper_432909089-monograph-image4.png|312px]]
  
 
'''Fig. 1.3''' Minimized Seismic Energy Penetration by Seismic Isolation and Energy Dissipation
 
'''Fig. 1.3''' Minimized Seismic Energy Penetration by Seismic Isolation and Energy Dissipation
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image5.jpeg|372px]] </div>
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  [[Image:draft_Samper_432909089-monograph-image5.jpeg|372px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-image6.png|336px]] '''</big>
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|  style="vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-monograph-image6.png|336px]] '''</big>
  
 
'''Fig. 1.5 '''Löscher-type Timber Bridge [Courtesy of the British Constructional Steelwork Association, Ltd]
 
'''Fig. 1.5 '''Löscher-type Timber Bridge [Courtesy of the British Constructional Steelwork Association, Ltd]
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image7.jpeg|288px]]
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|  style="vertical-align: top;width: 53%;"|[[Image:draft_Samper_432909089-monograph-image7.jpeg|288px]]
  
 
'''Fig. 1.6 '''Niagara Falls Bridge [Courtesy of the Niagara Falls Bridge Commission]
 
'''Fig. 1.6 '''Niagara Falls Bridge [Courtesy of the Niagara Falls Bridge Commission]
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-image8.jpeg|270px]] </span>
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|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-monograph-image8.jpeg|270px]] </span>
  
 
'''Fig. 1.7 '''Brooklyn Bridge [from [http://www.elclubdigital.com] www.elclubdigital.com]]
 
'''Fig. 1.7 '''Brooklyn Bridge [from [http://www.elclubdigital.com] www.elclubdigital.com]]
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{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
 
|-
 
|-
|  style="vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image9.jpeg|294px]]
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|  style="vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image9.jpeg|294px]]
  
 
'''Fig. 1.8''' The Bridge over the Donzère Canal, France [photo: J. Kerisel]
 
'''Fig. 1.8''' The Bridge over the Donzère Canal, France [photo: J. Kerisel]
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However, the real development came from Germany with papers published by Franz Dischinger and with the famous series of steel bridges crossing the river Rhine, as the Oberkassel Bridge, in Düsseldorf, Germany (Fig. 1.9).
 
However, the real development came from Germany with papers published by Franz Dischinger and with the famous series of steel bridges crossing the river Rhine, as the Oberkassel Bridge, in Düsseldorf, Germany (Fig. 1.9).
 
|-
 
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-image10.jpeg|288px]] </span>
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|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-monograph-image10.jpeg|288px]] </span>
  
 
'''Fig. 1.9 '''Oberkassel Bridge, Düsseldorf, Germany  
 
'''Fig. 1.9 '''Oberkassel Bridge, Düsseldorf, Germany  
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-image11.jpeg|282px]] </span>
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|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-monograph-image11.jpeg|282px]] </span>
  
 
'''Fig. 1.10''' Maracaibo Bridge, Venezuela [from en.structurae.de]
 
'''Fig. 1.10''' Maracaibo Bridge, Venezuela [from en.structurae.de]
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-image12.jpeg|300px]] </span>
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|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-monograph-image12.jpeg|300px]] </span>
  
 
'''Fig. 1.11 '''Barrios de Luna Bridge, Spain [from en.structurae.de]
 
'''Fig. 1.11 '''Barrios de Luna Bridge, Spain [from en.structurae.de]
|  style="text-align: center;vertical-align: top;width: 47%;"| [[Image:draft_Samper_432909089-image13.png|180px]]
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|  style="text-align: center;vertical-align: top;width: 46%;"|[[Image:draft_Samper_432909089-monograph-image13.png|180px]]
  
 
'''Fig. 1.12''' Yang Pu Bridge, China [photo: M. Virlogeux]
 
'''Fig. 1.12''' Yang Pu Bridge, China [photo: M. Virlogeux]
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-image14.jpeg|228px]] </span>
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|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-monograph-image14.jpeg|228px]] </span>
  
 
'''Fig. 1.13''' Normandie Bridge, France [from fr.structurae.de]
 
'''Fig. 1.13''' Normandie Bridge, France [from fr.structurae.de]
|  style="text-align: center;vertical-align: top;width: 47%;"| [[Image:draft_Samper_432909089-image15.jpeg|288px]]
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|  style="text-align: center;vertical-align: top;width: 46%;"|[[Image:draft_Samper_432909089-monograph-image15.jpeg|288px]]
  
 
'''Fig. 1.14''' Tatara Bridge,  Japan [from [http://www.answers.com] www.answers.com]]
 
'''Fig. 1.14''' Tatara Bridge,  Japan [from [http://www.answers.com] www.answers.com]]
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image16.png|600px]] </div>
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  [[Image:draft_Samper_432909089-monograph-image16.png|600px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image17.png|192px]]
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|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image17.png|192px]]
  
 
'''Fig. 1.16 '''Millau Bridge, France
 
'''Fig. 1.16 '''Millau Bridge, France
| style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-image18.jpeg|312px]] '''
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| style="vertical-align: top;width: 50%;"|[[Image:Draft_Samper_432909089_8898_monograph-image18.png|312px]]
  
 
'''Fig. 1.17''' Sutong Bridge, Nantong, China  
 
'''Fig. 1.17''' Sutong Bridge, Nantong, China  
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;width: 56%;"| [[Image:draft_Samper_432909089-image19-c.png|336px]]
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|  style="vertical-align: top;width: 54%;"|[[Image:draft_Samper_432909089-monograph-image19-c.png|336px]]
  
 
'''Fig. 1.18''' Rion-Antirion Viscous Dampers [Courtesy of ''FIP'' Industriale, Italy]
 
'''Fig. 1.18''' Rion-Antirion Viscous Dampers [Courtesy of ''FIP'' Industriale, Italy]
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 56%;"| [[Image:draft_Samper_432909089-image20.png|384px]]
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|  style="text-align: center;vertical-align: top;width: 55%;"|[[Image:draft_Samper_432909089-monograph-image20.png|384px]]
  
 
'''Fig. 1.19''' Dongting Lake Bridge, China
 
'''Fig. 1.19''' Dongting Lake Bridge, China
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|}
 
|}
  
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==Chapter 2. Fluid Viscous Damping Technology==
  
<div style="text-align: right; direction: ltr; margin-left: 1em;">
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:<big>2.1 General Overview </big>
<big>Chapter 2</big></div>
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<div style="text-align: right; direction: ltr; margin-left: 1em;">
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<big>Fluid Viscous Damping Technology</big></div>
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:<big>1.1 General Overview </big>
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Structures situated on seismic areas must be designed to resist earthquake ground motions. A fundamental rule regarding the seismic design of structures, express that higher damping implies lower induced seismic forces. For conventional constructions, the induced earthquake energy is dissipated by the structural components of the system designed to resist gravity loads. It is well known that damping level during the elastic seismic behaviour is generally very low, which implies not much dissipated energy. During strong ground motion, energy dissipation can be reached through damage of important structural elements, and considering only the resulting response forces within the structure due to an earthquake leads to massive structural dimensions, stiff structures with enormous local energy accumulation and plastic hinges. This strengthening method combined with usual bearing arrangements permits plastic deformations by way of leading to yield stress and cracks. In this sense, structural repair after an important seismic event is generally very expensive, the structure is set temporarily out of service and sometimes a lot of damaged structures must be demolished [Alvarez, 2004].
 
Structures situated on seismic areas must be designed to resist earthquake ground motions. A fundamental rule regarding the seismic design of structures, express that higher damping implies lower induced seismic forces. For conventional constructions, the induced earthquake energy is dissipated by the structural components of the system designed to resist gravity loads. It is well known that damping level during the elastic seismic behaviour is generally very low, which implies not much dissipated energy. During strong ground motion, energy dissipation can be reached through damage of important structural elements, and considering only the resulting response forces within the structure due to an earthquake leads to massive structural dimensions, stiff structures with enormous local energy accumulation and plastic hinges. This strengthening method combined with usual bearing arrangements permits plastic deformations by way of leading to yield stress and cracks. In this sense, structural repair after an important seismic event is generally very expensive, the structure is set temporarily out of service and sometimes a lot of damaged structures must be demolished [Alvarez, 2004].
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|-
|  style="vertical-align: top;width: 56%;"| [[Image:draft_Samper_432909089-image21-c.png|318px]]
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|  style="vertical-align: top;width: 55%;"|[[Image:draft_Samper_432909089-monograph-image21-c.png|318px]]
  
 
'''Fig. 2.1 '''Block Diagram of Passive Control System [Symans and Constantinou, 1999]
 
'''Fig. 2.1 '''Block Diagram of Passive Control System [Symans and Constantinou, 1999]
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The new tendencies regarding the seismic analysis and design of fluid viscous dampers capture the frequency dependence of such devices [Singh ''et al'', 2003]; the earthquake response of non-linear fluid viscous dampers [Peckan ''et al'', 1999; Lin and Chopra, 2002]; the seismic performance and behaviour of these devices during near-field ground motion [Tan ''et al'', 2005; Xu ''et al'', 2007] and the performance-based design of viscous dampers [Kim ''et al'', 2003; Li and Liang, 2007]. A state-of-the-art review can be found in the works of Lee and Taylor (2001) and Symans ''et al'' (2008).
 
The new tendencies regarding the seismic analysis and design of fluid viscous dampers capture the frequency dependence of such devices [Singh ''et al'', 2003]; the earthquake response of non-linear fluid viscous dampers [Peckan ''et al'', 1999; Lin and Chopra, 2002]; the seismic performance and behaviour of these devices during near-field ground motion [Tan ''et al'', 2005; Xu ''et al'', 2007] and the performance-based design of viscous dampers [Kim ''et al'', 2003; Li and Liang, 2007]. A state-of-the-art review can be found in the works of Lee and Taylor (2001) and Symans ''et al'' (2008).
  
:<big>1.2 Technological Aspects</big>
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:<big>2.2 Technological Aspects</big>
  
:<big>1.2.1 Historical Background</big>
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:<big>2.2.1 Historical Background</big>
  
 
As with many other types of engineered components, the requirements, needs and available funds from the military allowed rapid design evolution of fluid dampers to satisfy the needs of armed forces. Early fluid damping devices operated by viscous effects, where the operating medium was sheared by vanes or plates within the damper. Designs of this type were mere laboratory curiosities, since the maximum pressure available from shearing a fluid is limited by the onset of cavitation, which generally occurs at between 0.06 N/mm<sup>2</sup> and 0.1 N/mm<sup>2</sup>, depending on the viscosity of the fluid. This operating pressure was so low that for any given output level, a viscous damper was much larger and more costly than other types [Taylor, 1996].
 
As with many other types of engineered components, the requirements, needs and available funds from the military allowed rapid design evolution of fluid dampers to satisfy the needs of armed forces. Early fluid damping devices operated by viscous effects, where the operating medium was sheared by vanes or plates within the damper. Designs of this type were mere laboratory curiosities, since the maximum pressure available from shearing a fluid is limited by the onset of cavitation, which generally occurs at between 0.06 N/mm<sup>2</sup> and 0.1 N/mm<sup>2</sup>, depending on the viscosity of the fluid. This operating pressure was so low that for any given output level, a viscous damper was much larger and more costly than other types [Taylor, 1996].
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With the end of the Cold War in the late 80`s, much of this fully developed defence technology became available for civilian applications. In this context, demonstration of the benefits of damping technology on structures could take place immediately, using existing dampers and the seismic test facilities available at U.S. university research centres. In this sense, application of fluid viscous dampers as part of seismic energy dissipation systems was experimentally and analytically studied, being validated by extensive testing on one-sixth to one-half scale building and bridge models in the period 1990 – 1993 at the Multidisciplinary Centre for Earthquake Engineering Research (MCEER), located on the campus of the State University of New York at Buffalo in USA. Thus, implementation of fluid viscous damping technology began relatively swiftly, with wind protection usage beginning in 1993, and seismic protection usage beginning in 1995 [Taylor and Duflot, 2002].
 
With the end of the Cold War in the late 80`s, much of this fully developed defence technology became available for civilian applications. In this context, demonstration of the benefits of damping technology on structures could take place immediately, using existing dampers and the seismic test facilities available at U.S. university research centres. In this sense, application of fluid viscous dampers as part of seismic energy dissipation systems was experimentally and analytically studied, being validated by extensive testing on one-sixth to one-half scale building and bridge models in the period 1990 – 1993 at the Multidisciplinary Centre for Earthquake Engineering Research (MCEER), located on the campus of the State University of New York at Buffalo in USA. Thus, implementation of fluid viscous damping technology began relatively swiftly, with wind protection usage beginning in 1993, and seismic protection usage beginning in 1995 [Taylor and Duflot, 2002].
  
:<big>1.2.2 General Behaviour</big>
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:<big>2.2.2 General Behaviour</big>
  
 
Fluid viscous dampers operate on the principle of fluid flow through orifices. A stainless steel piston travels through chambers that are filled with silicone oil. The silicone oil is inert, non flammable, non toxic and stable for extremely long periods of time. The pressure difference between the two chambers cause silicone oil to flow through an orifice in the piston head and seismic energy is transformed into heat, which dissipates into the atmosphere. This associated temperature increase can be significant, particularly when the damper is subjected to long-duration or large-amplitude motions [Makris 1998; Makris ''et al'', 1998]. Mechanisms are available to compensate for the temperature rise such that the influence on the damper behaviour is relatively minor [Soong and Dargush, 1997]. However, the increase in temperature may be of concern due to the potential for heat-induced damage to the damper seals. In this case, the temperature rise can be reduced by reducing the pressure differential across the piston head (e.g., by employing a damper with a larger piston head) [Makris ''et al'', 1998]. Interestingly, although the damper is called a ''fluid viscous damper'', the fluid typically has a relatively low viscosity (e.g., silicone oil with a kinematic viscosity on the order of 0.001 m<sup>2</sup> /s at 20°C). The term ''fluid viscous damper ''is associated with the macroscopic behaviour of the damper which is essentially the same as that of an ideal linear or nonlinear viscous dashpot (i.e., the resisting force is directly related to the velocity). Generally, the fluid damper includes a double-ended piston rod (i.e., the piston rod projects outward from both sides of the piston head and exits the damper at both ends of the main cylinder). Such configurations are useful for minimizing the development of restoring forces (stiffness) due to fluid compression [Symans ''et al'', 2008]. The force/velocity relationship for this kind of damper can be characterized as ''F = C.V<sup>α</sup>'' where ''F'' is the output force, ''V'' the relative velocity across the damper; ''C'' is the damping coefficient and ''α'' is a constant exponent which is usually a value between 0.1 and 1.0 for earthquake protection, although at the present time some manufactures begin to apply dampers with very low damping coefficients, typically in the order of 0.02. Fluid viscous dampers can operate over temperature fluctuations ranging from –40°C to +70°C, and they have the unique ability to simultaneously reduce both stress and deflection within a structure subjected to a transient. This is because a fluid viscous damper varies its force only with velocity, which provides a response that is inherently out-of-phase with stresses due to flexing of the structure [Taylor and Duflot, 2002].
 
Fluid viscous dampers operate on the principle of fluid flow through orifices. A stainless steel piston travels through chambers that are filled with silicone oil. The silicone oil is inert, non flammable, non toxic and stable for extremely long periods of time. The pressure difference between the two chambers cause silicone oil to flow through an orifice in the piston head and seismic energy is transformed into heat, which dissipates into the atmosphere. This associated temperature increase can be significant, particularly when the damper is subjected to long-duration or large-amplitude motions [Makris 1998; Makris ''et al'', 1998]. Mechanisms are available to compensate for the temperature rise such that the influence on the damper behaviour is relatively minor [Soong and Dargush, 1997]. However, the increase in temperature may be of concern due to the potential for heat-induced damage to the damper seals. In this case, the temperature rise can be reduced by reducing the pressure differential across the piston head (e.g., by employing a damper with a larger piston head) [Makris ''et al'', 1998]. Interestingly, although the damper is called a ''fluid viscous damper'', the fluid typically has a relatively low viscosity (e.g., silicone oil with a kinematic viscosity on the order of 0.001 m<sup>2</sup> /s at 20°C). The term ''fluid viscous damper ''is associated with the macroscopic behaviour of the damper which is essentially the same as that of an ideal linear or nonlinear viscous dashpot (i.e., the resisting force is directly related to the velocity). Generally, the fluid damper includes a double-ended piston rod (i.e., the piston rod projects outward from both sides of the piston head and exits the damper at both ends of the main cylinder). Such configurations are useful for minimizing the development of restoring forces (stiffness) due to fluid compression [Symans ''et al'', 2008]. The force/velocity relationship for this kind of damper can be characterized as ''F = C.V<sup>α</sup>'' where ''F'' is the output force, ''V'' the relative velocity across the damper; ''C'' is the damping coefficient and ''α'' is a constant exponent which is usually a value between 0.1 and 1.0 for earthquake protection, although at the present time some manufactures begin to apply dampers with very low damping coefficients, typically in the order of 0.02. Fluid viscous dampers can operate over temperature fluctuations ranging from –40°C to +70°C, and they have the unique ability to simultaneously reduce both stress and deflection within a structure subjected to a transient. This is because a fluid viscous damper varies its force only with velocity, which provides a response that is inherently out-of-phase with stresses due to flexing of the structure [Taylor and Duflot, 2002].
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|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image22.png|282px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image22.png|282px]]
  
 
'''Fig 2.2 '''General view of a Fluid Viscous Damper [Courtesy of ''FIP ''Industriale s.P.a., Italy]
 
'''Fig 2.2 '''General view of a Fluid Viscous Damper [Courtesy of ''FIP ''Industriale s.P.a., Italy]
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image23.jpeg|252px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image23.jpeg|252px]]
  
 
'''Fig. 2.3 '''Fluid Viscous Dampers for De Las Piedras-High Speed Railway Bridge, Spain [Courtesy of ''Maurer Sönhe'', Germany]
 
'''Fig. 2.3 '''Fluid Viscous Dampers for De Las Piedras-High Speed Railway Bridge, Spain [Courtesy of ''Maurer Sönhe'', Germany]
Line 647: Line 619:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 60%;"| [[Image:draft_Samper_432909089-image24.png|372px]]
+
|  style="text-align: center;vertical-align: top;width: 59%;"|[[Image:draft_Samper_432909089-monograph-image24.png|372px]]
  
 
'''Fig. 2.4''' Typical Viscous Damper [Lee and Taylor, 2001]
 
'''Fig. 2.4''' Typical Viscous Damper [Lee and Taylor, 2001]
Line 668: Line 640:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image25.png|264px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image25.png|264px]]
  
 
'''Fig. 2.5''' Typical Plot of Base Shear Against Interstorey Drift [Lee and Taylor, 2001]
 
'''Fig. 2.5''' Typical Plot of Base Shear Against Interstorey Drift [Lee and Taylor, 2001]
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image26.png|240px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image26.png|240px]]
  
 
'''Fig. 2.6''' Base Shear Against Interstorey Drift with Added Dampers [Lee and Taylor, 2001]
 
'''Fig. 2.6''' Base Shear Against Interstorey Drift with Added Dampers [Lee and Taylor, 2001]
Line 710: Line 682:
  
 
|  style="vertical-align: top;"|In terms of the efficiency, the damping coefficient ξ relates to the efficiency η according to:
 
|  style="vertical-align: top;"|In terms of the efficiency, the damping coefficient ξ relates to the efficiency η according to:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image27.png|54px]] [Eq. 2.1]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">\xi =\frac{2}{\pi }\eta </math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.1]  
+
|}
+
 
This ends up in a maximum efficiency ''η = 96% ''for fluid viscous dampers.
 
This ends up in a maximum efficiency ''η = 96% ''for fluid viscous dampers.
 
|}
 
|}
Line 743: Line 709:
 
:* Non-toxic, not inflammable and not ageing fluids are applied.
 
:* Non-toxic, not inflammable and not ageing fluids are applied.
  
:<big>1.2.3 Application to Bridges</big>
+
:<big>2.2.3 Application to Bridges</big>
  
 
Decks for viaducts and long-span bridges require adequate expansion joints for large displacements under service conditions to absorb the effects of creep and thermal expansion. A common structural layout used in Europe, consists of continuous deck supported by ''POT'' devices [Priestley ''et al'', 1996]. By this way, the idea of employing devices with an insignificant response under long-period displacements and at the same time, capable of dissipating much induced seismic energy, was developed.
 
Decks for viaducts and long-span bridges require adequate expansion joints for large displacements under service conditions to absorb the effects of creep and thermal expansion. A common structural layout used in Europe, consists of continuous deck supported by ''POT'' devices [Priestley ''et al'', 1996]. By this way, the idea of employing devices with an insignificant response under long-period displacements and at the same time, capable of dissipating much induced seismic energy, was developed.
Line 754: Line 720:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<big>''' [[Image:draft_Samper_432909089-image28-c.png|432px]] '''</big></div>
+
<big>''' [[Image:draft_Samper_432909089-monograph-image28-c.png|432px]] '''</big></div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Line 761: Line 727:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image29.jpeg|276px]]
+
|  style="vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image29.jpeg|276px]]
  
 
'''Fig. 2.8''' Fluid Viscous Dampers at G4-Egnatia Motorway Bridge, Greece [Courtesy of ''Maurer Sönhe,'' Germany]
 
'''Fig. 2.8''' Fluid Viscous Dampers at G4-Egnatia Motorway Bridge, Greece [Courtesy of ''Maurer Sönhe,'' Germany]
|  style="vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image30.png|228px]]
+
|  style="vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image30.png|228px]]
  
 
'''Fig. 2.9''' 850 kN Capacity Damper for the Chun-Su Bridge, South Korea [Courtesy of ''FIP ''Industriale s.P.a., Italy]
 
'''Fig. 2.9''' 850 kN Capacity Damper for the Chun-Su Bridge, South Korea [Courtesy of ''FIP ''Industriale s.P.a., Italy]
Line 770: Line 736:
  
  
:<big>1.3 Mechanical Behaviour</big>
+
:<big>2.3 Mechanical Behaviour</big>
  
:<big>1.3.1 Energy Approach</big>
+
:<big>2.3.1 Energy Approach</big>
  
 
An earthquake is an energy phenomenon and therefore this energy character should be considered to achieve the best possible seismic protection for the structure. Without seismic protection system, the seismic energy is entering the structure very concentrated at the fixed axis. By means of shock transmission units the entering energy is distributed to several spots within the structure. In this case the energy input into the structure is still in same magnitude like without those devices, but now the energy is spread over the entire structure in more portions. By implementing additional energy dissipation capability, less energy is entering the structure, with the consequent response mitigation.
 
An earthquake is an energy phenomenon and therefore this energy character should be considered to achieve the best possible seismic protection for the structure. Without seismic protection system, the seismic energy is entering the structure very concentrated at the fixed axis. By means of shock transmission units the entering energy is distributed to several spots within the structure. In this case the energy input into the structure is still in same magnitude like without those devices, but now the energy is spread over the entire structure in more portions. By implementing additional energy dissipation capability, less energy is entering the structure, with the consequent response mitigation.
Line 778: Line 744:
 
The principles of physics that govern the effects of dissipation on the control of dynamic phenomena were studied more than two centuries ago [D`Alembert, Traité de Dynamique, 1743]. Nonetheless, their practical application has come about much later and within a much different time-frame in several sectors of engineering. As was previously exposed, the sector that was the first to adopt such damping technology was the military [France, 1897], followed by the automobile industry. In 1956 Housner already suggested an energy-based design of structures. Kato and Akiyama (1975) and Uang and Bertero (1990) made a valuable contribution to the development of the aspects of an energy-based approach, which presently meets with great concensus.
 
The principles of physics that govern the effects of dissipation on the control of dynamic phenomena were studied more than two centuries ago [D`Alembert, Traité de Dynamique, 1743]. Nonetheless, their practical application has come about much later and within a much different time-frame in several sectors of engineering. As was previously exposed, the sector that was the first to adopt such damping technology was the military [France, 1897], followed by the automobile industry. In 1956 Housner already suggested an energy-based design of structures. Kato and Akiyama (1975) and Uang and Bertero (1990) made a valuable contribution to the development of the aspects of an energy-based approach, which presently meets with great concensus.
  
The dynamic equation of a single-degree-of-freedom structure with mass ''m<sub>s</sub>'' damping coefficient ''c<sub>s</sub>'', stiffness ''k<sub>s</sub> ''and control force ''u'', subject to ground acceleration '' [[Image:draft_Samper_432909089-image31.png|36px]] ''is:
+
The dynamic equation of a single-degree-of-freedom structure with mass ''m<sub>s</sub>'' damping coefficient ''c<sub>s</sub>'', stiffness ''k<sub>s</sub> ''and control force ''u'', subject to ground acceleration '' [[Image:draft_Samper_432909089-monograph-image31.png|36px]] ''is:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image32.png|234px]] [Eq. 2.2]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
where  [[Image:draft_Samper_432909089-monograph-image33.png|30px]] [[Image:draft_Samper_432909089-monograph-image34.png|30px]] and  [[Image:draft_Samper_432909089-monograph-image35.png|30px]] are the displacement, velocity and acceleration responses respectively. The involved parameters are clearly explained in Fig. 2.10, which shows a simplified scheme for a single-degree-of-freedom system. Of course, each term in Eq. 2.2 is a force.
|-
+
| <math display="inline">m_s\overset{..}{x}(t)+c_s\overset{.}{x}(t)+k_sx(t)+</math><math>u=-m_s\overset{..}{x_g}(t)</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.2]  
+
|}
+
where  <math display="inline">x(t)</math> <math display="inline">\overset{.}{x}(t)</math> and  <math display="inline">\overset{..}{x}(t)</math> are the displacement, velocity and acceleration responses respectively. The involved parameters are clearly explained in Fig. 2.10, which shows a simplified scheme for a single-degree-of-freedom system. Of course, each term in Eq. 2.2 is a force.
+
  
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 45%;"| [[Image:draft_Samper_432909089-image36.png|228px]]
+
|  style="text-align: center;vertical-align: top;width: 44%;"|[[Image:draft_Samper_432909089-monograph-image36.png|228px]]
  
 
'''Fig. 2.10''' Complex Bridge Structure Explained with a Simplified Single Oscillation Mass
 
'''Fig. 2.10''' Complex Bridge Structure Explained with a Simplified Single Oscillation Mass
 
|  style="vertical-align: top;"|Integrating Eq. 2.2 with respect to ''x:''
 
|  style="vertical-align: top;"|Integrating Eq. 2.2 with respect to ''x:''
  
<math display="inline">\int_0^xm_s\overset{..}{x}(t)dx+\int_0^xc_s\overset{.}{x}(t)dx+</math><math>\int_0^xk_sx(t)dx+\int_0^xudx=-\int_0^xm_s\overset{..}{x}(t)dx</math>
+
[[Image:draft_Samper_432909089-monograph-image37.png|354px]]
  
 
where each term is now an energy component. Thus, we can define:
 
where each term is now an energy component. Thus, we can define:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image38.png|276px]] [Eq. 2.3]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
[[Image:draft_Samper_432909089-monograph-image39.png|240px]] [Eq. 2.4]
|-
+
| <math display="inline">\int_0^xm_s\ddot{x}dx=\int_0^xm_s\frac{d\dot{x}}{dt}dx=</math><math>\int_0^xm_s\dot{x}d\dot{x}=\frac{1}{2}m_s{\dot{x}}^2=</math><math>E_k</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.3]  
+
|}
+
<math display="inline">\int_0^xc_s\dot{x}dx=\int_0^xc_s\frac{dx}{dt}\frac{dt}{dt}dx=</math><math>\int_0^tc_s{\dot{x}}^2dt=E_v</math> [Eq. 2.4]
+
 
|}
 
|}
  
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image40.png|138px]] [Eq. 2.5]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
[[Image:draft_Samper_432909089-monograph-image41.png|72px]] [Eq. 2.6]
|-
+
 
| <math display="inline">\int_0^xk_sxdx=\frac{1}{2}k_sx^2=E_e</math>
+
[[Image:draft_Samper_432909089-monograph-image42.png|102px]] [Eq. 2.7]
|}
+
 
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.5]  
+
|}
+
<math display="inline">\int_0^xudx=E_h</math> [Eq. 2.6]
+
{| class='formulaSCP' style='width: 100%;'
+
|-
+
|
+
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">-\int_0^xm_s{\ddot{x}}_gdx=E_i</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.7]  
+
|}
+
 
An energy balance equation can be proposed in terms of the above defined:
 
An energy balance equation can be proposed in terms of the above defined:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image43.png|240px]] [Eq. 2.8]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">E_i\leq E_e+E_k+E_h+E_v=-\int_0^xm_s{\ddot{x}}_gdx</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.8]  
+
|}
+
 
where:
 
where:
  
Line 857: Line 793:
 
{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
 
|-
 
|-
|  style="vertical-align: top;width: 41%;"| [[Image:draft_Samper_432909089-image44.png|228px]]  
+
|  style="vertical-align: top;width: 40%;"|[[Image:draft_Samper_432909089-monograph-image44.png|228px]]  
 
|  style="vertical-align: top;"|'''Fig. 2.11 '''Concept of Energy Approach Considering the Energy Exchange Between Structure and Environment
 
|  style="vertical-align: top;"|'''Fig. 2.11 '''Concept of Energy Approach Considering the Energy Exchange Between Structure and Environment
 
|}
 
|}
  
  
<math display="inline">E_s=E_e+E_k</math> : Stored energy within structure
+
[[Image:draft_Samper_432909089-monograph-image45.png|84px]] : Stored energy within structure
  
<math display="inline">E_d=E_h+E_v</math> : Dissipated energy within structure
+
[[Image:draft_Samper_432909089-monograph-image46.png|90px]] : Dissipated energy within structure
  
 
Thus:
 
Thus:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image47.png|84px]] [Eq. 2.9]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">E_i\leq E_s+E_d</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.9]  
+
|}
+
 
The control force ''u'' by non-linear viscous dampers with damping coefficient ''c<sub>d</sub>''  is expressed as
 
The control force ''u'' by non-linear viscous dampers with damping coefficient ''c<sub>d</sub>''  is expressed as
  
<div style="text-align: right; direction: ltr; margin-left: 1em;">
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
  [[Image:draft_Samper_432909089-image48.png|120px]] [Eq. 2.10]</div>
+
  [[Image:draft_Samper_432909089-monograph-image48.png|120px]] [Eq. 2.10]</div>
  
 
In Eq. 2.10, the exponent ''N'' controls the damper nonlinearity and has typical values in the range of 0.10 to 1.0 for seismic applications. For the special case of ''N = 1'', Eq. 2.10 represents the force applied by linear viscous dampers. In the case of ''N = 0, ''Eq. 2.10 changes to a friction damper as follows:
 
In Eq. 2.10, the exponent ''N'' controls the damper nonlinearity and has typical values in the range of 0.10 to 1.0 for seismic applications. For the special case of ''N = 1'', Eq. 2.10 represents the force applied by linear viscous dampers. In the case of ''N = 0, ''Eq. 2.10 changes to a friction damper as follows:
  
<div style="text-align: right; direction: ltr; margin-left: 1em;">
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
  [[Image:draft_Samper_432909089-image49.png|90px]] [Eq. 2.11]</div>
+
  [[Image:draft_Samper_432909089-monograph-image49.png|90px]] [Eq. 2.11]</div>
  
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 56%;"| [[Image:draft_Samper_432909089-image50-c.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 55%;"|[[Image:draft_Samper_432909089-monograph-image50-c.png|312px]]
  
 
'''Fig. 2.12''' Plot of Force Against Velocity for Several Values of Damping Exponent ''N''  
 
'''Fig. 2.12''' Plot of Force Against Velocity for Several Values of Damping Exponent ''N''  
Line 898: Line 828:
  
 
Applying the force – velocity relationship expressed in Eq. 2.10 to Eq. 2.6 results:
 
Applying the force – velocity relationship expressed in Eq. 2.10 to Eq. 2.6 results:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image51.png|168px]] [Eq. 2.12]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">E_h=\int_0^xudx={\int_0^tc_d\overset{.}{\vert x\vert }}^{1+N}dt</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.12]  
+
|}
+
 
which represents the dissipated energy for a non-linear fluid viscous damper. The hysteretic behaviour of fluid viscous dampers can be plotted and shown in Fig. 2.13.
 
which represents the dissipated energy for a non-linear fluid viscous damper. The hysteretic behaviour of fluid viscous dampers can be plotted and shown in Fig. 2.13.
  
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 66%;"| [[Image:draft_Samper_432909089-image52.png|264px]]
+
|  style="text-align: center;vertical-align: top;width: 65%;"|[[Image:draft_Samper_432909089-monograph-image52.png|264px]]
  
 
'''Fig. 2.13''' Hysteresis Loops for Linear and Non-linear Fluid Viscous Dampers [Lee and Taylor, 2001]
 
'''Fig. 2.13''' Hysteresis Loops for Linear and Non-linear Fluid Viscous Dampers [Lee and Taylor, 2001]
Line 921: Line 845:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<big>''' [[Image:draft_Samper_432909089-image53.png|486px]] '''</big></div>
+
<big>''' [[Image:draft_Samper_432909089-monograph-image53.png|486px]] '''</big></div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
'''Fig. 2.14''' Force – Displacement Hysteretic Diagram of a Viscous Damper, ''N = 0.15'' [Courtesy of ''FIP ''Industriale, s.P.a., Italy]</div>
 
'''Fig. 2.14''' Force – Displacement Hysteretic Diagram of a Viscous Damper, ''N = 0.15'' [Courtesy of ''FIP ''Industriale, s.P.a., Italy]</div>
  
:<big>1.3.2 Effect of the Damper Parameters</big>
+
:<big>2.3.2 Effect of the Damper Parameters</big>
  
:''<sub>1.3.2.1 </sub>Damping coefficient c<sub>d</sub>''
+
:''2.3.2.1 Damping coefficient c<sub>d</sub>''
  
 
In general terms, for viscous dampers, ''c<sub>d</sub>'' does not affect the shape of the hysteretic force – displacement cycle; however, an increase of the value of this parameter increases the energy dissipation capacity and the maximum force in the device [Guerreiro, 2006]. In this sense, the work carried out by Virtuoso ''et al ''(2000) studies the modelling of the seismic behaviour of bridges with added viscous dampers, analyzing the effects of the constant ''c<sub>d</sub>'' (here called ''C'')''.'' To allow an analysis on the influence of that parameter on the structural response, values of the constant ''C ''between 0.10 and 10 were considered, since those values, together with the values considered for the parameter ''N ''(here called ''α''), will cover forces corresponding to seismic coefficient varying from 1% to 50% of the weight. In this study a set of five artificial accelerograms compatible with the response spectrum defined in Eurocode 8 – Part 2 [CEN, 1998b] with a peak ground acceleration of 0.30g, type B soil and 30 sec total duration of the series, were used. Two extreme cases were considered: a solution without elastic stiffness (deck totally free over the piers) and a solution with elastic stiffness (low stiffness elastic connection between the piers and the deck). Also, in this research the configurations of the force – velocity relation curves were presented for different values of ''α'', corresponding to the linear branch, which, were defined by the origin and the point corresponding to 10% of the maximum velocity and force corresponding to the defined seismic action and obtained without the consideration of the linear branch. Figs. 2.15 and 2.16 show maximum forces and displacements in the viscous damper without and with elastic stiffness respectively. They show that solution involving a higher displacement control always lead to higher force levels in the device. It is also possible to observe that the more efficient solutions, with better displacement control for the same force level, generally corresponds to low ''α'' values. Likewise, for device solutions with low values of the constant ''C'', the elastic stiffness of the structure has an important contribution on the displacement control. It is important to notice that the contribution of the elastic force is out of phase with the one transmitted by the devices, what means that, in a solution of this type there is always a force restraining the movement of the deck. The problem is that the forces transmitted to the structure must be controlled to limit the contribution of the piers to values lower than their elastic limit.
 
In general terms, for viscous dampers, ''c<sub>d</sub>'' does not affect the shape of the hysteretic force – displacement cycle; however, an increase of the value of this parameter increases the energy dissipation capacity and the maximum force in the device [Guerreiro, 2006]. In this sense, the work carried out by Virtuoso ''et al ''(2000) studies the modelling of the seismic behaviour of bridges with added viscous dampers, analyzing the effects of the constant ''c<sub>d</sub>'' (here called ''C'')''.'' To allow an analysis on the influence of that parameter on the structural response, values of the constant ''C ''between 0.10 and 10 were considered, since those values, together with the values considered for the parameter ''N ''(here called ''α''), will cover forces corresponding to seismic coefficient varying from 1% to 50% of the weight. In this study a set of five artificial accelerograms compatible with the response spectrum defined in Eurocode 8 – Part 2 [CEN, 1998b] with a peak ground acceleration of 0.30g, type B soil and 30 sec total duration of the series, were used. Two extreme cases were considered: a solution without elastic stiffness (deck totally free over the piers) and a solution with elastic stiffness (low stiffness elastic connection between the piers and the deck). Also, in this research the configurations of the force – velocity relation curves were presented for different values of ''α'', corresponding to the linear branch, which, were defined by the origin and the point corresponding to 10% of the maximum velocity and force corresponding to the defined seismic action and obtained without the consideration of the linear branch. Figs. 2.15 and 2.16 show maximum forces and displacements in the viscous damper without and with elastic stiffness respectively. They show that solution involving a higher displacement control always lead to higher force levels in the device. It is also possible to observe that the more efficient solutions, with better displacement control for the same force level, generally corresponds to low ''α'' values. Likewise, for device solutions with low values of the constant ''C'', the elastic stiffness of the structure has an important contribution on the displacement control. It is important to notice that the contribution of the elastic force is out of phase with the one transmitted by the devices, what means that, in a solution of this type there is always a force restraining the movement of the deck. The problem is that the forces transmitted to the structure must be controlled to limit the contribution of the piers to values lower than their elastic limit.
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image54.png|366px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image54.png|366px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Line 939: Line 863:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image55.png|366px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image55.png|366px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;width: 62%;"| [[Image:draft_Samper_432909089-image56.png|234px]]
+
|  style="vertical-align: top;width: 61%;"|[[Image:draft_Samper_432909089-monograph-image56.png|234px]]
  
 
'''Fig. 2.17 '''Maximum Forces in the Structure – Solution with Dampers and Elastic Stiffness [Virtuoso ''et al'', 2000]
 
'''Fig. 2.17 '''Maximum Forces in the Structure – Solution with Dampers and Elastic Stiffness [Virtuoso ''et al'', 2000]
Line 959: Line 883:
  
  
:''1.3.2.2 Velocity exponent N''
+
:''2.3.2.2 Velocity exponent N''
  
 
The damping exponent ''N'' represents the essence of the non-linear behaviour of fluid viscous dampers. On the contrary of the damping coefficient ''c<sub>d</sub>'', this parameter does not affect the size of the hysteretic force-displacement cycle and for that reason incidence of this parameter on the seismic response is not decisive as occurs with the damping coefficient, aspect enlarged in 2.3.2.1. Changes in the ''N-''exponent imply changes in the shape of the hysteretic force-displacement cycle, as was explained in Fig. 2.13. Low damping exponents tend to expose rectangular force-displacement hysteresis, as well as linear behaviour implies more elliptical force-displacement hysteresis cycles. The more practical incidence of the ''N-''exponent relates with the damper forces, depending on the relative velocities.
 
The damping exponent ''N'' represents the essence of the non-linear behaviour of fluid viscous dampers. On the contrary of the damping coefficient ''c<sub>d</sub>'', this parameter does not affect the size of the hysteretic force-displacement cycle and for that reason incidence of this parameter on the seismic response is not decisive as occurs with the damping coefficient, aspect enlarged in 2.3.2.1. Changes in the ''N-''exponent imply changes in the shape of the hysteretic force-displacement cycle, as was explained in Fig. 2.13. Low damping exponents tend to expose rectangular force-displacement hysteresis, as well as linear behaviour implies more elliptical force-displacement hysteresis cycles. The more practical incidence of the ''N-''exponent relates with the damper forces, depending on the relative velocities.
Line 965: Line 889:
 
If we consider the force at the dampers ''F'' as a function of the exponent ''N'', we can write
 
If we consider the force at the dampers ''F'' as a function of the exponent ''N'', we can write
  
<math display="inline">F(N)=c_d{\dot{x}}^N</math> where ''c<sub>d</sub>'' is a constant.
+
[[Image:draft_Samper_432909089-monograph-image57.png|90px]] where ''c<sub>d</sub>'' is a constant.
  
If ''c<sub>d</sub>'' is constant, ''F'' is maximum if '' <math display="inline">{\dot{x}}^N</math> ''is maximum.
+
If ''c<sub>d</sub>'' is constant, ''F'' is maximum if '' [[Image:draft_Samper_432909089-monograph-image58.png|18px]] ''is maximum.
  
Let  <math display="inline">f(N)={\dot{x}}^N</math> .Maximizing ''f'':
+
Let  [[Image:draft_Samper_432909089-monograph-image59.png|78px]] .Maximizing ''f'':
  
<math display="inline">f\grave (N)={\dot{x}}^Nlog\dot{x}=0</math> if and only if   
+
[[Image:draft_Samper_432909089-monograph-image60.png|144px]] if and only if   
 
{|
 
{|
 
|-
 
|-
| <math display="inline">{\dot{x}}^N=0\mbox{ }or\mbox{ }log\dot{x}=0</math>
+
| [[Image:draft_Samper_432909089-monograph-image61.png|126px]]
| <math display="inline">{\dot{x}}^N\not =0\mbox{ }\forall \mbox{ }N\geq 0\mbox{​}\wedge \mbox{​}\dot{x}\not =</math><math>0\mbox{​}</math>
+
| [[Image:draft_Samper_432909089-monograph-image62.png|center|150px]]
 
|}
 
|}
  
  
<math display="inline">log\dot{x}=0</math> if and only if  <math display="inline">\dot{x}=1</math> which implies a constant force ''F = c<sub>d</sub>''
+
[[Image:draft_Samper_432909089-monograph-image63.png|60px]] if and only if  [[Image:draft_Samper_432909089-monograph-image64.png|36px]] which implies a constant force ''F = c<sub>d</sub>''
  
 
Analyzing ''f'' in its domain:
 
Analyzing ''f'' in its domain:
  
:(1) If  <math display="inline">\dot{x}>1</math> then ''f'' is maximum if ''N'' is maximum, that is to say, if  <math display="inline">N\rightarrow \infty </math>
+
:(1) If  [[Image:draft_Samper_432909089-monograph-image65.png|36px]] then ''f'' is maximum if ''N'' is maximum, that is to say, if  [[Image:draft_Samper_432909089-monograph-image66.png|54px]]
  
:(2) If  <math display="inline">0\leq \dot{x}<1</math> then  <math display="inline">f(N)={\dot{x}}^N</math> can be written as <math display="inline">f(N)={\left(\frac{1}{m}\right)}^N=\frac{1}{m^N};\mbox{ }m\in \mathbb{R}</math> .
+
:(2) If  [[Image:draft_Samper_432909089-monograph-image67.png|60px]] then  [[Image:draft_Samper_432909089-monograph-image68.png|78px]] can be written as [[Image:draft_Samper_432909089-monograph-image69.png|192px]] .
  
Then, ''f'' is maximum if  <math display="inline">m^N</math> is little, which implies <math display="inline">N\rightarrow 0</math> .
+
Then, ''f'' is maximum if  [[Image:draft_Samper_432909089-monograph-image70.png|24px]] is little, which implies [[Image:draft_Samper_432909089-monograph-image71.png|48px]] .
  
This analytical approach shows that the critical point is <math display="inline">\dot{x}=1</math> . Being the damper velocities larger than 1, the maximum damper forces are obtained for high values of the damping exponent, on the contrary of the case where the damper velocities are lower than 1, in which the maximum damper forces are obtained when ''N'' is close to zero, that is to say, for non-linear dampers. Graphically, the above-mentioned can be clearly exposed in Fig. 2.18. Fig. 2.18 exposes variations of the dampers forces with the velocity exponent ''N'' for some common damper velocities. From these results, it is necessary to be cautious if velocity pulses are considered in the presence of linear dampers or dampers with ''N'' > 1. Likewise, similar considerations are necessary to take into account if non-linear dampers are considered in the presence of low velocities.
+
This analytical approach shows that the critical point is [[Image:draft_Samper_432909089-monograph-image72.png|36px]] . Being the damper velocities larger than 1, the maximum damper forces are obtained for high values of the damping exponent, on the contrary of the case where the damper velocities are lower than 1, in which the maximum damper forces are obtained when ''N'' is close to zero, that is to say, for non-linear dampers. Graphically, the above-mentioned can be clearly exposed in Fig. 2.18. Fig. 2.18 exposes variations of the dampers forces with the velocity exponent ''N'' for some common damper velocities. From these results, it is necessary to be cautious if velocity pulses are considered in the presence of linear dampers or dampers with ''N'' > 1. Likewise, similar considerations are necessary to take into account if non-linear dampers are considered in the presence of low velocities.
  
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 62%;"| [[Image:draft_Samper_432909089-image73.png|390px]]
+
|  style="text-align: center;vertical-align: top;width: 61%;"|[[Image:draft_Samper_432909089-monograph-image73.png|390px]]
  
 
'''Fig. 2.18''' Plot of Damper Forces as Function of the ''N''-exponent for Several Velocities and ''c<sub>d</sub>'' = 10 MN/(m/s)<sup>N</sup>
 
'''Fig. 2.18''' Plot of Damper Forces as Function of the ''N''-exponent for Several Velocities and ''c<sub>d</sub>'' = 10 MN/(m/s)<sup>N</sup>
Line 1,128: Line 1,052:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 47%;"| [[Image:draft_Samper_432909089-image74-c.png|240px]]
+
|  style="text-align: center;vertical-align: top;width: 47%;"|[[Image:draft_Samper_432909089-monograph-image74-c.png|240px]]
  
 
'''Fig. 2.19''' Extra-low Damping for Viscous Damper with ''N''=0.015
 
'''Fig. 2.19''' Extra-low Damping for Viscous Damper with ''N''=0.015
Line 1,137: Line 1,061:
  
  
:<big>1.3.3 Non-linear Viscous Dampers</big>
+
:<big>2.3.3 Non-linear Viscous Dampers</big>
  
:''1.3.3.1 Earthquake response''
+
:''2.3.3.1 Earthquake response''
  
 
Numerous experimental and analytical investigations have focused on linear fluid viscous dampers, because they can be modelled simply by a linear dashpot. While being effective in reducing seismic demands on the structure, linear viscous dampers may develop excessive damper forces in applications where large structural velocities can occur, as for example in long period structures subjected to intense ground shaking, especially in the near-fault region. Recently, some researchers and earthquake engineering professionals have begun to focus on fluid viscous dampers exhibiting non-linear Force-Velocity relationship because of their ability to limit the peak damper force at large structural velocities while still providing sufficient supplemental damping [Lin and Chopra, 2002; Symans ''et al'', 2008].
 
Numerous experimental and analytical investigations have focused on linear fluid viscous dampers, because they can be modelled simply by a linear dashpot. While being effective in reducing seismic demands on the structure, linear viscous dampers may develop excessive damper forces in applications where large structural velocities can occur, as for example in long period structures subjected to intense ground shaking, especially in the near-fault region. Recently, some researchers and earthquake engineering professionals have begun to focus on fluid viscous dampers exhibiting non-linear Force-Velocity relationship because of their ability to limit the peak damper force at large structural velocities while still providing sufficient supplemental damping [Lin and Chopra, 2002; Symans ''et al'', 2008].
Line 1,148: Line 1,072:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<sub> [[Image:draft_Samper_432909089-image75.png|600px]] </sub></div>
+
<sub> [[Image:draft_Samper_432909089-monograph-image75.png|600px]] </sub></div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;width: 58%;"| [[Image:draft_Samper_432909089-image76.png|336px]]
+
|  style="vertical-align: top;width: 57%;"|[[Image:draft_Samper_432909089-monograph-image76.png|336px]]
  
 
'''Fig. 2.21''' Influence of Damper Non-linearity on Mean Peak Responses, ''r'': Deformation, Relative Velocity, and Total Acceleration for Systems with ''ξ<sub>d</sub> = 30%'' [Lin and Chopra, 2002].
 
'''Fig. 2.21''' Influence of Damper Non-linearity on Mean Peak Responses, ''r'': Deformation, Relative Velocity, and Total Acceleration for Systems with ''ξ<sub>d</sub> = 30%'' [Lin and Chopra, 2002].
|  style="vertical-align: top;width: 43%;"| [[Image:draft_Samper_432909089-image77.png|228px]]
+
|  style="vertical-align: top;width: 42%;"|[[Image:draft_Samper_432909089-monograph-image77.png|228px]]
  
 
'''Fig. 2.22 '''Response History for Deformation of a SDF System (''T<sub>n</sub> = 1 sec, ξ = 5%'') with ''ξ<sub>d</sub> = 15%'' [Lin and Chopra, 2002].
 
'''Fig. 2.22 '''Response History for Deformation of a SDF System (''T<sub>n</sub> = 1 sec, ξ = 5%'') with ''ξ<sub>d</sub> = 15%'' [Lin and Chopra, 2002].
Line 1,167: Line 1,091:
  
 
It is important to say that for a given force and displacement amplitude, the energy dissipated per cycle for a nonlinear fluid damper is larger, by a factor λ/π (where λ is a parameter whose value depends exclusively on the velocity exponent), than that for the linear case and increases monotonically with reducing velocity exponent (up to a theoretical limit of 4/π=1.27 which corresponds to a velocity exponent of zero); however, the additional energy dissipation afforded by the nonlinear dampers is minimal. For a given frequency of motion, ''ω'', and displacement amplitude, ''x<sub>0</sub>'', to dissipate the same amount of energy per cycle, the damping coefficient of the nonlinear damper, ''c<sub>dNL</sub>'', must be larger than that of the linear damper, ''c<sub>dL</sub>'', as given by
 
It is important to say that for a given force and displacement amplitude, the energy dissipated per cycle for a nonlinear fluid damper is larger, by a factor λ/π (where λ is a parameter whose value depends exclusively on the velocity exponent), than that for the linear case and increases monotonically with reducing velocity exponent (up to a theoretical limit of 4/π=1.27 which corresponds to a velocity exponent of zero); however, the additional energy dissipation afforded by the nonlinear dampers is minimal. For a given frequency of motion, ''ω'', and displacement amplitude, ''x<sub>0</sub>'', to dissipate the same amount of energy per cycle, the damping coefficient of the nonlinear damper, ''c<sub>dNL</sub>'', must be larger than that of the linear damper, ''c<sub>dL</sub>'', as given by
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image78.png|138px]] [Eq. 2.13]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">c_{dNL}=c_{dL}\frac{\pi }{\lambda }{\left(\omega x_0\right)}^{1-\alpha }</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.13]  
+
|}
+
 
As an example, for a frequency of 1.0 Hz and displacement amplitude of 5 cm, the damping coefficient of a nonlinear damper with velocity exponent of 0.5 must be approximately three times larger than that of a linear damper to dissipate the same amount of energy per cycle. Conversely, if nonlinear dampers are used to limit the damper force, a reduction in energy dissipation capacity as compared to the case of linear dampers would be accepted [Symans ''et al'', 2008].
 
As an example, for a frequency of 1.0 Hz and displacement amplitude of 5 cm, the damping coefficient of a nonlinear damper with velocity exponent of 0.5 must be approximately three times larger than that of a linear damper to dissipate the same amount of energy per cycle. Conversely, if nonlinear dampers are used to limit the damper force, a reduction in energy dissipation capacity as compared to the case of linear dampers would be accepted [Symans ''et al'', 2008].
  
 
A last aspect to consider regarding the earthquake response of non-linear fluid viscous dampers, is that the earthquake-induced force in a non-linear viscous damper can be estimated from the damper force in a corresponding system with linear viscous damping, its peak deformation, and peak relative velocity; however, the relative velocity should not be approximated by the pseudo-velocity as this approximation introduces a large error in the damper force. In fact, if spectral pseudo-velocities are used, they are based on design displacements ''(S<sub>v</sub> = ω<sub>0</sub>S<sub>d</sub>)''. It is well known that effectiveness of non-linear viscous dampers is highly dependent on operating velocities, being necessary to have reliable estimates of the true velocity in the device [Pekcan ''et al'', 1999; Lin and Chopra, 2002].
 
A last aspect to consider regarding the earthquake response of non-linear fluid viscous dampers, is that the earthquake-induced force in a non-linear viscous damper can be estimated from the damper force in a corresponding system with linear viscous damping, its peak deformation, and peak relative velocity; however, the relative velocity should not be approximated by the pseudo-velocity as this approximation introduces a large error in the damper force. In fact, if spectral pseudo-velocities are used, they are based on design displacements ''(S<sub>v</sub> = ω<sub>0</sub>S<sub>d</sub>)''. It is well known that effectiveness of non-linear viscous dampers is highly dependent on operating velocities, being necessary to have reliable estimates of the true velocity in the device [Pekcan ''et al'', 1999; Lin and Chopra, 2002].
  
:''1.3.3.2 Equivalent linear viscous damping''
+
:''2.3.3.2 Equivalent linear viscous damping''
  
 
The energy dissipation capacity of a fluid viscous damper can be characterized by the supplemental damping ratio'' ξ<sub>d</sub>'' and its non-linearity by the parameter ''N; ''and it is found that the structural response is most effectively investigated in terms of these parameters because they are dimensionless and independent, and the structural response varies linearly with the excitation intensity [Lin and Chopra, 2002]. In this sense, a system with non-linear dampers is usually replaced by an equivalent linear system, with its properties determined using different methods: equalling the energy dissipated in the two systems [Jacobsen, 1930; Fabunmi, 1985]; equalling power consumption in the two systems [Pekcan ''et al'', 1999]; replacing the non-linear viscous damping by an array of frequency and amplitude-dependent linear viscous model [Rakheja and Sankar, 1986]; random vibration theory [Caughey, 1963; Roberts, 1976], and more recently, applying closed-form formulas based on probabilistic concept to obtain fundamental modal damping ratio without carrying out structural analysis [Lee ''et al'', 2004].
 
The energy dissipation capacity of a fluid viscous damper can be characterized by the supplemental damping ratio'' ξ<sub>d</sub>'' and its non-linearity by the parameter ''N; ''and it is found that the structural response is most effectively investigated in terms of these parameters because they are dimensionless and independent, and the structural response varies linearly with the excitation intensity [Lin and Chopra, 2002]. In this sense, a system with non-linear dampers is usually replaced by an equivalent linear system, with its properties determined using different methods: equalling the energy dissipated in the two systems [Jacobsen, 1930; Fabunmi, 1985]; equalling power consumption in the two systems [Pekcan ''et al'', 1999]; replacing the non-linear viscous damping by an array of frequency and amplitude-dependent linear viscous model [Rakheja and Sankar, 1986]; random vibration theory [Caughey, 1963; Roberts, 1976], and more recently, applying closed-form formulas based on probabilistic concept to obtain fundamental modal damping ratio without carrying out structural analysis [Lee ''et al'', 2004].
Line 1,186: Line 1,104:
 
Thus, equalling the energy dissipated in a vibration cycle of the non-linear system to that of equivalent viscous system [Pekcan ''et al'', 1999] and considering equation 2.10:
 
Thus, equalling the energy dissipated in a vibration cycle of the non-linear system to that of equivalent viscous system [Pekcan ''et al'', 1999] and considering equation 2.10:
  
[[Image:draft_Samper_432909089-image48.png|120px]]
+
[[Image:draft_Samper_432909089-monograph-image48.png|120px]]
  
 
Soong and Constantinou (1994) have shown that the work done (dissipated energy) in one cycle of sinusoidal loading can be written as
 
Soong and Constantinou (1994) have shown that the work done (dissipated energy) in one cycle of sinusoidal loading can be written as
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
|
+
[[Image:draft_Samper_432909089-monograph-image79.png|84px]] [Eq. 2.14]</div>
{| style='margin:auto;width: 100%; text-align:center;'
+
 
|-
+
that is basically the same equation as 2.6. Here, ''T<sub>0</sub> = 2π/ω<sub>0</sub>, ''where ''ω<sub>0</sub>'' is the circular frequency of the system and'' [[Image:draft_Samper_432909089-monograph-image80.png|90px]] .''
| <math display="inline">W_d=\int_0^{T_0}u\overset{.}{x}dt</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.14]  
+
|}
+
that is basically the same equation as 2.6. Here, ''T<sub>0</sub> = 2π/ω<sub>0</sub>, ''where ''ω<sub>0</sub>'' is the circular frequency of the system and'' <math display="inline">\overset{.}{x}=x_0sin{\omega }_0t</math> .''
+
  
 
Equation 2.14 can be integrated to give
 
Equation 2.14 can be integrated to give
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
|
+
[[Image:draft_Samper_432909089-monograph-image81.png|228px]] [Eq. 2.15]</div>
{| style='margin:auto;width: 100%; text-align:center;'
+
 
|-
+
| <math display="inline">W_d=2^{N+2}\frac{{\Gamma }^2(1+N/2)}{\Gamma (2+N)}c_dx_0{}^{1+N}{\omega }_0^N</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.15]  
+
|}
+
 
where ''Г( ) ''is the gamma function.
 
where ''Г( ) ''is the gamma function.
  
 
The equivalent (added) damping is calculated by equating equation 2.15 and the energy dissipated in equivalent viscous damping:
 
The equivalent (added) damping is calculated by equating equation 2.15 and the energy dissipated in equivalent viscous damping:
  
<div style="text-align: right; direction: ltr; margin-left: 1em;">
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
''4πξ<sub>d</sub>ω<sub>0</sub>E<sub>s</sub> = W<sub>d</sub>'' [Eq. 2.16]</div>
+
'''4πξ<sub>d</sub>ω<sub>0</sub>E<sub>s</sub> = W<sub>d</sub>''' [Eq. 2.16]</div>
 +
 
  
 
in which strain energy ''E<sub>s</sub> = kx<sub>0</sub><sup>2</sup>/2. ''Solving Eq. 2.16 for equivalent damping ratio:
 
in which strain energy ''E<sub>s</sub> = kx<sub>0</sub><sup>2</sup>/2. ''Solving Eq. 2.16 for equivalent damping ratio:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
|
+
[[Image:draft_Samper_432909089-monograph-image82.png|234px]] [Eq. 2.17]</div>
{| style='margin:auto;width: 100%; text-align:center;'
+
 
|-
+
| <math display="inline">{\xi }_d=\frac{2^{1+N}c_dx_0^{N-1}{\omega }_0^{N-2}}{\pi M}\frac{{\Gamma }^2(1+N/2)}{\Gamma (2+N)}</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.17]  
+
|}
+
 
where ''M'' is the mass of the system, and ''x<sub>0</sub>'' the amplitude of harmonic motion at the undamped natural frequency ''ω<sub>0</sub>.''
 
where ''M'' is the mass of the system, and ''x<sub>0</sub>'' the amplitude of harmonic motion at the undamped natural frequency ''ω<sub>0</sub>.''
  
 
Of course, the additional damping that the passive system introduces to the structure can be obtained by its energy dissipation capacity in each hysteretic cycle. This dissipated energy for each cycle, can be obtained calculating the area of the cycle in the force – displacement relationship of the viscous damper. Thus, for a selected cycle, it is possible to assess the equivalent damping ratio as follows:
 
Of course, the additional damping that the passive system introduces to the structure can be obtained by its energy dissipation capacity in each hysteretic cycle. This dissipated energy for each cycle, can be obtained calculating the area of the cycle in the force – displacement relationship of the viscous damper. Thus, for a selected cycle, it is possible to assess the equivalent damping ratio as follows:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image83.png|114px]] [Eq. 2.18]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">{\xi }_d=\frac{\mbox{cycle area}}{2\pi u_{max}x_{max}}</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.18]  
+
|}
+
 
where ''u<sub>max</sub> ''and ''x<sub>max</sub>'' are the maximum force and maximum displacement at the damper respectively.
 
where ''u<sub>max</sub> ''and ''x<sub>max</sub>'' are the maximum force and maximum displacement at the damper respectively.
  
 
Pekcan ''et al ''(1999), proposed a simple method for making the transformation from the non-linear damper behaviour to equivalent viscous damping. They explain that for velocity-dependent systems such as viscous dampers, consideration of the ''rate'' of energy dissipation – that is power (rather than energy) – becomes more important in seeking the equivalent linear properties for these systems. The proposed equivalent damping is
 
Pekcan ''et al ''(1999), proposed a simple method for making the transformation from the non-linear damper behaviour to equivalent viscous damping. They explain that for velocity-dependent systems such as viscous dampers, consideration of the ''rate'' of energy dissipation – that is power (rather than energy) – becomes more important in seeking the equivalent linear properties for these systems. The proposed equivalent damping is
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
|
+
[[Image:draft_Samper_432909089-monograph-image84.png|126px]] [Eq. 2.19]</div>
{| style='margin:auto;width: 100%; text-align:center;'
+
 
|-
+
| <math display="inline">c_{eq}=\frac{2}{1+N}c_d{\overset{.}{x}}_0^{N-1}</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.19]  
+
|}
+
 
Given the customary definition of damping ratio (''ξ'') obtained from ''c = 2ξω<sub>0</sub>M'', equation 2.19 can be expressed as follows:
 
Given the customary definition of damping ratio (''ξ'') obtained from ''c = 2ξω<sub>0</sub>M'', equation 2.19 can be expressed as follows:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
|
+
[[Image:draft_Samper_432909089-monograph-image85.png|156px]] [Eq. 2.20]</div>
{| style='margin:auto;width: 100%; text-align:center;'
+
 
|-
+
| <math display="inline">{\xi }_d=\frac{1}{1+N}\frac{c_dx_0^{N-1}{\omega }_0^{N-2}}{M}</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.20]  
+
|}
+
 
This formulation, also called ''power equivalent approach'', predicts higher damping values compared with energy based method (Eq. 2.17). This difference is greater for low values of ''N'', and for that reason it is necessary to be cautious using any of the above formulations for small ''N'' powers (''N<0.1''), since the mechanism of the devices changes from viscous (velocity dependent) to Coulomb friction type (when ''N'' tends to zero).
 
This formulation, also called ''power equivalent approach'', predicts higher damping values compared with energy based method (Eq. 2.17). This difference is greater for low values of ''N'', and for that reason it is necessary to be cautious using any of the above formulations for small ''N'' powers (''N<0.1''), since the mechanism of the devices changes from viscous (velocity dependent) to Coulomb friction type (when ''N'' tends to zero).
  
:<big>1.3.4 Performance of Viscous Dampers During Near-field Ground Motions</big>
+
:<big>2.3.4 Performance of Viscous Dampers During Near-field Ground Motions</big>
  
 
Near-field earthquakes are characterized by short duration pulses of long period with large peak ground velocities and accelerations. It has been observed from recent earthquake records that motions in the fault-normal direction contain destructive long-period pulses with high peak ground velocities, aspect that negatively affects long-period structures such as cable-stayed bridges. A lot of approaches to model these pulses have been recently proposed [Makris, 1997; He, 2003; Mavroeidis ''et al'', 2004].
 
Near-field earthquakes are characterized by short duration pulses of long period with large peak ground velocities and accelerations. It has been observed from recent earthquake records that motions in the fault-normal direction contain destructive long-period pulses with high peak ground velocities, aspect that negatively affects long-period structures such as cable-stayed bridges. A lot of approaches to model these pulses have been recently proposed [Makris, 1997; He, 2003; Mavroeidis ''et al'', 2004].
Line 1,273: Line 1,161:
 
Although the damper non-linearity does not significantly influence the displacement response (As was demonstrated in the research by Lin and Chopra, 2002), in general terms non-linear viscous dampers are more advantageous than linear dampers in reducing peak structural displacements and peak input energies when a structure is subjected to pulse-type excitation with pulse period longer than the natural period of the structure.
 
Although the damper non-linearity does not significantly influence the displacement response (As was demonstrated in the research by Lin and Chopra, 2002), in general terms non-linear viscous dampers are more advantageous than linear dampers in reducing peak structural displacements and peak input energies when a structure is subjected to pulse-type excitation with pulse period longer than the natural period of the structure.
  
:<big>1.4 Analysis and Design Issues</big>
+
:<big>2.4 Analysis and Design Issues</big>
  
:<big>1.4.1 Structural Analysis Including Viscous Dampers</big>
+
:<big>2.4.1 Structural Analysis Including Viscous Dampers</big>
  
 
The first step in the analysis is to find out how added damping affects the structure. This is generally done with a simple stick model with one node for each storey. Adding global damping to the stick model provides a good indication of how damping elements can benefit the structure. The analyst will then construct a simple two-dimensional model of the structure. In this model the dampers are entered as discrete elements. At this point there are a number of variables to play with: force capacity of the dampers, location and number of dampers, damper coefficient and damper exponent. The analyst has the task of finding the best solution. This is generally a trial-and-error process but there are some general guidelines. It is always best to minimize the number of dampers and the number of bays that use dampers. Also, it is known from experience that approximately 20%–30% of critical damping is a desirable range, and that 5% of this can be structural, leaving 15%–25% for viscous damping. So the first objective of the analyst is to determine the smallest possible number of dampers to provide approximately 20% critical damping without overloading either the beams or the columns. Also, it is always best to start with linear dampers and then find out what happens with nonlinear dampers after the locations, number and characteristics of the dampers have been fairly well determined [Lee and Taylor, 2001].
 
The first step in the analysis is to find out how added damping affects the structure. This is generally done with a simple stick model with one node for each storey. Adding global damping to the stick model provides a good indication of how damping elements can benefit the structure. The analyst will then construct a simple two-dimensional model of the structure. In this model the dampers are entered as discrete elements. At this point there are a number of variables to play with: force capacity of the dampers, location and number of dampers, damper coefficient and damper exponent. The analyst has the task of finding the best solution. This is generally a trial-and-error process but there are some general guidelines. It is always best to minimize the number of dampers and the number of bays that use dampers. Also, it is known from experience that approximately 20%–30% of critical damping is a desirable range, and that 5% of this can be structural, leaving 15%–25% for viscous damping. So the first objective of the analyst is to determine the smallest possible number of dampers to provide approximately 20% critical damping without overloading either the beams or the columns. Also, it is always best to start with linear dampers and then find out what happens with nonlinear dampers after the locations, number and characteristics of the dampers have been fairly well determined [Lee and Taylor, 2001].
Line 1,283: Line 1,171:
 
Finally, it can be important to say that in the present days exist good and powerful computing tools that permit to solve non-linear structures equipped with linear/non-linear energy dissipation devices such as fluid viscous dampers. Commercial computing codes such as ''ANSYS ''[Ansys Inc, 2005] or ''SAP2000'' [Computers & Structures, 2007] include the option of applying non-linear energy dissipation devices. However, modelling of some damping elements (e.g. dampers with temperature-dependent or frequency-dependent) can be more challenging or, in some cases, not possible with a given program. When the modelling of such behaviour is not possible, the expected response may be bounded by analyzing the structure over a range of behaviours. Fortunately, for majority of fluid viscous dampers actually manufactured, properties are largely independent with respect to frequency and temperature [Symans ''et al'', 2008].
 
Finally, it can be important to say that in the present days exist good and powerful computing tools that permit to solve non-linear structures equipped with linear/non-linear energy dissipation devices such as fluid viscous dampers. Commercial computing codes such as ''ANSYS ''[Ansys Inc, 2005] or ''SAP2000'' [Computers & Structures, 2007] include the option of applying non-linear energy dissipation devices. However, modelling of some damping elements (e.g. dampers with temperature-dependent or frequency-dependent) can be more challenging or, in some cases, not possible with a given program. When the modelling of such behaviour is not possible, the expected response may be bounded by analyzing the structure over a range of behaviours. Fortunately, for majority of fluid viscous dampers actually manufactured, properties are largely independent with respect to frequency and temperature [Symans ''et al'', 2008].
  
:<big>1.4.2 Design Issues for Viscous Dampers</big>
+
:<big>2.4.2 Design Issues for Viscous Dampers</big>
  
 
The peak force ''f<sub>D0</sub>(N)'' in the non-linear fluid viscous damper with known non-linear parameter ''N ''can be expressed as
 
The peak force ''f<sub>D0</sub>(N)'' in the non-linear fluid viscous damper with known non-linear parameter ''N ''can be expressed as
  
<div style="text-align: right; direction: ltr; margin-left: 1em;">
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
  [[Image:draft_Samper_432909089-image86.png|168px]] [Eq. 2.21]</div>
+
  [[Image:draft_Samper_432909089-monograph-image86.png|168px]] [Eq. 2.21]</div>
  
 
where ''V = ω<sub>0</sub>x<sub>0</sub>'' is the spectral pseudo-velocity for the SDF system; ''c<sub>1</sub>'' is the damping coefficient of the linear system and ''β<sub>N </sub>''is a constant defined as
 
where ''V = ω<sub>0</sub>x<sub>0</sub>'' is the spectral pseudo-velocity for the SDF system; ''c<sub>1</sub>'' is the damping coefficient of the linear system and ''β<sub>N </sub>''is a constant defined as
  
<div style="text-align: right; direction: ltr; margin-left: 1em;">
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
  [[Image:draft_Samper_432909089-image87.png|156px]] [Eq. 2.22]</div>
+
  [[Image:draft_Samper_432909089-monograph-image87.png|156px]] [Eq. 2.22]</div>
 +
 
 +
The non-linear damper force can be computed from Eq. 2.21 if ''x<sub>0</sub>'' and '' [[Image:draft_Samper_432909089-monograph-image88.png|18px]] ''are known. They can be estimated as the peak values of deformation and relative velocity of the corresponding linear system. Equation 2.21 is almost exact in the velocity-sensitive region of the spectrum, overestimates the damper force in the acceleration-sensitive region (by at most 15%); and underestimates in the displacement-sensitive region (by at most 7%). Moreover, the accuracy of Eq. 2.21 deteriorates slightly with the increase of the equivalent damping ''ξ<sub>d</sub>. ''However, the actual velocity '' [[Image:draft_Samper_432909089-monograph-image88.png|12px]] ''of the corresponding linear system required in Eq. 2.21 and to compute ''f<sub>D0 </sub>(N=1) = c<sub>1</sub> [[Image:draft_Samper_432909089-monograph-image88.png|12px]] ''is not readily available, because the velocity spectrum is not plotted routinely. If the velocity '' [[Image:draft_Samper_432909089-monograph-image88.png|18px]] ''is replaced by the pseudo-velocity, Eq. 2.21 changes to
 +
 
 +
<div style="text-align: left; direction: ltr; margin-left: 1em;">
 +
[[Image:draft_Samper_432909089-monograph-image89.png|186px]] [Eq. 2.23]</div>
  
The non-linear damper force can be computed from Eq. 2.21 if ''x<sub>0</sub>'' and '' <math display="inline">\overset{.}{x_0}</math> ''are known. They can be estimated as the peak values of deformation and relative velocity of the corresponding linear system. Equation 2.21 is almost exact in the velocity-sensitive region of the spectrum, overestimates the damper force in the acceleration-sensitive region (by at most 15%); and underestimates in the displacement-sensitive region (by at most 7%). Moreover, the accuracy of Eq. 2.21 deteriorates slightly with the increase of the equivalent damping ''ξ<sub>d</sub>. ''However, the actual velocity '' <math display="inline">\overset{.}{x_0}</math> ''of the corresponding linear system required in Eq. 2.21 and to compute ''f<sub>D0 </sub>(N=1) = c<sub>1</sub> <math display="inline">\overset{.}{x_0}</math> ''is not readily available, because the velocity spectrum is not plotted routinely. If the velocity '' <math display="inline">\overset{.}{x_0}</math> ''is replaced by the pseudo-velocity, Eq. 2.21 changes to
 
{| class='formulaSCP' style='width: 100%;'
 
|-
 
|
 
{| style='margin:auto;width: 100%; text-align:center;'
 
|-
 
| <math display="inline">f_{D0}{\left(N\right)}_{approx}=\frac{f_{D0}(N=1)}{{\beta }_N}</math>
 
|}
 
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.23]
 
|}
 
 
The resulting estimate of the damper force is not accurate, which increase with the system’s period, damper non-linearity and supplemental damping ratio. Thus, velocity should not be approximated by the pseudo-velocity [Lin and Chopra, 2002].
 
The resulting estimate of the damper force is not accurate, which increase with the system’s period, damper non-linearity and supplemental damping ratio. Thus, velocity should not be approximated by the pseudo-velocity [Lin and Chopra, 2002].
  
 
Another important point regarding the design of non-linear fluid viscous dampers is how to select the properties ''c<sub>d</sub>'' and ''N'' to satisfy a design requirement. As was previously explained, the structural deformation is essentially unaffected by the damper non-linearity parameter ''N'' and it is essentially the same as that for the corresponding linear system. The total damping capacity that must be provided in the system to limit the deformation of a linear system to a design value can be determined directly from the design spectrum. Subtracting the inherent damping in the structure from the total damping required gives ''ξ<sub>d</sub>'', the necessary supplemental damping. Many different non-linear fluid viscous dampers can be chosen to provide the required supplemental damping ratio ''ξ<sub>d</sub>''. Thus, for a selected value of ''N:''
 
Another important point regarding the design of non-linear fluid viscous dampers is how to select the properties ''c<sub>d</sub>'' and ''N'' to satisfy a design requirement. As was previously explained, the structural deformation is essentially unaffected by the damper non-linearity parameter ''N'' and it is essentially the same as that for the corresponding linear system. The total damping capacity that must be provided in the system to limit the deformation of a linear system to a design value can be determined directly from the design spectrum. Subtracting the inherent damping in the structure from the total damping required gives ''ξ<sub>d</sub>'', the necessary supplemental damping. Many different non-linear fluid viscous dampers can be chosen to provide the required supplemental damping ratio ''ξ<sub>d</sub>''. Thus, for a selected value of ''N:''
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
<div style="text-align: left; direction: ltr; margin-left: 1em;">
|
+
[[Image:draft_Samper_432909089-monograph-image90.png|168px]] [Eq. 2.24]</div>
{| style='margin:auto;width: 100%; text-align:center;'
+
 
|-
+
| <math display="inline">c_d=\frac{2M{\xi }_d{\omega }_0}{{\beta }_N}{\left({\omega }_0D\right)}^{1-N}</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. 2.24]  
+
|}
+
 
where ''M ''is the mass of the system and ''D'' is the allowable deformation.
 
where ''M ''is the mass of the system and ''D'' is the allowable deformation.
  
 
Fig. 2.21 suggests that the selected damper defined by Eq. 2.24 should satisfy the design constraint reasonably well. Also, the structural deformation should be very close to the allowable value in the velocity-sensitive region, less than the allowable value in the acceleration–sensitive spectral region, but exceed slightly the allowable value in the displacement-sensitive spectral region [Lin and Chopra, 2002].
 
Fig. 2.21 suggests that the selected damper defined by Eq. 2.24 should satisfy the design constraint reasonably well. Also, the structural deformation should be very close to the allowable value in the velocity-sensitive region, less than the allowable value in the acceleration–sensitive spectral region, but exceed slightly the allowable value in the displacement-sensitive spectral region [Lin and Chopra, 2002].
  
:<big>1.5 Practical Applications</big>
+
:<big>2.5 Practical Applications</big>
  
:<big>1.5.1 Study Case 1: Rion-Antirion Bridge, Greece</big>
+
:<big>2.5.1 Study Case 1: Rion-Antirion Bridge, Greece</big>
  
 
Amongst long-span cable-stayed bridges that incorporate additional passive seismic protection, the recently inaugurated Rion – Antirion Bridge in the Gulf of Corinth, Greece, is one of the most interesting bridges located in a high seismicity zone generated by active local faults.
 
Amongst long-span cable-stayed bridges that incorporate additional passive seismic protection, the recently inaugurated Rion – Antirion Bridge in the Gulf of Corinth, Greece, is one of the most interesting bridges located in a high seismicity zone generated by active local faults.
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image91.jpeg|372px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image91.jpeg|372px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image92.png|600px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image92.png|600px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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|-
 
|-
 
|  style="vertical-align: top;"|The damping system consisted in fuses and viscous dampers acting in parallel, connecting the deck with the pylons in the transverse direction. The fuses were designed to work as rigid connections to resist low-to-moderate intensity earthquakes as well as high wind loads. For the design earthquake, the fuses were calculated to fail allowing energy dissipation through the fluid viscous dampers.
 
|  style="vertical-align: top;"|The damping system consisted in fuses and viscous dampers acting in parallel, connecting the deck with the pylons in the transverse direction. The fuses were designed to work as rigid connections to resist low-to-moderate intensity earthquakes as well as high wind loads. For the design earthquake, the fuses were calculated to fail allowing energy dissipation through the fluid viscous dampers.
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image93.jpeg|300px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image93.jpeg|300px]]
  
 
'''Fig. 2.25'''  Response Design Spectrum [Combault ''et al,'' 2000]
 
'''Fig. 2.25'''  Response Design Spectrum [Combault ''et al,'' 2000]
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;width: 49%;"| [[Image:draft_Samper_432909089-image94.jpeg|198px]]
+
|  style="vertical-align: top;width: 46%;"|[[Image:draft_Samper_432909089-monograph-image94.jpeg|198px]]
  
 
'''Fig. 2.26''' Isolation System in the Antirion Approach Viaduct [Infanti ''et al'', 2004]
 
'''Fig. 2.26''' Isolation System in the Antirion Approach Viaduct [Infanti ''et al'', 2004]
|  style="text-align: center;vertical-align: top;width: 56%;"| [[Image:draft_Samper_432909089-image95-c.png|282px]]
+
|  style="text-align: center;vertical-align: top;width: 53%;"|[[Image:draft_Samper_432909089-monograph-image95-c.png|282px]]
  
 
'''Fig. 2.27''' Fuse Restraint [Infanti ''et al,'' 2003]
 
'''Fig. 2.27''' Fuse Restraint [Infanti ''et al,'' 2003]
Line 1,387: Line 1,265:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 56%;"| [[Image:draft_Samper_432909089-image96.jpeg|228px]]
+
|  style="text-align: center;vertical-align: top;width: 54%;"|[[Image:draft_Samper_432909089-monograph-image96.jpeg|228px]]
  
 
'''Fig. 2.28''' Full-Scale Viscous Damper Prototype Testing [Infanti ''et al, ''2004]
 
'''Fig. 2.28''' Full-Scale Viscous Damper Prototype Testing [Infanti ''et al, ''2004]
|  style="text-align: center;vertical-align: top;width: 46%;"| [[Image:draft_Samper_432909089-image97.jpeg|138px]]
+
|  style="text-align: center;vertical-align: top;width: 45%;"|[[Image:draft_Samper_432909089-monograph-image97.jpeg|138px]]
  
 
'''Fig. 2.29''' Fuse Element During Fatigue Test [Infanti'' et al,'' 2004]
 
'''Fig. 2.29''' Fuse Element During Fatigue Test [Infanti'' et al,'' 2004]
Line 1,398: Line 1,276:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image98.jpeg|156px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image98.jpeg|156px]]
  
 
'''Fig. 2.30 '''Internal Hydraulic Damper [Lecinq ''et al,'' 2003]
 
'''Fig. 2.30 '''Internal Hydraulic Damper [Lecinq ''et al,'' 2003]
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image99.jpeg|126px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image99.jpeg|126px]]
  
 
'''Fig. 2.31''' External Hydraulic Damper on Normandy Bridge [Lecinq ''et al,'' 2003]
 
'''Fig. 2.31''' External Hydraulic Damper on Normandy Bridge [Lecinq ''et al,'' 2003]
Line 1,409: Line 1,287:
 
In his MSc Thesis, Morgenthal (1999) carries out a detailed research on the seismic behaviour of the Rion-Antirion Bridge. He describes the bridge and exposes analytical modelling using finite elements to study the seismic control strategies incorporating different seismic protection devices, such as structural fuses, hydraulic dampers, seismic connectors and elasto-plastic isolators. Finally, a parametric analysis of the seismic behaviour of different deck isolation devices is exposed.
 
In his MSc Thesis, Morgenthal (1999) carries out a detailed research on the seismic behaviour of the Rion-Antirion Bridge. He describes the bridge and exposes analytical modelling using finite elements to study the seismic control strategies incorporating different seismic protection devices, such as structural fuses, hydraulic dampers, seismic connectors and elasto-plastic isolators. Finally, a parametric analysis of the seismic behaviour of different deck isolation devices is exposed.
  
:<big>1.5.2 Study Case 2: Tempozan Bridge, Japan</big>
+
:<big>2.5.2 Study Case 2: Tempozan Bridge, Japan</big>
  
 
Due to severe damage to bridges caused by the Hyogo-ken-Nanbu earthquake in 1995, very high ground motion was required according to the bridge design specifications set in 1996 [Japan Road Association, 1996], in addition to the relatively frequent earthquake motion specifications by which old structures were designed and constructed. Hence, seismic safety of cable-stayed bridges that were built prior to that specification was reviewed, and seismic retrofit was performed. In order to study the effectiveness of passive control to the seismic retrofit of a cable-stayed bridge, a numerical analysis on a model of a cable-stayed bridge was carried out. An existing cable-stayed bridge with fixed-hinge connections between deck and towers was modelled and its connections were replaced by isolation bearings and dampers. The isolation bearings were assumed to be of the elastic and hysteretic type. The dampers were linear and variable. The objective was to increase the damping ratio of the bridge by using passive control technologies. The chosen bridge model was the Tempozan Bridge, located in Osaka, Japan.
 
Due to severe damage to bridges caused by the Hyogo-ken-Nanbu earthquake in 1995, very high ground motion was required according to the bridge design specifications set in 1996 [Japan Road Association, 1996], in addition to the relatively frequent earthquake motion specifications by which old structures were designed and constructed. Hence, seismic safety of cable-stayed bridges that were built prior to that specification was reviewed, and seismic retrofit was performed. In order to study the effectiveness of passive control to the seismic retrofit of a cable-stayed bridge, a numerical analysis on a model of a cable-stayed bridge was carried out. An existing cable-stayed bridge with fixed-hinge connections between deck and towers was modelled and its connections were replaced by isolation bearings and dampers. The isolation bearings were assumed to be of the elastic and hysteretic type. The dampers were linear and variable. The objective was to increase the damping ratio of the bridge by using passive control technologies. The chosen bridge model was the Tempozan Bridge, located in Osaka, Japan.
Line 1,417: Line 1,295:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-image100.jpeg|228px]] </span>
+
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-monograph-image100.jpeg|228px]] </span>
  
 
'''Fig. 2.32''' Tempozan Bridge [from en.structurae.de]
 
'''Fig. 2.32''' Tempozan Bridge [from en.structurae.de]
|  style="vertical-align: top;width: 65%;"| [[Image:draft_Samper_432909089-image101.png|384px]]
+
|  style="vertical-align: top;width: 61%;"|[[Image:draft_Samper_432909089-monograph-image101.png|384px]]
  
 
'''Fig. 2.33''' Side View of the Tempozan Bridge [Iemura and Pradono, 2003]  
 
'''Fig. 2.33''' Side View of the Tempozan Bridge [Iemura and Pradono, 2003]  
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image102.png|432px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image102.png|432px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image103.png|324px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image103.png|324px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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As a conclusion remark, it is clear that additional viscous damping to control the seismic response of cable-stayed bridges is beneficial, reducing the seismic forces on members as well as their displacements by increasing the structural damping.
 
As a conclusion remark, it is clear that additional viscous damping to control the seismic response of cable-stayed bridges is beneficial, reducing the seismic forces on members as well as their displacements by increasing the structural damping.
  
<div style="text-align: right; direction: ltr; margin-left: 1em;">
+
==Chapter 3. Seismic Response. Parametric Analysis==
<big>Chapter 3</big></div>
+
 
+
<div style="text-align: right; direction: ltr; margin-left: 1em;">
+
<big>Seismic Response. Parametric Analysis</big></div>
+
  
:<big>1.1 Introduction</big>
+
:<big>3.1 Introduction</big>
  
 
As a starting point, the seismic analysis without the incorporation of additional damping devices is presented. In order to compare the analysis results of the bridge models considering the incorporation of additional passive devices, this part is essential, and must be employed as basic configuration (reference structures).
 
As a starting point, the seismic analysis without the incorporation of additional damping devices is presented. In order to compare the analysis results of the bridge models considering the incorporation of additional passive devices, this part is essential, and must be employed as basic configuration (reference structures).
Line 1,542: Line 1,416:
 
The last point of this chapter corresponds to the seismic response of the selected bridges considering far-fault and near-fault ground motions respectively. In this part, nonlinear time history analysis is conducted taking into account the geometric and material nonlinearities of the structures and the cable vibration effects. The input ground motions are acceleration time histories: five artificial three-orthogonal component acceleration records for far-fault analysis and five real three-orthogonal component acceleration records for near-fault analysis. The seismic response of the bridges considers the displacement, velocity and acceleration time histories and the response of the deck, cables and towers. Because of the complex nature of the nonlinear time history analysis and according to the recommendations of Eurocode 8 Part 2 [CEN, 1998b], the average of the maximum response parameters in the assessment of the structural response for the record selection is considered.
 
The last point of this chapter corresponds to the seismic response of the selected bridges considering far-fault and near-fault ground motions respectively. In this part, nonlinear time history analysis is conducted taking into account the geometric and material nonlinearities of the structures and the cable vibration effects. The input ground motions are acceleration time histories: five artificial three-orthogonal component acceleration records for far-fault analysis and five real three-orthogonal component acceleration records for near-fault analysis. The seismic response of the bridges considers the displacement, velocity and acceleration time histories and the response of the deck, cables and towers. Because of the complex nature of the nonlinear time history analysis and according to the recommendations of Eurocode 8 Part 2 [CEN, 1998b], the average of the maximum response parameters in the assessment of the structural response for the record selection is considered.
  
:<big>1.2 Structural Modelling</big>
+
 
 +
:<big>3.2 Structural Modelling</big>
  
 
The seismic response analysis of the concrete cable-stayed bridges takes into consideration eight ''3-D'' symmetric bridge models for an adequate study. The chosen bridges were taken from the specialized literature, and specifically, from Walter’s Bridges [Walter, 1999] including the recommendations of Aparicio and Casas (2000) and Priestley ''at al'' (1996). The examination is based on a symmetric multi-stay reference cable-stayed bridge, having two pylons, double-plane cable layout and a main span length of about 200 m. Two stay cable layouts were selected: fan-type and harp-type. The semi-harp pattern was rejected because this typology is an intermediate pattern, and both harp and fan patterns are enough for an adequate analysis. The main span lengths of the bridges are 217 m and 204.60 m, depending on the stay spacing. In this sense, long-span cable-stayed bridges have experienced adequate performance during recent earthquake events, and it is expected that short-to-medium spans bridges show a worse seismic performance, mainly if near-source effects are considered. That is the main reason to select the proposed span lengths.  Moreover, the deck pattern considers two cases: a slab-type deck and a hollow-box type deck. The first one, due to its inherent flexibility, considers a stay spacing of 6.20 m. In the second case, 12.40 m - stay spacing is considered. The selected tower, for all cases, is a concrete frame-type tower, with deck levels of 30 and 60 m from bottom. The height of the towers is 81 m and 111 m respectively. All these dimensions were taken from Walter’s recommendations.
 
The seismic response analysis of the concrete cable-stayed bridges takes into consideration eight ''3-D'' symmetric bridge models for an adequate study. The chosen bridges were taken from the specialized literature, and specifically, from Walter’s Bridges [Walter, 1999] including the recommendations of Aparicio and Casas (2000) and Priestley ''at al'' (1996). The examination is based on a symmetric multi-stay reference cable-stayed bridge, having two pylons, double-plane cable layout and a main span length of about 200 m. Two stay cable layouts were selected: fan-type and harp-type. The semi-harp pattern was rejected because this typology is an intermediate pattern, and both harp and fan patterns are enough for an adequate analysis. The main span lengths of the bridges are 217 m and 204.60 m, depending on the stay spacing. In this sense, long-span cable-stayed bridges have experienced adequate performance during recent earthquake events, and it is expected that short-to-medium spans bridges show a worse seismic performance, mainly if near-source effects are considered. That is the main reason to select the proposed span lengths.  Moreover, the deck pattern considers two cases: a slab-type deck and a hollow-box type deck. The first one, due to its inherent flexibility, considers a stay spacing of 6.20 m. In the second case, 12.40 m - stay spacing is considered. The selected tower, for all cases, is a concrete frame-type tower, with deck levels of 30 and 60 m from bottom. The height of the towers is 81 m and 111 m respectively. All these dimensions were taken from Walter’s recommendations.
Line 1,550: Line 1,425:
 
Regarding the bridge modelling, the analysis is carried out considering the use of beam and cable elements for all the bridges. The deck is modelled using a single spine to avoid the use of shell elements, with the incorporation of transverse rigid-links to simulate the anchor of cables. In fact, the use of beam elements can be more useful to assess forces on members, with clear graphical results and a considerable decrease of the computing time, especially when non-linear time history analysis is applied. Moreover, the non-linear analysis takes into consideration the geometric non-linearities that are present in almost all cable-stayed bridges. These non-linearities due to high compressions in the deck and pylons are considered by the axial – bending interaction. Non-linear behaviour of cables is considered by a multi-element cable formulation (''tension-only'' elements), in order to take into account the spatial vibrations of them. Likewise, non-linearities due to large displacements in the overall geometry are considered too. Spatial variability is not considered because of the main span length of the bridges (of about 200 m), that not recommends this effect according to Eurocode 8.
 
Regarding the bridge modelling, the analysis is carried out considering the use of beam and cable elements for all the bridges. The deck is modelled using a single spine to avoid the use of shell elements, with the incorporation of transverse rigid-links to simulate the anchor of cables. In fact, the use of beam elements can be more useful to assess forces on members, with clear graphical results and a considerable decrease of the computing time, especially when non-linear time history analysis is applied. Moreover, the non-linear analysis takes into consideration the geometric non-linearities that are present in almost all cable-stayed bridges. These non-linearities due to high compressions in the deck and pylons are considered by the axial – bending interaction. Non-linear behaviour of cables is considered by a multi-element cable formulation (''tension-only'' elements), in order to take into account the spatial vibrations of them. Likewise, non-linearities due to large displacements in the overall geometry are considered too. Spatial variability is not considered because of the main span length of the bridges (of about 200 m), that not recommends this effect according to Eurocode 8.
  
:<big>1.2.1 Geometric Layout</big>
+
 
 +
:<big>3.2.1 Geometric Layout</big>
  
 
The longitudinal layout of the cable stays is one of the fundamental items in the design of cable-stayed bridges. It influences, in fact, not only the structural performance of the bridge, but also the method of erection and the economics.
 
The longitudinal layout of the cable stays is one of the fundamental items in the design of cable-stayed bridges. It influences, in fact, not only the structural performance of the bridge, but also the method of erection and the economics.
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{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
 
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|-
 
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image106.jpeg|210px]]  
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|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image107.jpeg|192px]]  
 
|}
 
|}
  
==Fig. 3.1 Longitudinal Layouts for Fan Pattern (dimensions in metres)==
+
'''Fig. 3.1'' Longitudinal Layouts for Fan Pattern (dimensions in metres)
  
 
{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
 
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|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image108.jpeg|192px]]  
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|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image108.jpeg|192px]]  
|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image109.jpeg|186px]]  
+
|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image109.jpeg|186px]]  
 
|-
 
|-
|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image110.jpeg|198px]]  
+
|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image110.jpeg|198px]]  
|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image111.jpeg|198px]]  
+
|  style="border: 1pt solid black;text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image111.jpeg|198px]]  
 
|}
 
|}
  
  
==Fig. 3.2 Longitudinal Layouts for Harp Pattern (dimensions in metres)==
+
'''Fig. 3.2''' Longitudinal Layouts for Harp Pattern (dimensions in metres)
  
 
In the transverse direction, the majority of existing structures consist of two planes of cables, generally on the edge of the structure. However, several bridges have been successfully built recently with only one central plane of cables. In principle, it is quite possible to envisage solutions using three or more planes, with the aim of reducing the cross-sectional forces when the deck is very wide, but this possibility has been rarely exploited [Walter, 1999].
 
In the transverse direction, the majority of existing structures consist of two planes of cables, generally on the edge of the structure. However, several bridges have been successfully built recently with only one central plane of cables. In principle, it is quite possible to envisage solutions using three or more planes, with the aim of reducing the cross-sectional forces when the deck is very wide, but this possibility has been rarely exploited [Walter, 1999].
Line 1,591: Line 1,467:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-image112.png|174px]] '''
+
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image112.png|174px]] '''
  
 
'''Fig. 3.3''' Transverse Configuration
 
'''Fig. 3.3''' Transverse Configuration
Line 1,606: Line 1,482:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image113-c.png|234px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image113-c.png|234px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' A-type Pylon</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' A-type Pylon</span>
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image114-c.png|234px]] '''</span>
+
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image114-c.png|234px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' B-type Pylon</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' B-type Pylon</span>
Line 1,624: Line 1,500:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-image115-c.jpeg|312px]] '''
+
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image115-c.jpeg|312px]] '''
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Slab-Type Deck</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Slab-Type Deck</span>
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image116.jpeg|288px]] '''</span>
+
|  style="vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image116.jpeg|288px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Hollow-box Type Deck</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Hollow-box Type Deck</span>
 
|}
 
|}
  
 
+
'''Fig. 3.5''' Proposed Decks for the Analysis (dimensions in metres)
==Fig. 3.5 Proposed Decks for the Analysis (dimensions in metres)==
+
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Line 1,671: Line 1,546:
 
The cross-section areas for the slab-type and the hollow-box type are the same, which implies the same weight. This point is very important, because the main difference between both typologies is the inertia. The slab-type deck considers a very low inertia. The slenderness ratio in this case is ''h/L = 0.40/217 = 1/543 ''for the fan pattern, and ''h/L = 0.40/204.6 = 1/512'' for the harp pattern, which implies a very slender deck. Of course, in this case bending moments are expected to be low. The hollow-box type deck shows a high inertia. The slenderness ratio is ''h/L =2.25/217=1/96 ''for the fan pattern, and ''h/L = 2.25/204.6 = 1/91'' for the harp pattern, which implies a rigid deck. These differences involve the use of 6.2 m stay-spacing for the flexible deck (slab-type), and 12.4 m stay-spacing for the rigid deck (hollow-box type).
 
The cross-section areas for the slab-type and the hollow-box type are the same, which implies the same weight. This point is very important, because the main difference between both typologies is the inertia. The slab-type deck considers a very low inertia. The slenderness ratio in this case is ''h/L = 0.40/217 = 1/543 ''for the fan pattern, and ''h/L = 0.40/204.6 = 1/512'' for the harp pattern, which implies a very slender deck. Of course, in this case bending moments are expected to be low. The hollow-box type deck shows a high inertia. The slenderness ratio is ''h/L =2.25/217=1/96 ''for the fan pattern, and ''h/L = 2.25/204.6 = 1/91'' for the harp pattern, which implies a rigid deck. These differences involve the use of 6.2 m stay-spacing for the flexible deck (slab-type), and 12.4 m stay-spacing for the rigid deck (hollow-box type).
  
:<big>1.2.2 Basis of Design and Actions</big>
+
 
 +
:<big>3.2.2 Basis of Design and Actions</big>
  
 
Materials and their mechanical properties have been chosen according to the general specifications and regulations for bridge design, taking into account seismic considerations [Priestley ''et al'', 1996; Walter, 1999; Ministerio de Fomento, 2000; Aparicio and Casas, 2000].
 
Materials and their mechanical properties have been chosen according to the general specifications and regulations for bridge design, taking into account seismic considerations [Priestley ''et al'', 1996; Walter, 1999; Ministerio de Fomento, 2000; Aparicio and Casas, 2000].
Line 1,743: Line 1,619:
 
{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-image117.png|270px]] '''</big>
+
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-monograph-image117.png|270px]] '''</big>
  
 
'''Fig. 3.6''' Parallel-Strand Cables
 
'''Fig. 3.6''' Parallel-Strand Cables
Line 1,791: Line 1,667:
 
For the above mentioned, to combine the loads on the bridge models, it is necessary to add each action: ''q<sub>PL</sub> + q<sub>SPL</sub> + q<sub>E</sub>.''
 
For the above mentioned, to combine the loads on the bridge models, it is necessary to add each action: ''q<sub>PL</sub> + q<sub>SPL</sub> + q<sub>E</sub>.''
  
:<big>1.2.3 Nonlinearities</big>
+
 
 +
:<big>3.2.3 Nonlinearities</big>
  
 
Nonlinearities can be broadly divided into geometric and material nonlinearities. Material nonlinearities depend on the specific structure (materials used, loads acting, design assumptions). Although it is certain that the elastic-plastic effect tends to reduce the seismic response of long-span cable-stayed bridges [Ren and Obata, 1999], material nonlinearities depend highly on the characteristics of the input earthquake records. In general terms, cable-stayed bridges experience very long periods, and for that reason formation of plastic hinges at the supports can be difficult. In fact, EC8 – 2 [CEN, 1998b] recommends for a well-designed cable-stayed bridge a behaviour factor ''q = 1'', that is to say, an elastic seismic behaviour. Moreover, because of the high axial forces on the pylons, ductility of them can be questionable, and due to the importance of such structures, it is preferable an elastic behaviour of the materials, without formation of plastic hinges at the pylons. That is the main reason why the inelastic behaviour is not considered in this research. In this sense, dimensions and some special considerations for the selected bridge typologies take into account an elastic seismic behaviour of the materials. However, material nonlinearities due to the presence of additional viscous dampers as well as the ''tension-only'' nonlinear effect of the cables are considered.
 
Nonlinearities can be broadly divided into geometric and material nonlinearities. Material nonlinearities depend on the specific structure (materials used, loads acting, design assumptions). Although it is certain that the elastic-plastic effect tends to reduce the seismic response of long-span cable-stayed bridges [Ren and Obata, 1999], material nonlinearities depend highly on the characteristics of the input earthquake records. In general terms, cable-stayed bridges experience very long periods, and for that reason formation of plastic hinges at the supports can be difficult. In fact, EC8 – 2 [CEN, 1998b] recommends for a well-designed cable-stayed bridge a behaviour factor ''q = 1'', that is to say, an elastic seismic behaviour. Moreover, because of the high axial forces on the pylons, ductility of them can be questionable, and due to the importance of such structures, it is preferable an elastic behaviour of the materials, without formation of plastic hinges at the pylons. That is the main reason why the inelastic behaviour is not considered in this research. In this sense, dimensions and some special considerations for the selected bridge typologies take into account an elastic seismic behaviour of the materials. However, material nonlinearities due to the presence of additional viscous dampers as well as the ''tension-only'' nonlinear effect of the cables are considered.
Line 1,805: Line 1,682:
 
Those nonlinearities are especially considered in this research, because in some sense they govern the behaviour of this kind of structures, as was explained before. Geometric nonlinearity can be considered on a step-by-step basis in nonlinear static and direct integration time history analysis, and incorporated in the stiffness matrix for linear analysis.
 
Those nonlinearities are especially considered in this research, because in some sense they govern the behaviour of this kind of structures, as was explained before. Geometric nonlinearity can be considered on a step-by-step basis in nonlinear static and direct integration time history analysis, and incorporated in the stiffness matrix for linear analysis.
  
:<big>1.2.4 Modelling</big>
 
  
:''1.2.4.1 Tower modelling ''
+
:<big>3.2.4 Modelling</big>
 +
 
 +
:''3.2.4.1 Tower modelling ''
  
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 41%;"| [[Image:draft_Samper_432909089-image118-c.png|246px]]
+
|  style="text-align: center;vertical-align: top;width: 40%;"|[[Image:draft_Samper_432909089-monograph-image118-c.png|246px]]
  
 
'''Fig. 3.7 '''Modelling of the Towers
 
'''Fig. 3.7 '''Modelling of the Towers
Line 1,824: Line 1,702:
 
For the adequate consideration of the vibrational characteristics of the towers, the distribution of lumped masses shown in Fig. 3.7 was considered. The masses were obtained from the elements using the mass density of the materials and the volume of the elements. These uncoupled masses are equal for each of the three translational degrees-of-freedom. In the case of the code RAM Advanse, these masses need to be explicitly added in the selected joints. In SAP2000, these masses are automatically added at the end joints of the elements. However, it was necessary to place additional masses to take into account an adequate distribution of lumped masses in order to have a more accurate dynamic model. The lumped masses involves no mass coupling between degrees of freedom at a joint or between different joints
 
For the adequate consideration of the vibrational characteristics of the towers, the distribution of lumped masses shown in Fig. 3.7 was considered. The masses were obtained from the elements using the mass density of the materials and the volume of the elements. These uncoupled masses are equal for each of the three translational degrees-of-freedom. In the case of the code RAM Advanse, these masses need to be explicitly added in the selected joints. In SAP2000, these masses are automatically added at the end joints of the elements. However, it was necessary to place additional masses to take into account an adequate distribution of lumped masses in order to have a more accurate dynamic model. The lumped masses involves no mass coupling between degrees of freedom at a joint or between different joints
  
:''1.2.4.2 Deck Modelling''
+
:''3.2.4.2 Deck Modelling''
  
 
As was previously mentioned, the decks of the bridges are of two kinds: a slab-type deck for a stay spacing of 6.2 m and a hollow box-type deck for a stay spacing of 12.40 m. To simplify the computing process, the decks were modelled using a single spine passing through the centroid of the cross-sections, and applying linear beam elements. To simulate the exact stiffness and masses of the decks, an accurate definition of the geometry of the cross sections was developed with the computational code RAM Advanse and also exported to SAP2000. That was possible modifying the internal code in RAM Advanse, in order to generate the desired sections. A ''3-D ''rendering of the deck models can be seen in Fig. 3.8.
 
As was previously mentioned, the decks of the bridges are of two kinds: a slab-type deck for a stay spacing of 6.2 m and a hollow box-type deck for a stay spacing of 12.40 m. To simplify the computing process, the decks were modelled using a single spine passing through the centroid of the cross-sections, and applying linear beam elements. To simulate the exact stiffness and masses of the decks, an accurate definition of the geometry of the cross sections was developed with the computational code RAM Advanse and also exported to SAP2000. That was possible modifying the internal code in RAM Advanse, in order to generate the desired sections. A ''3-D ''rendering of the deck models can be seen in Fig. 3.8.
Line 1,830: Line 1,708:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image119.png|174px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image119.png|174px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Slab-type deck</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Slab-type deck</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image120.png|198px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image120.png|198px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Hollow-box type deck</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Hollow-box type deck</span>
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;width: 39%;"| [[Image:draft_Samper_432909089-image121-c.png|222px]]
+
|  style="vertical-align: top;width: 38%;"|[[Image:draft_Samper_432909089-monograph-image121-c.png|222px]]
  
 
'''Fig. 3.9''' Deck Modelling
 
'''Fig. 3.9''' Deck Modelling
Line 1,855: Line 1,733:
 
As for the non-structural components, such as parapets and beacons, their contribution to the structural rigidity is expected to be quite insignificant and therefore is ignored in the modelling. Likewise, since the cross sections of the deck are rigid (especially the hollow-box type deck), the corresponding warping constants are very large. Consequently, no cross-sectional warping is anticipated.
 
As for the non-structural components, such as parapets and beacons, their contribution to the structural rigidity is expected to be quite insignificant and therefore is ignored in the modelling. Likewise, since the cross sections of the deck are rigid (especially the hollow-box type deck), the corresponding warping constants are very large. Consequently, no cross-sectional warping is anticipated.
  
:''1.2.4.3 Stay cable model''
+
:''3.2.4.3 Stay cable model''
  
 
In a first stage, truss-elements for the cables were applied to compute the prestressing forces on the stays. The strategy of modelling stay cables using truss-elements has been widely used, in which the nonlinear behaviour of the cables is accounted by linearizing the cable stiffness using the concept of an equivalent modulus of elasticity [Ernst, 1965]. However, prediction of the tri-dimensional vibration of the cables is not possible, aspect that can be important for an accurate dynamic analysis, especially if the cable-deck interaction plays an important role. For that reason, this strategy was only considered for the evaluation of the cable forces, through the nonlinear static analysis applying the code RAM Advanse.
 
In a first stage, truss-elements for the cables were applied to compute the prestressing forces on the stays. The strategy of modelling stay cables using truss-elements has been widely used, in which the nonlinear behaviour of the cables is accounted by linearizing the cable stiffness using the concept of an equivalent modulus of elasticity [Ernst, 1965]. However, prediction of the tri-dimensional vibration of the cables is not possible, aspect that can be important for an accurate dynamic analysis, especially if the cable-deck interaction plays an important role. For that reason, this strategy was only considered for the evaluation of the cable forces, through the nonlinear static analysis applying the code RAM Advanse.
Line 1,868: Line 1,746:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image122.png|528px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image122.png|528px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;"|'''Table 3.7''' Nominal Diameters of the Stays
+
|  style="vertical-align: top;width: 40%;"|'''Table 3.7''' Nominal Diameters of the Stays
  
[[Image:draft_Samper_432909089-image123-c.png|294px]]  
+
[[Image:draft_Samper_432909089-monograph-image123-c.png|294px]]  
|  style="vertical-align: top;"|A last aspect in the stay cable design is the fact that cables on each bridge were designed uniformly, that is to say, for the worse stress condition. Table 3.7 shows the pre-design of the stays for each bridge model
+
|  style="vertical-align: top;width: 60%;"|A last aspect in the stay cable design is the fact that cables on each bridge were designed uniformly, that is to say, for the worse stress condition. Table 3.7 shows the pre-design of the stays for each bridge model
 
|}
 
|}
  
  
:''1.2.4.4 Connections and boundary conditions''
+
:''3.2.4.4 Connections and boundary conditions''
  
 
If the deck is connected with very flexible bearings to the towers, the induced seismic forces will be kept to minimum values but the deck may have a large displacement response. On the other hand, a very stiff connection between the deck and the towers will result in lower deck displacement response but will attract much higher seismic forces during an earthquake [Iemura and Pradono, 2003]. The influence of different support conditions on the mode distribution has been investigated by Ali and Abdel-Ghaffar (1995) and Tuladhar and Dilger (1999). Movable supports lead to a more flexible structure, and of course, the decision upon the support conditions of the deck is usually governed by serviceability as well as earthquake considerations. A restrained deck will avoid excessive movements due to traffic and wind loading, however, in the case of an earthquake a restrained deck will generate high axial forces which are applied to the pier-pylon system. Elastic supports for the deck at the towers give very low deck displacements and deck bending moments compared to pinned or fixed connections. Roller supports also cause the bridge have very low first longitudinal direction mode frequency, indicating that the bridge is very flexible in that direction.
 
If the deck is connected with very flexible bearings to the towers, the induced seismic forces will be kept to minimum values but the deck may have a large displacement response. On the other hand, a very stiff connection between the deck and the towers will result in lower deck displacement response but will attract much higher seismic forces during an earthquake [Iemura and Pradono, 2003]. The influence of different support conditions on the mode distribution has been investigated by Ali and Abdel-Ghaffar (1995) and Tuladhar and Dilger (1999). Movable supports lead to a more flexible structure, and of course, the decision upon the support conditions of the deck is usually governed by serviceability as well as earthquake considerations. A restrained deck will avoid excessive movements due to traffic and wind loading, however, in the case of an earthquake a restrained deck will generate high axial forces which are applied to the pier-pylon system. Elastic supports for the deck at the towers give very low deck displacements and deck bending moments compared to pinned or fixed connections. Roller supports also cause the bridge have very low first longitudinal direction mode frequency, indicating that the bridge is very flexible in that direction.
Line 1,961: Line 1,839:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-image124-c.png|246px]] '''
+
|  style="text-align: center;vertical-align: top;width: 40%"|''' [[Image:draft_Samper_432909089-monograph-image124-c.png|246px]] '''
  
 
'''Fig. 3.11''' Modelling of the Tower-deck Connection
 
'''Fig. 3.11''' Modelling of the Tower-deck Connection
|  style="vertical-align: top;"|In the present investigation, connection between deck and towers is supported by the lower strut through vertical rigid links with pinned bearings at the end-joints, in the connection with the deck (Fig. 3.11). These rigid links were idealized using linear massless springs with all the directional degrees-of-freedom fixed. For the abutment-to-deck connection, sliding bearings were used, in order to permit free longitudinal displacements of the structure due to normal expansions, and free rotations about the transverse axes (Fig. 3.12).  
+
|  style="vertical-align: top;width: 60%"|In the present investigation, connection between deck and towers is supported by the lower strut through vertical rigid links with pinned bearings at the end-joints, in the connection with the deck (Fig. 3.11). These rigid links were idealized using linear massless springs with all the directional degrees-of-freedom fixed. For the abutment-to-deck connection, sliding bearings were used, in order to permit free longitudinal displacements of the structure due to normal expansions, and free rotations about the transverse axes (Fig. 3.12).  
 
|}
 
|}
  
Line 1,973: Line 1,851:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
''' [[Image:draft_Samper_432909089-image125-c.png|390px]] '''</div>
+
''' [[Image:draft_Samper_432909089-monograph-image125-c.png|390px]] '''</div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 45%;"| [[Image:draft_Samper_432909089-image126-c.png|282px]]
+
|  style="text-align: center;vertical-align: top;width: 45%;"|[[Image:draft_Samper_432909089-monograph-image126-c.png|282px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) ''' ''AB1'' pattern</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) ''' ''AB1'' pattern</span>
  
  
|  style="text-align: center;vertical-align: top;width: 55%;"| [[Image:draft_Samper_432909089-image127-c.png|264px]]
+
|  style="text-align: center;vertical-align: top;width: 54%;"|[[Image:draft_Samper_432909089-monograph-image127-c.png|264px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) ''' ''AB4 ''pattern</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) ''' ''AB4 ''pattern</span>
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 45%;"| [[Image:draft_Samper_432909089-image128.png|270px]]
+
|  style="text-align: center;vertical-align: top;width: 45%;"|[[Image:draft_Samper_432909089-monograph-image128.png|270px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(c) ''' ''AR1 ''pattern</span>
 
<span style="text-align: center; font-size: 75%;">'''(c) ''' ''AR1 ''pattern</span>
|  style="text-align: center;vertical-align: top;width: 55%;"| [[Image:draft_Samper_432909089-image129.png|246px]]
+
|  style="text-align: center;vertical-align: top;width: 54%;"|[[Image:draft_Samper_432909089-monograph-image129.png|246px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(d) ''' ''AR4'' Pattern</span>
 
<span style="text-align: center; font-size: 75%;">'''(d) ''' ''AR4'' Pattern</span>
Line 2,001: Line 1,879:
 
'''Fig. 3.13''' Complete ''3-D'' Finite Element Models of Some Bridges</div>
 
'''Fig. 3.13''' Complete ''3-D'' Finite Element Models of Some Bridges</div>
  
:<big>1.3 Nonlinear Static Analysis under Service Loads</big>
+
 
 +
:<big>3.3 Nonlinear Static Analysis under Service Loads</big>
  
 
As a first step of the nonlinear seismic analysis, a nonlinear static analysis under gravity loads and stay prestressing forces was carried out. Nonlinearities in this stage include the stay cable sag effect, axial force-bending moment interaction, large displacements effect and the material nonlinearity due to the ''tension only'' formulation of the cable elements. Of course, the nonlinear static analysis was performed for each bridge model, considering the actualized data of the stay prestressing forces.
 
As a first step of the nonlinear seismic analysis, a nonlinear static analysis under gravity loads and stay prestressing forces was carried out. Nonlinearities in this stage include the stay cable sag effect, axial force-bending moment interaction, large displacements effect and the material nonlinearity due to the ''tension only'' formulation of the cable elements. Of course, the nonlinear static analysis was performed for each bridge model, considering the actualized data of the stay prestressing forces.
Line 2,011: Line 1,890:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-image130-c.png|246px]] '''
+
|  style="vertical-align: top;width: 40%"|''' [[Image:draft_Samper_432909089-monograph-image130-c.png|246px]] '''
  
 
'''Fig. 3.14''' Location of Measured Displacements
 
'''Fig. 3.14''' Location of Measured Displacements
|  style="vertical-align: top;"|Fig. 3.14 shows the location of the measured displacements on the bridges. ''∆<sub>1-L</sub>'' corresponds to the longitudinal displacement of the tower-top, and ''∆<sub>1-T</sub>''corresponds to the transverse displacement of the same point. ''∆<sub>3-V</sub>'' is the vertical displacement of the deck at the mid-span, and finally, ''∆<sub>4-L</sub>''corresponds to the longitudinal displacement of the deck-ends.
+
|  style="vertical-align: top;width: 60%"|Fig. 3.14 shows the location of the measured displacements on the bridges. ''∆<sub>1-L</sub>'' corresponds to the longitudinal displacement of the tower-top, and ''∆<sub>1-T</sub>''corresponds to the transverse displacement of the same point. ''∆<sub>3-V</sub>'' is the vertical displacement of the deck at the mid-span, and finally, ''∆<sub>4-L</sub>''corresponds to the longitudinal displacement of the deck-ends.
 
|}
 
|}
  
Line 2,085: Line 1,964:
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> Positive value implies a displacement toward the mid-span</span>
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> Positive value implies a displacement toward the mid-span</span>
  
<sup>b</sup> Negative value implies a descending
+
<span style="text-align: center; font-size: 75%;"><sup>b</sup> Negative value implies a descending</span>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Line 2,177: Line 2,056:
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In the bridge plane <sup>e</sup> Out-of-plane</span>
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In the bridge plane <sup>e</sup> Out-of-plane</span>
  
<sup>b</sup> At the tower-deck level <sup>d </sup>At the upper strut level - Implies compression
+
<span style="text-align: center; font-size: 75%;"><sup>b</sup> At the tower-deck level <sup>d </sup>At the upper strut level - Implies compression</span>
  
 
Values of displacements and forces shown in Tables 3.8 and 3.9 are in accordance with the structures. Although a static parametric analysis is not the aim of this investigation, it can be interesting to observe that maximum displacements at the deck-ends are associated with the harp pattern. Because of the loads and geometric symmetry of the structures, transverse deflections at the tower-deck level as well as the longitudinal and transverse deflections of the decks at the mid-span are zero.
 
Values of displacements and forces shown in Tables 3.8 and 3.9 are in accordance with the structures. Although a static parametric analysis is not the aim of this investigation, it can be interesting to observe that maximum displacements at the deck-ends are associated with the harp pattern. Because of the loads and geometric symmetry of the structures, transverse deflections at the tower-deck level as well as the longitudinal and transverse deflections of the decks at the mid-span are zero.
Line 2,187: Line 2,066:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 48%;"| [[Image:draft_Samper_432909089-image131-c.png|324px]]
+
|  style="text-align: center;vertical-align: top;width: 48%;"|[[Image:draft_Samper_432909089-monograph-image131-c.png|324px]]
  
 
<span style="text-align: center; font-size: 75%;">(a) In-plane bending moments of the deck</span>
 
<span style="text-align: center; font-size: 75%;">(a) In-plane bending moments of the deck</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image132.png|258px]]
+
|  style="text-align: center;vertical-align: top;width: 51%;"|[[Image:draft_Samper_432909089-monograph-image132.png|258px]]
  
 
<span style="text-align: center; font-size: 75%;">(b) Axial forces of the deck</span>
 
<span style="text-align: center; font-size: 75%;">(b) Axial forces of the deck</span>
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 48%;"| [[Image:draft_Samper_432909089-image133.png|246px]]
+
|  style="text-align: center;vertical-align: top;width: 48%;"|[[Image:draft_Samper_432909089-monograph-image133.png|246px]]
  
 
<span style="text-align: center; font-size: 75%;">(c) In-plane bending moments of the pylon</span>
 
<span style="text-align: center; font-size: 75%;">(c) In-plane bending moments of the pylon</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image134.png|264px]]
+
|  style="text-align: center;vertical-align: top;width: 51%;"|[[Image:draft_Samper_432909089-monograph-image134.png|264px]]
  
 
<span style="text-align: center; font-size: 75%;">(d) Axial forces of the pylon</span>
 
<span style="text-align: center; font-size: 75%;">(d) Axial forces of the pylon</span>
Line 2,208: Line 2,087:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-image135.png|258px]] '''</big>
+
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-monograph-image135.png|258px]] '''</big>
  
 
<span style="text-align: center; font-size: 75%;">(a) In-plane bending moments of the deck</span>
 
<span style="text-align: center; font-size: 75%;">(a) In-plane bending moments of the deck</span>
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-image136.png|258px]] '''</big>
+
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-monograph-image136.png|258px]] '''</big>
  
 
<span style="text-align: center; font-size: 75%;">(b) Axial forces of the deck</span>
 
<span style="text-align: center; font-size: 75%;">(b) Axial forces of the deck</span>
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-image137.png|234px]] '''</big>
+
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-monograph-image137.png|234px]] '''</big>
  
 
<span style="text-align: center; font-size: 75%;">(c) In-plane bending moments of the pylon</span>
 
<span style="text-align: center; font-size: 75%;">(c) In-plane bending moments of the pylon</span>
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-image138.png|246px]] '''</big>
+
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-monograph-image138.png|246px]] '''</big>
  
 
<span style="text-align: center; font-size: 75%;">(d) Axial forces of the pylon</span>
 
<span style="text-align: center; font-size: 75%;">(d) Axial forces of the pylon</span>
Line 2,270: Line 2,149:
 
From Table 3.10, it is clear that in this case ''P-Δ'' and large-displacement effects are not very significant on the longitudinal deck displacements. More important differences can be found for the bending moments of the deck, with differences up to 30% comparing the case of only cable sag effect and the case including all the geometric nonlinearities. Differences for the axial force of the towers, axial force of the deck and axial force of the cables are less sensitive, which implies that ''P-Δ'' and large-displacement effects are not very important in such cases. However, it can be supposed that those differences could be more important on longer bridges, and of course, in the nonlinear dynamic analyses. For that reason it is important to consider all those nonlinearities in the static case as starting point for the nonlinear seismic analyses [Abdel-Ghaffar, 1991; Morgenthal, 1999].
 
From Table 3.10, it is clear that in this case ''P-Δ'' and large-displacement effects are not very significant on the longitudinal deck displacements. More important differences can be found for the bending moments of the deck, with differences up to 30% comparing the case of only cable sag effect and the case including all the geometric nonlinearities. Differences for the axial force of the towers, axial force of the deck and axial force of the cables are less sensitive, which implies that ''P-Δ'' and large-displacement effects are not very important in such cases. However, it can be supposed that those differences could be more important on longer bridges, and of course, in the nonlinear dynamic analyses. For that reason it is important to consider all those nonlinearities in the static case as starting point for the nonlinear seismic analyses [Abdel-Ghaffar, 1991; Morgenthal, 1999].
  
:<big>1.4 Modal Analysis</big>
 
  
:<big>1.4.1 Natural Frequencies and Modal Shapes</big>
+
:<big>3.4 Modal Analysis</big>
 +
 
 +
:<big>3.4.1 Natural Frequencies and Modal Shapes</big>
  
 
The dynamic response of a structure can be well characterized by a modal analysis. Because of the complex nature of these structures and their seismic response, a two-dimensional analysis is not adequate to capture the three-dimensionality of the system, and for that reason, a ''3D'' analysis is always recommended for the nonlinear static/dynamic analysis of cable-stayed bridges, and of course, for a modal analysis.
 
The dynamic response of a structure can be well characterized by a modal analysis. Because of the complex nature of these structures and their seismic response, a two-dimensional analysis is not adequate to capture the three-dimensionality of the system, and for that reason, a ''3D'' analysis is always recommended for the nonlinear static/dynamic analysis of cable-stayed bridges, and of course, for a modal analysis.
Line 2,278: Line 2,158:
 
As first approach of the seismic response of these structures, a characterization of natural frequencies, modal shapes, modal participation factors as well as damping mechanism is highly recommended, as was explained in the state-of-the-art review. In order to obtain the modified stiffness matrix considering all nonlinearities available, a nonlinear static analysis was performed first using the proposed finite element modelling. The dynamic free vibration analysis is then performed based on the deformed configuration, and considering as previous stage the nonlinear static analysis. As a consequence, the modal analysis can be used as the basis for modal superposition in the response-spectrum analysis.
 
As first approach of the seismic response of these structures, a characterization of natural frequencies, modal shapes, modal participation factors as well as damping mechanism is highly recommended, as was explained in the state-of-the-art review. In order to obtain the modified stiffness matrix considering all nonlinearities available, a nonlinear static analysis was performed first using the proposed finite element modelling. The dynamic free vibration analysis is then performed based on the deformed configuration, and considering as previous stage the nonlinear static analysis. As a consequence, the modal analysis can be used as the basis for modal superposition in the response-spectrum analysis.
  
A lot of strategies to find the natural frequencies have been proposed. The Eigenvector analysis determines the undamped free-vibration mode shapes and frequencies of the system, solving the generalized eigenvalue problem  <math display="inline">[-{\omega }^2M+K]\phi =0</math> , where M is the diagonal mass matrix, K is the stiffness matrix, ω<sup>2 </sup>is the diagonal matrix of eigenvalues, and Φ is the matrix of the corresponding eigenvectors (modal shapes). This problem can be solved using the classical tools of linear algebra, or applying approximate strategies to speed-up the solution, as for example the Stodola-Vianella method.
+
A lot of strategies to find the natural frequencies have been proposed. The Eigenvector analysis determines the undamped free-vibration mode shapes and frequencies of the system, solving the generalized eigenvalue problem  [[Image:draft_Samper_432909089-monograph-image139.png|120px]] , where M is the diagonal mass matrix, K is the stiffness matrix, ω<sup>2 </sup>is the diagonal matrix of eigenvalues, and Φ is the matrix of the corresponding eigenvectors (modal shapes). This problem can be solved using the classical tools of linear algebra, or applying approximate strategies to speed-up the solution, as for example the Stodola-Vianella method.
  
 
The Ritz-vector analysis is another strategy to solve this problem, in which the analysis seeks to find modes that are excited by a particular loading. In this sense, it has been demonstrated [Wilson et al, 1982] that dynamic analyses based on a special set of load-dependent Ritz vectors yield more accurate results than the use of the same number of natural mode shapes. The reason why the Ritz vectors yield good results is that they are generated by taking into account the spatial distribution of the dynamic loading, whereas the direct use of the natural mode shapes neglects this very important information. In addition, the Ritz-vector algorithm automatically includes the advantages of the proven numerical techniques of static condensation, Guyan reduction and static correction due to higher-mode truncation. The spatial distribution of the dynamic load vector serves as a starting load vector to initiate the procedure. The first Ritz vector is the static displacement vector corresponding to the starting load vector. The remaining vectors are generated from a recurrence relationship in which the mass matrix is multiplied by the previously obtained Ritz vector and used as the load vector for the next static solution.
 
The Ritz-vector analysis is another strategy to solve this problem, in which the analysis seeks to find modes that are excited by a particular loading. In this sense, it has been demonstrated [Wilson et al, 1982] that dynamic analyses based on a special set of load-dependent Ritz vectors yield more accurate results than the use of the same number of natural mode shapes. The reason why the Ritz vectors yield good results is that they are generated by taking into account the spatial distribution of the dynamic loading, whereas the direct use of the natural mode shapes neglects this very important information. In addition, the Ritz-vector algorithm automatically includes the advantages of the proven numerical techniques of static condensation, Guyan reduction and static correction due to higher-mode truncation. The spatial distribution of the dynamic load vector serves as a starting load vector to initiate the procedure. The first Ritz vector is the static displacement vector corresponding to the starting load vector. The remaining vectors are generated from a recurrence relationship in which the mass matrix is multiplied by the previously obtained Ritz vector and used as the load vector for the next static solution.
Line 2,286: Line 2,166:
 
{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
 
|-
 
|-
|  style="vertical-align: top;"|'''Table 3.11''' Natural Periods for the Longest Cables
+
|  style="vertical-align: top;width: 60%;"|'''Table 3.11''' Natural Periods for the Longest Cables
  
 
{| style="width: 55%;border-collapse: collapse;"  
 
{| style="width: 55%;border-collapse: collapse;"  
Line 2,328: Line 2,208:
  
  
|  style="vertical-align: top;"|Table 3.11 shows predominant periods for the longest cables of the bridges that depend on the tension forces (''T''), the angle of the cable sag with regard to the horizontal plane (''α<sub>0</sub>''), the unitary mass of the cables (''m'') and the horizontal projection of the cables (''L).''
+
|  style="vertical-align: top;width: 40%;"|Table 3.11 shows predominant periods for the longest cables of the bridges that depend on the tension forces (''T''), the angle of the cable sag with regard to the horizontal plane (''α<sub>0</sub>''), the unitary mass of the cables (''m'') and the horizontal projection of the cables (''L).''
 
|}
 
|}
  
Line 2,703: Line 2,583:
 
|-
 
|-
 
|  style="border-top: 1pt solid black;border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AB1'''</span>
 
|  style="border-top: 1pt solid black;border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AB1'''</span>
|  style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image140.png|162px]]
+
|  style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image140.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.806 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.806 sec</span>
|  style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image141.png|162px]]
+
|  style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image141.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.663 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.663 sec</span>
|  style="border-top: 1pt solid black;border-left: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image142.png|168px]]
+
|  style="border-top: 1pt solid black;border-left: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image142.png|168px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.585sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.585sec</span>
 
|-
 
|-
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AB2'''</span>
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AB2'''</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image143.png|162px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image143.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.694 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.694 sec</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image144.png|180px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image144.png|180px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.539 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.539 sec</span>
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image145.png|186px]]
+
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image145.png|186px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.064 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.064 sec</span>
 
|-
 
|-
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AB3'''</span>
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AB3'''</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image146.png|162px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image146.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.801 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.801 sec</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image147.png|162px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image147.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.798 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.798 sec</span>
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image148.png|186px]]
+
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image148.png|186px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.689 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.689 sec</span>
 
|-
 
|-
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AB4'''</span>
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AB4'''</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image149.png|162px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image149.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.804 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.804 sec</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image150.png|180px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image150.png|180px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.662 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.662 sec</span>
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image151.png|186px]]
+
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image151.png|186px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.156 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.156 sec</span>
 
|-
 
|-
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AR1'''</span>
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AR1'''</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image152.png|162px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image152.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.992 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.992 sec</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image153.png|180px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image153.png|180px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.900 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.900 sec</span>
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image154.png|186px]]
+
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image154.png|186px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.733 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.733 sec</span>
 
|-
 
|-
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AR2'''</span>
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AR2'''</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image155.png|162px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image155.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.938 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.938 sec</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image156.png|168px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image156.png|168px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.568 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.568 sec</span>
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image157.png|186px]]
+
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image157.png|186px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.130 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.130 sec</span>
 
|-
 
|-
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AR3'''</span>
 
|  style="border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AR3'''</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image158.png|162px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image158.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.020 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.020 sec</span>
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image159.png|162px]]
+
|  style="border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image159.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.795 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.795 sec</span>
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image160.png|186px]]
+
|  style="border-left: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image160.png|186px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 1.772 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 1.772 sec</span>
 
|-
 
|-
 
|  style="border-bottom: 2pt solid black;border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AR4'''</span>
 
|  style="border-bottom: 2pt solid black;border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''AR4'''</span>
|  style="border-left: 1pt solid black;border-bottom: 2pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image161.png|162px]]
+
|  style="border-left: 1pt solid black;border-bottom: 2pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image161.png|162px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.845 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.845 sec</span>
|  style="border-left: 1pt solid black;border-bottom: 2pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image162.png|180px]]
+
|  style="border-left: 1pt solid black;border-bottom: 2pt solid black;border-right: 1pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image162.png|180px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.535 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.535 sec</span>
|  style="border-left: 1pt solid black;border-bottom: 2pt solid black;text-align: center;width: 0%;"| [[Image:draft_Samper_432909089-image163.png|186px]]
+
|  style="border-left: 1pt solid black;border-bottom: 2pt solid black;text-align: center;width: 0%;"|[[Image:draft_Samper_432909089-monograph-image163.png|186px]]
  
 
<span style="text-align: center; font-size: 75%;">T = 2.080 sec</span>
 
<span style="text-align: center; font-size: 75%;">T = 2.080 sec</span>
Line 3,264: Line 3,144:
  
  
:<big>1.4.2 Damping</big>
+
:<big>3.4.2 Damping</big>
  
 
Damping on cable-stayed bridges is low, of about 2%, according to Morgenthal (1999). Kawashima and Unjoh (1991) found that critical damping ratio was dependent with the excitation amplitude and modal shape, aspect that makes the damping estimation very complex. Of course, this approach is not by the side of this research, and constant values for the damping ratio are suggested for each bridge, depending on the modal shape. In this sense, the approximation by Kawashima and Unjoh (1991), in which the critical damping ratio for the main modes is correlated with the natural frequencies of the bridges, can be applied here. Table 3.17 shows results of this formulation. Here, ξ<sup>BV</sup>, ξ<sup>BH</sup> and ξ<sup>T</sup> are the critical damping ratios for vertical bending oscillations, transverse bending oscillations and torsional oscillations, respectively.
 
Damping on cable-stayed bridges is low, of about 2%, according to Morgenthal (1999). Kawashima and Unjoh (1991) found that critical damping ratio was dependent with the excitation amplitude and modal shape, aspect that makes the damping estimation very complex. Of course, this approach is not by the side of this research, and constant values for the damping ratio are suggested for each bridge, depending on the modal shape. In this sense, the approximation by Kawashima and Unjoh (1991), in which the critical damping ratio for the main modes is correlated with the natural frequencies of the bridges, can be applied here. Table 3.17 shows results of this formulation. Here, ξ<sup>BV</sup>, ξ<sup>BH</sup> and ξ<sup>T</sup> are the critical damping ratios for vertical bending oscillations, transverse bending oscillations and torsional oscillations, respectively.
Line 3,321: Line 3,201:
 
Finally, it is important to say that results of the damping analysis are in accordance with real measures on bridges, as can be seen in the works of Garevski and Savern (1992, 1993); Yamaguchi and Manabu (1997) and Atkins and Wilson (2000).
 
Finally, it is important to say that results of the damping analysis are in accordance with real measures on bridges, as can be seen in the works of Garevski and Savern (1992, 1993); Yamaguchi and Manabu (1997) and Atkins and Wilson (2000).
  
:<big>1.5 Seismic Response Analysis Applying the Response Spectrum Method</big>
+
 
 +
:<big>3.5 Seismic Response Analysis Applying the Response Spectrum Method</big>
  
 
The response spectrum method for the seismic analysis of structures is a useful and powerful tool, well implemented in the current seismic regulations. This methodology computes the maximum seismic response of a structure using modal superposition, on the basis of a modal analysis previously performed, and applying an elastic design response spectrum as seismic input. Although this strategy is questionable in the case of structures with nonlinear seismic behaviour, it can be applicable as first approach for the seismic analysis, in order to compare the maximum seismic responses with those obtained from the nonlinear time history analysis considering similar conditions of soil and effective ground acceleration.
 
The response spectrum method for the seismic analysis of structures is a useful and powerful tool, well implemented in the current seismic regulations. This methodology computes the maximum seismic response of a structure using modal superposition, on the basis of a modal analysis previously performed, and applying an elastic design response spectrum as seismic input. Although this strategy is questionable in the case of structures with nonlinear seismic behaviour, it can be applicable as first approach for the seismic analysis, in order to compare the maximum seismic responses with those obtained from the nonlinear time history analysis considering similar conditions of soil and effective ground acceleration.
Line 3,401: Line 3,282:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
| ''' [[Image:draft_Samper_432909089-image164-c.png|294px]] '''
+
| ''' [[Image:draft_Samper_432909089-monograph-image164-c.png|294px]] '''
  
 
'''Fig. 3.17''' Design Acceleration Response Spectra
 
'''Fig. 3.17''' Design Acceleration Response Spectra
|  style="text-align: center;"|''' [[Image:draft_Samper_432909089-image165-c.png|276px]] '''
+
|  style="text-align: center;"|''' [[Image:draft_Samper_432909089-monograph-image165-c.png|276px]] '''
  
 
'''Fig. 3.18''' Design Velocity Response Spectra
 
'''Fig. 3.18''' Design Velocity Response Spectra
Line 3,416: Line 3,297:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image166.png|204px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image166.png|204px]]
  
 
'''Fig. 3.19 '''Maximum Seismic Longitudinal Displacements for 81 m - height  Towers
 
'''Fig. 3.19 '''Maximum Seismic Longitudinal Displacements for 81 m - height  Towers
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image167.png|198px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image167.png|198px]]
  
 
'''Fig. 3.20 '''Maximum Seismic Longitudinal Displacements for 111 m - height Towers
 
'''Fig. 3.20 '''Maximum Seismic Longitudinal Displacements for 111 m - height Towers
Line 3,427: Line 3,308:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-image168.png|204px]] '''
+
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image168.png|204px]] '''
  
 
'''Fig. 3.21''' Maximum Tower Longitudinal Displacements for ''AB1'' Bridge
 
'''Fig. 3.21''' Maximum Tower Longitudinal Displacements for ''AB1'' Bridge
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image169.png|204px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image169.png|204px]]
  
 
'''Fig. 3.22''' Maximum Tower Longitudinal Displacements for ''AR4'' Bridge
 
'''Fig. 3.22''' Maximum Tower Longitudinal Displacements for ''AR4'' Bridge
Line 3,442: Line 3,323:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image170.png|288px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image170.png|288px]]
  
 
'''Fig. 3.23''' Maximum Vertical Seismic Displacements – Slab-type Deck  
 
'''Fig. 3.23''' Maximum Vertical Seismic Displacements – Slab-type Deck  
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image171.png|282px]]
+
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image171.png|282px]]
  
 
'''Fig. 3.24''' Maximum Vertical Seismic Displacements – Hollow box-type Deck
 
'''Fig. 3.24''' Maximum Vertical Seismic Displacements – Hollow box-type Deck
Line 3,453: Line 3,334:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image172.png|288px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image172.png|288px]]
  
 
'''Fig. 3.25''' Maximum Vertical Deck Displacements for ''AB1'' Bridge
 
'''Fig. 3.25''' Maximum Vertical Deck Displacements for ''AB1'' Bridge
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image173.png|282px]]
+
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image173.png|282px]]
  
 
'''Fig. 3.26''' Maximum Vertical Deck Displacements for ''AR4'' Bridge
 
'''Fig. 3.26''' Maximum Vertical Deck Displacements for ''AR4'' Bridge
Line 3,468: Line 3,349:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 49%;"| [[Image:draft_Samper_432909089-image174-c.png|210px]]
+
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image174-c.png|210px]]
  
 
'''Fig. 3.27''' Envelope of Maximum Seismic Compressive Forces for 81 m-Height Towers
 
'''Fig. 3.27''' Envelope of Maximum Seismic Compressive Forces for 81 m-Height Towers
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image175-c.png|204px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image175-c.png|204px]]
  
 
'''Fig. 3.28''' Envelope of Maximum Seismic Compressive Forces for 111 m-Height Towers
 
'''Fig. 3.28''' Envelope of Maximum Seismic Compressive Forces for 111 m-Height Towers
Line 3,483: Line 3,364:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image176-c.png|186px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image176-c.png|186px]]
  
 
'''Fig. 3.29''' Envelope of Maximum Tower Axial Forces for ''AB1'' Bridge
 
'''Fig. 3.29''' Envelope of Maximum Tower Axial Forces for ''AB1'' Bridge
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image177-c.png|192px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image177-c.png|192px]]
  
 
'''Fig. 3.30''' Envelope of Maximum Tower Axial Forces for ''AR4'' Bridge
 
'''Fig. 3.30''' Envelope of Maximum Tower Axial Forces for ''AR4'' Bridge
Line 3,494: Line 3,375:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image178-c.png|294px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image178-c.png|294px]]
  
 
'''Fig. 3.31 '''Envelope of Seismic Axial Forces for Decks – 81 m Tower-Height
 
'''Fig. 3.31 '''Envelope of Seismic Axial Forces for Decks – 81 m Tower-Height
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image179-c.png|288px]]
+
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image179-c.png|288px]]
  
 
'''Fig. 3.32 '''Envelope of Seismic Axial Forces for Decks – 111 m Tower-Height
 
'''Fig. 3.32 '''Envelope of Seismic Axial Forces for Decks – 111 m Tower-Height
Line 3,509: Line 3,390:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image180-c.png|300px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image180-c.png|300px]]
  
 
'''Fig. 3.33''' Response Comparison for Axial Forces – Deck of ''AB1'' Bridge
 
'''Fig. 3.33''' Response Comparison for Axial Forces – Deck of ''AB1'' Bridge
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image181-c.png|294px]]
+
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image181-c.png|294px]]
  
 
'''Fig. 3.34''' Response Comparison for Axial Forces – Deck of ''AR4'' Bridge
 
'''Fig. 3.34''' Response Comparison for Axial Forces – Deck of ''AR4'' Bridge
Line 3,524: Line 3,405:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 53%;"| [[Image:draft_Samper_432909089-image182.png|264px]]
+
|  style="text-align: center;vertical-align: top;width: 52%;"|[[Image:draft_Samper_432909089-monograph-image182.png|264px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Tower moments of ''AB1'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Tower moments of ''AB1'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 47%;"| [[Image:draft_Samper_432909089-image183-c.png|276px]]
+
|  style="text-align: center;vertical-align: top;width: 47%;"|[[Image:draft_Samper_432909089-monograph-image183-c.png|276px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Deck moments of ''AB1'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Deck moments of ''AB1'' bridge</span>
 
|-
 
|-
|  style="vertical-align: top;width: 53%;"| [[Image:draft_Samper_432909089-image184.png|294px]]
+
|  style="vertical-align: top;width: 52%;"|[[Image:draft_Samper_432909089-monograph-image184.png|294px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Tower moments of ''AB3'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Tower moments of ''AB3'' bridge</span>
|  style="vertical-align: top;width: 47%;"| [[Image:draft_Samper_432909089-image185.png|264px]]
+
|  style="vertical-align: top;width: 47%;"|[[Image:draft_Samper_432909089-monograph-image185.png|264px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Deck moments of ''AB3'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Deck moments of ''AB3'' bridge</span>
 
|-
 
|-
|  style="vertical-align: top;width: 53%;"| [[Image:draft_Samper_432909089-image186.jpeg|288px]]
+
|  style="vertical-align: top;width: 52%;"|[[Image:draft_Samper_432909089-monograph-image186.jpeg|288px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(e)''' Tower moments of ''AR1'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(e)''' Tower moments of ''AR1'' bridge</span>
|  style="vertical-align: top;width: 47%;"| [[Image:draft_Samper_432909089-image187.jpeg|264px]]
+
|  style="vertical-align: top;width: 47%;"|[[Image:draft_Samper_432909089-monograph-image187.jpeg|264px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(f)''' Deck moments of ''AR1'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(f)''' Deck moments of ''AR1'' bridge</span>
 
|-
 
|-
|  style="vertical-align: top;width: 53%;"| [[Image:draft_Samper_432909089-image188.jpeg|294px]]
+
|  style="vertical-align: top;width: 52%;"|[[Image:draft_Samper_432909089-monograph-image188.jpeg|294px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(g)''' Tower moments of ''AR3'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(g)''' Tower moments of ''AR3'' bridge</span>
  
  
|  style="vertical-align: top;width: 47%;"| [[Image:draft_Samper_432909089-image189.jpeg|264px]]
+
|  style="vertical-align: top;width: 47%;"|[[Image:draft_Samper_432909089-monograph-image189.jpeg|264px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(h)''' Deck moments of ''AR3'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(h)''' Deck moments of ''AR3'' bridge</span>
Line 3,565: Line 3,446:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 49%;"| [[Image:draft_Samper_432909089-image190.png|276px]]
+
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image190.png|276px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Tower moments of ''AB2'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Tower moments of ''AB2'' bridge</span>
|  style="vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image191.png|282px]]
+
|  style="vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image191.png|282px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Deck moments of ''AB2'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Deck moments of ''AB2'' bridge</span>
 
|-
 
|-
|  style="vertical-align: top;width: 49%;"| [[Image:draft_Samper_432909089-image192.jpeg|282px]]
+
|  style="vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image192.jpeg|282px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Tower moments of ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Tower moments of ''AB4'' bridge</span>
  
  
|  style="vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image193.jpeg|288px]]
+
|  style="vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image193.jpeg|288px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Deck moments of ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Deck moments of ''AB4'' bridge</span>
Line 3,583: Line 3,464:
  
 
|-
 
|-
|  style="vertical-align: top;width: 49%;"| [[Image:draft_Samper_432909089-image194.jpeg|282px]]
+
|  style="vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image194.jpeg|282px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(e)''' Tower moments of ''AR2'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(e)''' Tower moments of ''AR2'' bridge</span>
  
  
|  style="vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image195.jpeg|294px]]
+
|  style="vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image195.jpeg|294px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(f)''' Deck moments of ''AR2'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(f)''' Deck moments of ''AR2'' bridge</span>
Line 3,594: Line 3,475:
  
 
|-
 
|-
|  style="vertical-align: top;width: 49%;"| [[Image:draft_Samper_432909089-image196.png|282px]]
+
|  style="vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image196.png|282px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(g)''' Tower moments of ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(g)''' Tower moments of ''AR4'' bridge</span>
  
  
|  style="vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image197.png|288px]]
+
|  style="vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image197.png|288px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(h)''' Deck moments of ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(h)''' Deck moments of ''AR4'' bridge</span>
Line 3,894: Line 3,775:
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> Near the mid-span <sup>e</sup> Out-of-plane</span>
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> Near the mid-span <sup>e</sup> Out-of-plane</span>
  
<sup>b</sup> At the tower-deck connection <sup>d</sup> In the bridge plane <sup>f</sup> At the upper strut level
+
<span style="text-align: center; font-size: 75%;"><sup>b</sup> At the tower-deck connection <sup>d</sup> In the bridge plane <sup>f</sup> At the upper strut level</span>
  
 
- Implies compression
 
- Implies compression
Line 4,004: Line 3,885:
  
  
:<big>1.6 Effect of Variation of the Stay Prestressing Forces on the Seismic Response of Cable-Stayed Bridges</big>
+
:<big>3.6 Effect of Variation of the Stay Prestressing Forces on the Seismic Response of Cable-Stayed Bridges</big>
  
 
In order to investigate if static variations of the stay prestressing forces of cable-stayed bridges are important regarding their seismic response, two bridge models were considered to simulate this effect. ''AB1'' and ''AR4 ''bridges were employed in this study, because they are representative of extreme cases of all the analyzed bridges in this work.
 
In order to investigate if static variations of the stay prestressing forces of cable-stayed bridges are important regarding their seismic response, two bridge models were considered to simulate this effect. ''AB1'' and ''AR4 ''bridges were employed in this study, because they are representative of extreme cases of all the analyzed bridges in this work.
Line 4,020: Line 3,901:
 
|  colspan='9'  style="border-top: 2pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Stay Prestressing Forces [kN].'''</span>
 
|  colspan='9'  style="border-top: 2pt solid black;border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Stay Prestressing Forces [kN].'''</span>
 
|-
 
|-
|  rowspan='8' style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image198.png|216px]] '''</span>
+
|  rowspan='8' style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image198.png|216px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(a) ''' ''AB1''- Original Load Condition</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) ''' ''AB1''- Original Load Condition</span>
Line 4,103: Line 3,984:
 
|  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|
 
|  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|
 
|-
 
|-
|  rowspan='8' style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image198.png|216px]] '''</span>
+
|  rowspan='8' style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image198.png|216px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AB1'' – Optimal Load Condition</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AB1'' – Optimal Load Condition</span>
Line 4,186: Line 4,067:
 
|  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|
 
|  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|
 
|-
 
|-
|  rowspan='8' style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-image199.png|210px]] '''
+
|  rowspan='8' style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image199.png|210px]] '''
  
 
<span style="text-align: center; font-size: 75%;">'''(c)''' '' AR4'' – Original Load Conditio'''n'''</span>
 
<span style="text-align: center; font-size: 75%;">'''(c)''' '' AR4'' – Original Load Conditio'''n'''</span>
Line 4,269: Line 4,150:
 
|  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|
 
|  style="border-left: 1pt solid black;border-bottom: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|
 
|-
 
|-
|  rowspan='8' style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-image199.png|210px]] '''
+
|  rowspan='8' style="border-top: 1pt solid black;border-left: 1pt solid black;border-right: 1pt solid black;text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image199.png|210px]] '''
  
 
<span style="text-align: center; font-size: 75%;">'''(d)''' '' AR4''  – Optimal Load Condition</span>
 
<span style="text-align: center; font-size: 75%;">'''(d)''' '' AR4''  – Optimal Load Condition</span>
Line 4,362: Line 4,243:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image200.png|192px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image200.png|192px]] '''</span>
  
 
'''Fig. 3.37''' Longitudinal displacements of the tower –''AB1'' Bridge
 
'''Fig. 3.37''' Longitudinal displacements of the tower –''AB1'' Bridge
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image201.png|198px]]
+
|  style="text-align: center;vertical-align: top;width: 51%;"|[[Image:draft_Samper_432909089-monograph-image201.png|198px]]
  
 
'''Fig. 3.38 '''Longitudinal displacements of the tower –''AR4'' Bridge
 
'''Fig. 3.38 '''Longitudinal displacements of the tower –''AR4'' Bridge
Line 4,373: Line 4,254:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image202.png|288px]]
+
|  style="text-align: center;vertical-align: top;width: 48%;"|[[Image:draft_Samper_432909089-monograph-image202.png|288px]]
  
 
'''Fig. 3.39''' Vertical displacements of the deck –''AB1'' Bridge
 
'''Fig. 3.39''' Vertical displacements of the deck –''AB1'' Bridge
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image203.png|306px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image203.png|306px]] '''</span>
  
 
'''Fig. 3.40''' Vertical displacements of the deck –''AR4'' Bridge
 
'''Fig. 3.40''' Vertical displacements of the deck –''AR4'' Bridge
Line 4,390: Line 4,271:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 53%;"| [[Image:draft_Samper_432909089-image204.png|192px]]
+
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image204.png|192px]]
  
 
'''Fig. 3.41''' Compressive forces of the tower –''AB1'' Bridge
 
'''Fig. 3.41''' Compressive forces of the tower –''AB1'' Bridge
  
  
|  colspan='2'  style="text-align: center;vertical-align: top;width: 53%;"| [[Image:draft_Samper_432909089-image205-c.png|192px]]
+
|  colspan='2'  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image205-c.png|192px]]
  
 
'''Fig. 3.42''' Compressive forces of the tower –''AR4'' Bridge
 
'''Fig. 3.42''' Compressive forces of the tower –''AR4'' Bridge
 
|-
 
|-
|  colspan='2'  style="text-align: center;vertical-align: top;width: 53%;"| [[Image:draft_Samper_432909089-image206-c.png|300px]]
+
|  colspan='2'  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image206-c.png|300px]]
  
 
'''Fig.  3.43''' Axial forces of the deck –''AB1'' Bridge
 
'''Fig.  3.43''' Axial forces of the deck –''AB1'' Bridge
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image207.png|300px]]
+
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image207.png|300px]]
  
 
'''Fig. 3.44''' Axial forces of the deck –''AR4'' Bridge
 
'''Fig. 3.44''' Axial forces of the deck –''AR4'' Bridge
Line 4,410: Line 4,291:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image208-c.png|288px]]
+
|  style="text-align: center;vertical-align: top;width: 48%;"|[[Image:draft_Samper_432909089-monograph-image208-c.png|288px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Bending moments of the towers – original load condition</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Bending moments of the towers – original load condition</span>
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image209.png|294px]]
+
|  style="text-align: center;vertical-align: top;width: 51%;"|[[Image:draft_Samper_432909089-monograph-image209.png|294px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Bending moments of the towers – optimal load condition</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Bending moments of the towers – optimal load condition</span>
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image210.png|306px]]
+
|  style="text-align: center;vertical-align: top;width: 48%;"|[[Image:draft_Samper_432909089-monograph-image210.png|306px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Deck bending moments – original load cond.</span>
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Deck bending moments – original load cond.</span>
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image211-c.png|300px]]
+
|  style="text-align: center;vertical-align: top;width: 51%;"|[[Image:draft_Samper_432909089-monograph-image211-c.png|300px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Deck bending moments – optimal load cond.</span>
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Deck bending moments – optimal load cond.</span>
Line 4,433: Line 4,314:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image212.png|306px]]
+
|  style="text-align: center;vertical-align: top;width: 48%;"|[[Image:draft_Samper_432909089-monograph-image212.png|306px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Tower bending moments – original load cond.</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Tower bending moments – original load cond.</span>
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image213.png|288px]]
+
|  style="text-align: center;vertical-align: top;width: 51%;"|[[Image:draft_Samper_432909089-monograph-image213.png|288px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Tower bending moments – optimal load cond.</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Tower bending moments – optimal load cond.</span>
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image214.png|306px]]
+
|  style="text-align: center;vertical-align: top;width: 48%;"|[[Image:draft_Samper_432909089-monograph-image214.png|306px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Deck bending moments – original load cond.</span>
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Deck bending moments – original load cond.</span>
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image215.png|288px]]
+
|  style="text-align: center;vertical-align: top;width: 51%;"|[[Image:draft_Samper_432909089-monograph-image215.png|288px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Deck bending moments – optimal load cond.</span>
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Deck bending moments – optimal load cond.</span>
Line 4,588: Line 4,469:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 60%;"| [[Image:draft_Samper_432909089-image216.png|336px]]
+
|  style="text-align: center;vertical-align: top;width: 58%;"|[[Image:draft_Samper_432909089-monograph-image216.png|336px]]
  
 
'''Fig. 3.47''' Average Variation of the Seismic Response
 
'''Fig. 3.47''' Average Variation of the Seismic Response
Line 4,599: Line 4,480:
 
As a conclusion, this study shows that low-to-moderate variations of the stay prestressing forces on cable-stayed bridges imply low variations of the seismic response. These variations of the seismic response are not very different if the stay cable layout, stay spacing or deck level is changed, and only specific differences regarding the shape of the internal forces or displacements can be found, and specially for the deck. The main variations of the seismic response come from the vertical deflections and internal forces of the deck, as long as variations for the seismic response of the towers are less sensitive, especially the longitudinal displacements and axial forces.
 
As a conclusion, this study shows that low-to-moderate variations of the stay prestressing forces on cable-stayed bridges imply low variations of the seismic response. These variations of the seismic response are not very different if the stay cable layout, stay spacing or deck level is changed, and only specific differences regarding the shape of the internal forces or displacements can be found, and specially for the deck. The main variations of the seismic response come from the vertical deflections and internal forces of the deck, as long as variations for the seismic response of the towers are less sensitive, especially the longitudinal displacements and axial forces.
  
:<big>1.7 Seismic Response Applying Nonlinear Direct Integration Time History Analysis</big>
 
  
:<big>1.7.1 General Considerations and Selected Models</big>
+
:<big>3.7 Seismic Response Applying Nonlinear Direct Integration Time History Analysis</big>
 +
 
 +
:<big>3.7.1 General Considerations and Selected Models</big>
  
 
All the subsequent analyses consider the application of step-by-step strategies to solve the cable-stayed bridge models of this work, considering that nonlinear direct integration time history analysis is the best alternative to accurately represent the complex nature of such structures. The structures are solved using the code SAP2000 [Computers & Structures, 2007], considering all the nonlinearities available and previously explained. In spite of the tremendous computational effort involved, this methodology is the best suitable, and application of the Hilber-Hughes-Taylor-α integration procedure seems to be more adequate [Hilber ''et al'', 1977], according to the explanations of Appendix A.
 
All the subsequent analyses consider the application of step-by-step strategies to solve the cable-stayed bridge models of this work, considering that nonlinear direct integration time history analysis is the best alternative to accurately represent the complex nature of such structures. The structures are solved using the code SAP2000 [Computers & Structures, 2007], considering all the nonlinearities available and previously explained. In spite of the tremendous computational effort involved, this methodology is the best suitable, and application of the Hilber-Hughes-Taylor-α integration procedure seems to be more adequate [Hilber ''et al'', 1977], according to the explanations of Appendix A.
Line 4,607: Line 4,489:
 
Response spectrum analysis of the bridge models left clear that the largest displacements are obtained with the highest deck level, and the worse condition for the internal forces is obtained with the fan pattern bridges. Although the modal analysis of the bridges showed that higher order modes can be important especially on bridges with low deck level, structures with high deck level may experience important geometric nonlinear effects, especially large nonlinear axial force – bending moment interaction for the tower and longitudinal girder elements as well as large nonlinear behaviour due to the geometric change caused by the large displacements on the whole structure. On the other hand, the deck level is generally conditioned by functional aspects, constituting a geometric parameter that cannot be modified. For those reasons, the worse condition occurs with bridges with high deck level, and only those structures are considered in the subsequent analyses. Even through the effect of the stay spacing can be important, the response spectrum analysis gave a good idea about the incidence of this parameter on the seismic response of the bridge models, and it is not necessary to consider its variation during the nonlinear time history analysis. Thus, according to the above explained and in order to avoid the large information generated and the excessive computer effort due to the application of the nonlinear time history analysis if all cases are considered, only two models are studied: bridges ''AB4 ''and ''AR4'', that is to say, fan and harp pattern bridges with 12.4 m-stay spacing (hollow box type deck) and 60 m-deck level. Those structures are considered as critical, especially the ''AB4 ''model. Although ''AR4'' bridge seems to be not very critical, its consideration permits an adequate analysis between fan and harp pattern layouts. Moreover, those structures contain less joints and elements than the other models, reducing the computer time that can be crucial in a nonlinear time history analysis. In some sense, influence of the bridge configuration on the seismic response was analyzed applying the response spectrum method. Thus, the next pages are focused on the nonlinear seismic response of the bridges for different input ground motion typologies, and taking into account the stay cable layout, with and without the incorporation of additional energy dissipation devices.
 
Response spectrum analysis of the bridge models left clear that the largest displacements are obtained with the highest deck level, and the worse condition for the internal forces is obtained with the fan pattern bridges. Although the modal analysis of the bridges showed that higher order modes can be important especially on bridges with low deck level, structures with high deck level may experience important geometric nonlinear effects, especially large nonlinear axial force – bending moment interaction for the tower and longitudinal girder elements as well as large nonlinear behaviour due to the geometric change caused by the large displacements on the whole structure. On the other hand, the deck level is generally conditioned by functional aspects, constituting a geometric parameter that cannot be modified. For those reasons, the worse condition occurs with bridges with high deck level, and only those structures are considered in the subsequent analyses. Even through the effect of the stay spacing can be important, the response spectrum analysis gave a good idea about the incidence of this parameter on the seismic response of the bridge models, and it is not necessary to consider its variation during the nonlinear time history analysis. Thus, according to the above explained and in order to avoid the large information generated and the excessive computer effort due to the application of the nonlinear time history analysis if all cases are considered, only two models are studied: bridges ''AB4 ''and ''AR4'', that is to say, fan and harp pattern bridges with 12.4 m-stay spacing (hollow box type deck) and 60 m-deck level. Those structures are considered as critical, especially the ''AB4 ''model. Although ''AR4'' bridge seems to be not very critical, its consideration permits an adequate analysis between fan and harp pattern layouts. Moreover, those structures contain less joints and elements than the other models, reducing the computer time that can be crucial in a nonlinear time history analysis. In some sense, influence of the bridge configuration on the seismic response was analyzed applying the response spectrum method. Thus, the next pages are focused on the nonlinear seismic response of the bridges for different input ground motion typologies, and taking into account the stay cable layout, with and without the incorporation of additional energy dissipation devices.
  
The geometry, structural modelling, mechanical properties, material data as well as the loads and analysis cases were well explained. Now, the seismic input is considered by use of acceleration time histories and taking into account the largest spectral velocity of each event in the period range of interest, applied to the principal direction of the structures as measure of the seismic hazard, in order to consider the velocity-sensitivity of the bridge models. The complex damping mechanism is simplified here and considered as only dependent on the modal shapes, according to Kawashima and Unjoh (1991). In this sense, proportional damping to stiffness and mass is considered in the direct integration analysis (Rayleigh’s damping), that is to say,  <math display="inline">C=\alpha M+\beta K</math> , in which ''C'' is the damping matrix, ''M'' is the mass matrix'', K'' is the stiffness matrix, ''α'' is the mass proportional coefficient, and ''β'' is the stiffness proportional coefficient. For the selected bridges, ''α'' = 0.0734 and ''β'' = 0.000513 values were used.
+
The geometry, structural modelling, mechanical properties, material data as well as the loads and analysis cases were well explained. Now, the seismic input is considered by use of acceleration time histories and taking into account the largest spectral velocity of each event in the period range of interest, applied to the principal direction of the structures as measure of the seismic hazard, in order to consider the velocity-sensitivity of the bridge models. The complex damping mechanism is simplified here and considered as only dependent on the modal shapes, according to Kawashima and Unjoh (1991). In this sense, proportional damping to stiffness and mass is considered in the direct integration analysis (Rayleigh’s damping), that is to say,  [[Image:draft_Samper_432909089-monograph-image217.png|102px]] , in which ''C'' is the damping matrix, ''M'' is the mass matrix'', K'' is the stiffness matrix, ''α'' is the mass proportional coefficient, and ''β'' is the stiffness proportional coefficient. For the selected bridges, ''α'' = 0.0734 and ''β'' = 0.000513 values were used.
  
 
For the step-by-step integration of equations of motion, 20 analysis cases involving more than 13 hours of computer time for the far-fault analysis, and more than 48 hours of computer time for the near-fault analysis were required. An estimation of more than 120 hours of computer time including trial-and-error tests, parameter adjusts and convergence trials were necessary for successful and accurate results. Although the Hilber-Hughes-Taylor-α method is unconditionally stable for linear analysis, in the case of the selected models sometimes the convergence was difficult, and a lot of error-and-trial tests were necessary to reach an adequate convergence. In this sense, the far-fault analysis was easier, with the exception of some analysis cases for ''AR4'' bridge''. ''For the case of the near-fault analysis, the convergence is more complicated, aspect that is reflected in the enormous computer time required, because of the modifications necessary in the convergence parameters. Those experiences reflect the highly nonlinear behaviour of the models and the strong incidence of the long-period velocity-pulses of the near-fault ground motions, aspect enlarged in Appendix A. In a recent publication, Chen and Ricles (2008b) expose the stability conditions of direct integration algorithms for nonlinear analysis.
 
For the step-by-step integration of equations of motion, 20 analysis cases involving more than 13 hours of computer time for the far-fault analysis, and more than 48 hours of computer time for the near-fault analysis were required. An estimation of more than 120 hours of computer time including trial-and-error tests, parameter adjusts and convergence trials were necessary for successful and accurate results. Although the Hilber-Hughes-Taylor-α method is unconditionally stable for linear analysis, in the case of the selected models sometimes the convergence was difficult, and a lot of error-and-trial tests were necessary to reach an adequate convergence. In this sense, the far-fault analysis was easier, with the exception of some analysis cases for ''AR4'' bridge''. ''For the case of the near-fault analysis, the convergence is more complicated, aspect that is reflected in the enormous computer time required, because of the modifications necessary in the convergence parameters. Those experiences reflect the highly nonlinear behaviour of the models and the strong incidence of the long-period velocity-pulses of the near-fault ground motions, aspect enlarged in Appendix A. In a recent publication, Chen and Ricles (2008b) expose the stability conditions of direct integration algorithms for nonlinear analysis.
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In order to control the convergence of the models, several time integration parameters were taken into account. The ''numerical damping'' of the Hilber-Hughes-Taylor-α method  was selected as -0.2 for all the analysis cases. This value adequately controls the instability due to the high frequency content, with an acceptable accuracy. To control the iteration and sub-stepping process, some important parameters were considered. The ''maximum sub-step size'' reflects the upper limit on the step size used for integration. The ''minimum sub-step size'' limits the smallest sub-step size, in order to stop the analysis bellow this limit indicating that convergence has failed. The ''maximum iterations per sub-step'' controls the number of iterations allowed in a step before the use of smaller sub-steps, a number usually higher for the analysis of cable structures. The ''iteration convergence tolerance'' compares the magnitude of force error with the magnitude of the force acting on the structure to guarantee that equilibrium is achieved at each step of the analysis, a value that usually decreases when large-displacement effects are considered. Anyway, for all the analysis cases in this work, time-step size of 0.02 sec was employed.
 
In order to control the convergence of the models, several time integration parameters were taken into account. The ''numerical damping'' of the Hilber-Hughes-Taylor-α method  was selected as -0.2 for all the analysis cases. This value adequately controls the instability due to the high frequency content, with an acceptable accuracy. To control the iteration and sub-stepping process, some important parameters were considered. The ''maximum sub-step size'' reflects the upper limit on the step size used for integration. The ''minimum sub-step size'' limits the smallest sub-step size, in order to stop the analysis bellow this limit indicating that convergence has failed. The ''maximum iterations per sub-step'' controls the number of iterations allowed in a step before the use of smaller sub-steps, a number usually higher for the analysis of cable structures. The ''iteration convergence tolerance'' compares the magnitude of force error with the magnitude of the force acting on the structure to guarantee that equilibrium is achieved at each step of the analysis, a value that usually decreases when large-displacement effects are considered. Anyway, for all the analysis cases in this work, time-step size of 0.02 sec was employed.
  
:<big>1.7.2 Input Ground Motions</big>
+
 
 +
:<big>3.7.2 Input Ground Motions</big>
  
 
In order to consider different characteristics of the input ground motions in this research, earthquake records were divided into far-fault and near-fault ground motions depending on the source distance. For the analysis, a collection of 10 earthquake records was selected as input ground motion considering three components for each one. The number of earthquake events was selected to take into consideration the average of the response parameter in the assessment of the structural response, and according to Eurocode 8, Part 1, a minimum of 5 accelerograms is necessary for each case. Because of the structures are founded on bedrock, time histories need to be recorded on rock or hard soil, and for that reason, soil-structure interaction is not considered here. Regarding the record selection and according to the previously exposed, a collection of 5 artificial accelerograms compatible with response spectra defined by Eurocode 8 Part 2, were generated in order to analyze the far-fault effects. For near-fault ground motions, it is preferable to employ real acceleration records, because in this case that option may reflect in a better way the basic aspects of the complex nature of the near-source effects, including the incidence of forward rupture directivity (velocity pulse-type earthquakes). In fact, near-fault effects cannot be adequately described by uniform scaling of a fixed response spectral shape; the shape of the spectrum must become richer at long periods as the level of the spectrum increases [Somerville, 1997]. Although there are some investigations about the formulation and application of near-fault design spectra, this strategy is not clearly defined for bridges, and a lot of uncertainties are involved. In this sense, the works of Li and Zhu (2003) and Akkar and Gülkan (2003) propose and explain the procedure for implementation of near-fault design spectra on building design codes. In this research, the record selection for near-fault ground motions was made choosing the current practice to carefully select records that reflect the expected magnitude, distance and other characteristics of the source of the events that are in some sense most likely to threaten the structure.
 
In order to consider different characteristics of the input ground motions in this research, earthquake records were divided into far-fault and near-fault ground motions depending on the source distance. For the analysis, a collection of 10 earthquake records was selected as input ground motion considering three components for each one. The number of earthquake events was selected to take into consideration the average of the response parameter in the assessment of the structural response, and according to Eurocode 8, Part 1, a minimum of 5 accelerograms is necessary for each case. Because of the structures are founded on bedrock, time histories need to be recorded on rock or hard soil, and for that reason, soil-structure interaction is not considered here. Regarding the record selection and according to the previously exposed, a collection of 5 artificial accelerograms compatible with response spectra defined by Eurocode 8 Part 2, were generated in order to analyze the far-fault effects. For near-fault ground motions, it is preferable to employ real acceleration records, because in this case that option may reflect in a better way the basic aspects of the complex nature of the near-source effects, including the incidence of forward rupture directivity (velocity pulse-type earthquakes). In fact, near-fault effects cannot be adequately described by uniform scaling of a fixed response spectral shape; the shape of the spectrum must become richer at long periods as the level of the spectrum increases [Somerville, 1997]. Although there are some investigations about the formulation and application of near-fault design spectra, this strategy is not clearly defined for bridges, and a lot of uncertainties are involved. In this sense, the works of Li and Zhu (2003) and Akkar and Gülkan (2003) propose and explain the procedure for implementation of near-fault design spectra on building design codes. In this research, the record selection for near-fault ground motions was made choosing the current practice to carefully select records that reflect the expected magnitude, distance and other characteristics of the source of the events that are in some sense most likely to threaten the structure.
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{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
 
|-
 
|-
|  style="vertical-align: top;"|'''Table 3.25''' Ground Motion Parameters for Far-Fault Events
+
|  style="vertical-align: top;width: 40%;"|'''Table 3.25''' Ground Motion Parameters for Far-Fault Events
  
 
{| style="width: 64%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 64%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
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|  style="vertical-align: top;"|''R<sub>e</sub>'' is the closest distance to the fault rupture surface; ''t<sub>e</sub>'' is the effective duration of the strong motion (obtained using the Arias Intensity of the earthquake events) and ''PGA, PGV'' and ''PGD'' are the peak ground acceleration, velocity and displacement respectively. All these ground motion parameters were obtained using the code SeismoSignal [Seismosoft, 2006].
+
|  style="vertical-align: top;width: 60%;"|''R<sub>e</sub>'' is the closest distance to the fault rupture surface; ''t<sub>e</sub>'' is the effective duration of the strong motion (obtained using the Arias Intensity of the earthquake events) and ''PGA, PGV'' and ''PGD'' are the peak ground acceleration, velocity and displacement respectively. All these ground motion parameters were obtained using the code SeismoSignal [Seismosoft, 2006].
 
|}
 
|}
  
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image218.png|438px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image218.png|438px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image219.png|438px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image219.png|438px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
'''Fig. 3.49''' Comparison of Horizontal Elastic Pseudo-velocity Response Spectra for Near-Fault Ground Motions</div>
 
'''Fig. 3.49''' Comparison of Horizontal Elastic Pseudo-velocity Response Spectra for Near-Fault Ground Motions</div>
  
:<big>1.7.3 Importance of Velocity Spectra on the Seismic Response of Long-Period Structures</big>
+
 
 +
:<big>3.7.3 Importance of Velocity Spectra on the Seismic Response of Long-Period Structures</big>
  
 
Traditionally, the employ of the ''PGA'' or the effective ground acceleration as measure of the seismic hazard has been widely applied in the seismic analysis of structures, and worldwide implemented in the current seismic regulations. It is known that this approximation is inaccurate because additional parameters such as the frequency content, strong motion duration of the earthquake input and some additional parameters involved with the source can be important. The approximation of the ''PGA'' as measure of the seismic hazard sometimes works good mainly on short-period structures. However, in the case of long-period structures this approximation can be wrong, and structures could be more affected by velocities or even displacements.
 
Traditionally, the employ of the ''PGA'' or the effective ground acceleration as measure of the seismic hazard has been widely applied in the seismic analysis of structures, and worldwide implemented in the current seismic regulations. It is known that this approximation is inaccurate because additional parameters such as the frequency content, strong motion duration of the earthquake input and some additional parameters involved with the source can be important. The approximation of the ''PGA'' as measure of the seismic hazard sometimes works good mainly on short-period structures. However, in the case of long-period structures this approximation can be wrong, and structures could be more affected by velocities or even displacements.
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image220.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image220.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Pseudo-acceleration </span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Pseudo-acceleration </span>
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image221.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 51%;"|[[Image:draft_Samper_432909089-monograph-image221.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Pseudo-velocity </span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Pseudo-velocity </span>
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-image222.png|312px]] '''</big>
+
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-monograph-image222.png|312px]] '''</big>
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Pseudo-acceleration</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Pseudo-acceleration</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image223.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image223.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Pseudo-velocity</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Pseudo-velocity</span>
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<span style="text-align: center; font-size: 75%;">Mw: Moment magnitude; Re: Closest distance to the fault rupture; t: Duration of the event</span>
 
<span style="text-align: center; font-size: 75%;">Mw: Moment magnitude; Re: Closest distance to the fault rupture; t: Duration of the event</span>
  
te: Effective duration of the strong-motion; PGA: Peak ground accel. PGV: Peak ground vel.
+
<span style="text-align: center; font-size: 75%;">te: Effective duration of the strong-motion; PGA: Peak ground accel. PGV: Peak ground vel.</span>
  
 
Table 3.27 shows that both stations were very close to the fault rupture, with short effective durations of the strong motion (obtained here using the Arias intensity of the earthquake components). Important ground accelerations for all the components are appreciated, and especially for San Fernando event, with ground accelerations higher than ''1.0g'' for the horizontal components. Important ground velocities are observed for the horizontal components, especially the components 260º (Landers) and 164º (San Fernando), coincidently with the components for which the maximum spectral velocities were observed. Those components experience ground velocities higher than 1 m/sec.
 
Table 3.27 shows that both stations were very close to the fault rupture, with short effective durations of the strong motion (obtained here using the Arias intensity of the earthquake components). Important ground accelerations for all the components are appreciated, and especially for San Fernando event, with ground accelerations higher than ''1.0g'' for the horizontal components. Important ground velocities are observed for the horizontal components, especially the components 260º (Landers) and 164º (San Fernando), coincidently with the components for which the maximum spectral velocities were observed. Those components experience ground velocities higher than 1 m/sec.
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image224.png|300px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image224.png|300px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Landers, Lucerne station</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Landers, Lucerne station</span>
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image225.png|300px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image225.png|300px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' San Fernando, Pacoima Dam Abut. station</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' San Fernando, Pacoima Dam Abut. station</span>
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image226.png|300px]]
+
|  style="text-align: center;vertical-align: top;width: 51%;"|[[Image:draft_Samper_432909089-monograph-image226.png|300px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Landers, Lucerne station</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Landers, Lucerne station</span>
|  style="text-align: center;vertical-align: top;width: 49%;"| [[Image:draft_Samper_432909089-image227.png|300px]]
+
|  style="text-align: center;vertical-align: top;width: 48%;"|[[Image:draft_Samper_432909089-monograph-image227.png|300px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' San Fernando, Pacoima Dam Abut. station</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' San Fernando, Pacoima Dam Abut. station</span>
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image228.png|300px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image228.png|300px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Landers, Lucerne station</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Landers, Lucerne station</span>
|  style="text-align: center;vertical-align: top;width: 52%;"| [[Image:draft_Samper_432909089-image229.png|300px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image229.png|300px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' San Fernando, Pacoima Dam Abut. station</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' San Fernando, Pacoima Dam Abut. station</span>
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Investigations regarding these matters have been focused on the need of having seismic design spectra for long-periods. It is known that velocity or displacement spectra obtained from direct conversion of acceleration spectra in most codes are unrealistic in both shape and amplitude, and for that reason, velocity or even displacement design spectra not obtained from acceleration spectra for long-periods have been proposed since the middle of the 90´s. The works of Trifunac (1995), Tolis (1999), Bommer (1999), Bommer ''et al'' (2000), Hu and Yu (2000) and Faccioli ''et al'' (2004) are some proposals of spectra for long-period structures.
 
Investigations regarding these matters have been focused on the need of having seismic design spectra for long-periods. It is known that velocity or displacement spectra obtained from direct conversion of acceleration spectra in most codes are unrealistic in both shape and amplitude, and for that reason, velocity or even displacement design spectra not obtained from acceleration spectra for long-periods have been proposed since the middle of the 90´s. The works of Trifunac (1995), Tolis (1999), Bommer (1999), Bommer ''et al'' (2000), Hu and Yu (2000) and Faccioli ''et al'' (2004) are some proposals of spectra for long-period structures.
  
:<big>1.7.4 Seismic Response Considering Far-fault Ground Motions</big>
+
 
 +
:<big>3.7.4 Seismic Response Considering Far-fault Ground Motions</big>
  
 
In the analysis of the bridge models considering far-fault ground motions, each orthogonal three-component event was applied with a time-step size of 0.02 sec. The time integration parameters to control the convergence were chosen in order to guarantee the stability conditions of the nonlinear analysis with an adequate accuracy. In this sense, the maximum and minimum sub-step size employed was zero, the maximum iterations per sub-step were 60 (for which an adequate accuracy using cable formulation was obtained in the nonlinear static analysis), and the iteration convergence tolerance was 1x10<sup>-4</sup>, for which adequate results were obtained in the large-displacement analyses. In general terms the convergence was easy to obtain in these cases, with the exception of some cases for the ''AR4'' model.
 
In the analysis of the bridge models considering far-fault ground motions, each orthogonal three-component event was applied with a time-step size of 0.02 sec. The time integration parameters to control the convergence were chosen in order to guarantee the stability conditions of the nonlinear analysis with an adequate accuracy. In this sense, the maximum and minimum sub-step size employed was zero, the maximum iterations per sub-step were 60 (for which an adequate accuracy using cable formulation was obtained in the nonlinear static analysis), and the iteration convergence tolerance was 1x10<sup>-4</sup>, for which adequate results were obtained in the large-displacement analyses. In general terms the convergence was easy to obtain in these cases, with the exception of some cases for the ''AR4'' model.
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image230.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image230.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image231.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image231.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image232.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image232.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image233.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image233.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,178: Line 5,063:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image234.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image234.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image235.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image235.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,194: Line 5,079:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image236.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image236.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image237.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image237.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,208: Line 5,093:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image238.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image238.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image239.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image239.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,226: Line 5,111:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image240.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image240.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image241.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image241.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,242: Line 5,127:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image242.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image242.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image243.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image243.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,256: Line 5,141:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image244.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image244.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image245.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image245.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,272: Line 5,157:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image246.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image246.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image247.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image247.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,286: Line 5,171:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image248.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image248.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image249.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image249.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,664: Line 5,549:
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In-plane</span>
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In-plane</span>
  
<sup>b</sup> At the tower-deck connection <sup>d</sup> Out-of-plane
+
<span style="text-align: center; font-size: 75%;"><sup>b</sup> At the tower-deck connection <sup>d</sup> Out-of-plane</span>
 +
 
  
:<big>1.7.5 Seismic Response Considering Near-Fault Ground Motions</big>
+
:<big>3.7.5 Seismic Response Considering Near-Fault Ground Motions</big>
  
 
Basically, the same considerations were used in the near-fault analysis for the models, including the time-step size, damping characterization and zero-time conditions among other things. Because of the inherent highly nonlinear behaviour involved in the near-source ground motion, some convergence troubles were experienced as was previously explained. In this sense, sometimes the integration parameters were strongly modified in order to reach the desired stability with the required accuracy. Numerical damping of -0.2 was enough to guarantee that the solution was invariant, with an adequate control of the high frequency content. Maximum sub-step size between 0 – 0.02 was employed, and a minimum sub-step size of zero was selected for all the bridge models. With regard to the maximum iterations per sub-step, different values were necessary to apply depending on the event and model. In the case of far-fault analysis, 60 iterations were enough; however, for near-fault ground motions, 120 iterations were a normal value, and sometimes up to 180 iterations were necessary (Landers event), with the obvious increment of computer time. To guarantee an adequate tolerance of the iterations, especially when large-displacement effects are considered, 1x10<sup>-3</sup> value was used.
 
Basically, the same considerations were used in the near-fault analysis for the models, including the time-step size, damping characterization and zero-time conditions among other things. Because of the inherent highly nonlinear behaviour involved in the near-source ground motion, some convergence troubles were experienced as was previously explained. In this sense, sometimes the integration parameters were strongly modified in order to reach the desired stability with the required accuracy. Numerical damping of -0.2 was enough to guarantee that the solution was invariant, with an adequate control of the high frequency content. Maximum sub-step size between 0 – 0.02 was employed, and a minimum sub-step size of zero was selected for all the bridge models. With regard to the maximum iterations per sub-step, different values were necessary to apply depending on the event and model. In the case of far-fault analysis, 60 iterations were enough; however, for near-fault ground motions, 120 iterations were a normal value, and sometimes up to 180 iterations were necessary (Landers event), with the obvious increment of computer time. To guarantee an adequate tolerance of the iterations, especially when large-displacement effects are considered, 1x10<sup>-3</sup> value was used.
Line 5,676: Line 5,562:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image250.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image250.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image251.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image251.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,690: Line 5,576:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image252.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image252.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image253.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image253.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,706: Line 5,592:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image254.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image254.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image255.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image255.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,722: Line 5,608:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image256.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image256.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image257.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image257.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,736: Line 5,622:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image258.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image258.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image259.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image259.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,756: Line 5,642:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image260.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image260.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image261.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image261.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,772: Line 5,658:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image262.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image262.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image263.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image263.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,786: Line 5,672:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image264.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image264.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image265.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image265.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,802: Line 5,688:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image266.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image266.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image267.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image267.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 5,816: Line 5,702:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image268.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image268.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image269.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image269.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 6,194: Line 6,080:
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In-plane</span>
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In-plane</span>
  
<sup>b</sup> At the tower-deck connection <sup>d</sup> Out-of-plane
+
<span style="text-align: center; font-size: 75%;"><sup>b</sup> At the tower-deck connection <sup>d</sup> Out-of-plane</span>
  
:<big>1.8 Comparative Results</big>
+
 
 +
:<big>3.8 Comparative Results</big>
  
 
The last point of this chapter exposes some comparisons between the obtained responses applying far-fault ground motions and near-fault ground motions. Moreover, in addition of the seismic responses obtained with the nonlinear time history analysis, maximum responses obtained with the response spectrum analysis are considered in order to compare results.
 
The last point of this chapter exposes some comparisons between the obtained responses applying far-fault ground motions and near-fault ground motions. Moreover, in addition of the seismic responses obtained with the nonlinear time history analysis, maximum responses obtained with the response spectrum analysis are considered in order to compare results.
Line 6,212: Line 6,099:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image270.png|258px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image270.png|258px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Longitudinal displacement of the deck</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Longitudinal displacement of the deck</span>
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image271.png|258px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image271.png|258px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Vertical deflection of the deck at the mid-span</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Vertical deflection of the deck at the mid-span</span>
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image272.png|258px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image272.png|258px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Transverse displacement of the deck at the mid-span</span>
 
<span style="text-align: center; font-size: 75%;">'''(c)''' Transverse displacement of the deck at the mid-span</span>
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image273.png|258px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image273.png|258px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Longitudinal displacement of the tower-top</span>
 
<span style="text-align: center; font-size: 75%;">'''(d)''' Longitudinal displacement of the tower-top</span>
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image274.png|258px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image274.png|258px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Compressive forces at the tower base</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Compressive forces at the tower base</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image275.png|258px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image275.png|258px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Compressive forces of the deck at the tower-deck connection</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Compressive forces of the deck at the tower-deck connection</span>
Line 6,249: Line 6,136:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image276.png|258px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image276.png|258px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-image277.png|258px]] '''</big>
+
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-monograph-image277.png|258px]] '''</big>
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' In-plane bending moments at the tower base</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' In-plane bending moments at the tower base</span>
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-image278.png|258px]] '''</big>
+
|  style="text-align: center;vertical-align: top;"|<big>''' [[Image:draft_Samper_432909089-monograph-image278.png|258px]] '''</big>
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Out-of-plane bending moments at the tower base</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Out-of-plane bending moments at the tower base</span>
Line 6,269: Line 6,156:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<big>''' [[Image:draft_Samper_432909089-image279.png|258px]] '''</big></div>
+
<big>''' [[Image:draft_Samper_432909089-monograph-image279.png|258px]] '''</big></div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Line 6,277: Line 6,164:
 
'''Fig. 3.77''' Average of the Maximum Bending Moments for ''AR4 ''Bridge</div>
 
'''Fig. 3.77''' Average of the Maximum Bending Moments for ''AR4 ''Bridge</div>
  
<div style="text-align: right; direction: ltr; margin-left: 1em;">
+
==Chapter 4. Seismic Protection. Application of Fluid Viscous Dampers==
<big>Chapter 4</big></div>
+
 
+
<div style="text-align: right; direction: ltr; margin-left: 1em;">
+
<big>Seismic Protection. Application of Fluid Viscous Dampers</big></div>
+
  
:<big>1.1 General Considerations</big>
+
:<big>4.1 General Considerations</big>
  
 
Fluid viscous dampers constitute an attractive methodology to protect structures against earthquakes. Application of those strategies on buildings and bridges has been widely used, but their incorporation on cable-stayed bridges has been slow. For that reason, this study is focused on the implementation of nonlinear viscous dampers as additional energy dissipation devices on such structures, with the purpose of analyze their seismic response in the presence of both far-fault and near-fault ground motions. In order to simplify this analysis and to consider the same nonlinear time history analyses cases previously discussed, ''AB4'' and ''AR4'' bridge models are studied. With this selection, the incidence of the stay cable layout on the seismic response and an adequate comparison with the undamped cases is possible.
 
Fluid viscous dampers constitute an attractive methodology to protect structures against earthquakes. Application of those strategies on buildings and bridges has been widely used, but their incorporation on cable-stayed bridges has been slow. For that reason, this study is focused on the implementation of nonlinear viscous dampers as additional energy dissipation devices on such structures, with the purpose of analyze their seismic response in the presence of both far-fault and near-fault ground motions. In order to simplify this analysis and to consider the same nonlinear time history analyses cases previously discussed, ''AB4'' and ''AR4'' bridge models are studied. With this selection, the incidence of the stay cable layout on the seismic response and an adequate comparison with the undamped cases is possible.
Line 6,293: Line 6,176:
 
The Chapter begins with the definition of the structural modelling of the viscous dampers, and the considered parameters. With the aim to select the optimal arrange of the dampers, five different damper layouts are studied, considering the worse conditions for both far-fault and near-fault ground motions. With this analysis, the definitive structural layout for the study of the bridges including additional viscous dampers is achieved, and the new dynamic characterization of the bridges is exposed, including evaluation of natural periods, modal shapes and modal damping. In order to select the optimal damper parameters, a parametric study is conducted with one of the bridge models and considering again the worse conditions for both far-fault and near-fault ground motions. After that, the influence of the velocity exponent of the dampers is analyzed. The nonlinear time history analysis applying the optimal damper parameters for both structures is then performed, considering all the analysis cases for both far-fault and near-fault ground motions. Finally, comparative studies between the optimal solutions, with and without the incorporation of additional dampers, are performed. Comparisons between far-fault and near-fault ground motions considering the effects of the stay cable layout are included, as well as an energy characterization of the problem.
 
The Chapter begins with the definition of the structural modelling of the viscous dampers, and the considered parameters. With the aim to select the optimal arrange of the dampers, five different damper layouts are studied, considering the worse conditions for both far-fault and near-fault ground motions. With this analysis, the definitive structural layout for the study of the bridges including additional viscous dampers is achieved, and the new dynamic characterization of the bridges is exposed, including evaluation of natural periods, modal shapes and modal damping. In order to select the optimal damper parameters, a parametric study is conducted with one of the bridge models and considering again the worse conditions for both far-fault and near-fault ground motions. After that, the influence of the velocity exponent of the dampers is analyzed. The nonlinear time history analysis applying the optimal damper parameters for both structures is then performed, considering all the analysis cases for both far-fault and near-fault ground motions. Finally, comparative studies between the optimal solutions, with and without the incorporation of additional dampers, are performed. Comparisons between far-fault and near-fault ground motions considering the effects of the stay cable layout are included, as well as an energy characterization of the problem.
  
:<big>1.2 Modelling of Nonlinear Fluid Viscous Dampers</big>
+
:<big>4.2 Modelling of Nonlinear Fluid Viscous Dampers</big>
  
 
Modelling of nonlinear fluid viscous dampers considers the use of nonlinear ''link ''elements according to the structural code SAP2000. Because of the inherent nonlinear behaviour of the dampers, during the analysis, the nonlinear force-deformation relationships are used at all degrees-of-freedom for which nonlinear properties were specified, and for that reason, linear effective stiffness and linear effective damping is not used for any nonlinear analysis. In this sense, nonlinear time-history analysis is absolutely necessary when nonlinear additional energy dissipation devices are added. This is the correct way to determine the effect of added dampers, since nonlinear time-history analysis does not use the effective damping values, and the energy dissipation in the elements is directly accounted for, as well as the effects of modal cross-coupling [Computers & Structures, 2007].
 
Modelling of nonlinear fluid viscous dampers considers the use of nonlinear ''link ''elements according to the structural code SAP2000. Because of the inherent nonlinear behaviour of the dampers, during the analysis, the nonlinear force-deformation relationships are used at all degrees-of-freedom for which nonlinear properties were specified, and for that reason, linear effective stiffness and linear effective damping is not used for any nonlinear analysis. In this sense, nonlinear time-history analysis is absolutely necessary when nonlinear additional energy dissipation devices are added. This is the correct way to determine the effect of added dampers, since nonlinear time-history analysis does not use the effective damping values, and the energy dissipation in the elements is directly accounted for, as well as the effects of modal cross-coupling [Computers & Structures, 2007].
Line 6,299: Line 6,182:
 
{| style="width: 100%;"  
 
{| style="width: 100%;"  
 
|-
 
|-
|  style="vertical-align: top;width: 30%;"| [[Image:draft_Samper_432909089-image280.png|102px]]
+
|  style="vertical-align: top;width: 30%;"|[[Image:draft_Samper_432909089-monograph-image280.png|102px]]
  
 
'''Fig. 4.1''' Maxwell Viscoelasticity Model for Nonlinear Dampers
 
'''Fig. 4.1''' Maxwell Viscoelasticity Model for Nonlinear Dampers
Line 6,306: Line 6,189:
 
The nonlinear force-deformation relationship is given by:
 
The nonlinear force-deformation relationship is given by:
  
  <math display="inline"></math>  <math display="inline">F=kd_k=c{\dot{d}}_c{}^N</math> [Eq. 4.1]
+
[[Image:draft_Samper_432909089-monograph-image281.png|12px]] [[Image:draft_Samper_432909089-monograph-image282.png|102px]] [Eq. 4.1]
  
where ''k'' is the spring constant, ''c'' is the damping coefficient, ''N'' is the velocity exponent of the damper, ''d<sub>k</sub>'' is the deformation across the spring, and  <math display="inline">{\dot{d}}_c</math> is the deformation rate of the damper.
+
where ''k'' is the spring constant, ''c'' is the damping coefficient, ''N'' is the velocity exponent of the damper, ''d<sub>k</sub>'' is the deformation across the spring, and  [[Image:draft_Samper_432909089-monograph-image283.png|18px]] is the deformation rate of the damper.
 
|}
 
|}
  
Line 6,320: Line 6,203:
 
Finally, it is important to say that, on the contrary of the viscoelastic dampers, frequency and temperature dependency is minimum [Symans ''et al'', 2008], which simplifies the mathematical modelling of the viscous dampers.
 
Finally, it is important to say that, on the contrary of the viscoelastic dampers, frequency and temperature dependency is minimum [Symans ''et al'', 2008], which simplifies the mathematical modelling of the viscous dampers.
  
:<big>1.3 Optimal Arrange of the Dampers</big>
+
:<big>4.3 Optimal Arrange of the Dampers</big>
  
 
One of the questions that designers need to respond is the best configuration of the dampers into the structure. The dampers are external devices, normally not affected by direct permanent loads, and located at places where the replacement or inspection is easy. In buildings, this task sometimes can be complicated, because of the numerous possibilities in which the dampers can be located; and for that reason, optimization procedures can be an excellent tool that can help designers in those decisions. In the case of bridges, possibilities for the location of the external devices are much more limited, and normally the dampers need to be installed at the deck-ends (abutment-deck connection) and/or at the pylon/tower-deck connection.
 
One of the questions that designers need to respond is the best configuration of the dampers into the structure. The dampers are external devices, normally not affected by direct permanent loads, and located at places where the replacement or inspection is easy. In buildings, this task sometimes can be complicated, because of the numerous possibilities in which the dampers can be located; and for that reason, optimization procedures can be an excellent tool that can help designers in those decisions. In the case of bridges, possibilities for the location of the external devices are much more limited, and normally the dampers need to be installed at the deck-ends (abutment-deck connection) and/or at the pylon/tower-deck connection.
Line 6,326: Line 6,209:
 
In order to investigate the best locations of the damper devices for this research, a brief study was conducted considering five analysis cases, all applied to the ''AB4'' model. The first case considered locations of the dampers at the deck-ends only, in the longitudinal direction. Case 2 considered longitudinal dampers at the deck-ends and at the tower-deck connections, in which the fixed-hinge of the tower-deck connections was changed by roller supports plus the dampers. Case 3 considered dampers at the deck-ends plus longitudinal damper in one of the tower-deck connections, and the replacement of the associated tower-deck connection by roller supports. Case 4 considered dampers at the deck-ends and transverse dampers plus roller supports at the tower-deck connections. Finally, case 5 considered dampers at the deck-ends and at the tower-deck connections for both directions, plus the corresponding replacement of the tower-deck connections by roller supports. It is obvious that cases 4 and 5 are an attempt of exploring the tri-dimensional response of the bridges in the presence of additional damping devices selected to control the longitudinal and transverse responses. Table 4.1 summarizes the five analysis cases.
 
In order to investigate the best locations of the damper devices for this research, a brief study was conducted considering five analysis cases, all applied to the ''AB4'' model. The first case considered locations of the dampers at the deck-ends only, in the longitudinal direction. Case 2 considered longitudinal dampers at the deck-ends and at the tower-deck connections, in which the fixed-hinge of the tower-deck connections was changed by roller supports plus the dampers. Case 3 considered dampers at the deck-ends plus longitudinal damper in one of the tower-deck connections, and the replacement of the associated tower-deck connection by roller supports. Case 4 considered dampers at the deck-ends and transverse dampers plus roller supports at the tower-deck connections. Finally, case 5 considered dampers at the deck-ends and at the tower-deck connections for both directions, plus the corresponding replacement of the tower-deck connections by roller supports. It is obvious that cases 4 and 5 are an attempt of exploring the tri-dimensional response of the bridges in the presence of additional damping devices selected to control the longitudinal and transverse responses. Table 4.1 summarizes the five analysis cases.
  
==Table 4.1 Layout of the Tower-Deck Connections and Dampers for the Analysis Cases==
+
'''Table 4.1''' Layout of the Tower-Deck Connections and Dampers for the Analysis Cases
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 6,429: Line 6,312:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image284.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image284.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image285.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image285.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 6,438: Line 6,321:
  
  
==Fig. 4.2 Longitudinal Displacement of the Deck==
+
'''Fig. 4.2''' Longitudinal Displacement of the Deck
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image286.png|312px]] '''</span>
+
|  style="text-align: right;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image286.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image287.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image287.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 6,451: Line 6,334:
  
  
==Fig. 4.3 Transverse Displacement of the Deck at the Mid-Span==
+
'''Fig. 4.3''' Transverse Displacement of the Deck at the Mid-Span
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image288.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image288.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image289.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image289.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 6,464: Line 6,347:
  
  
==Fig. 4.4 Axial Force at the Tower Base==
+
'''Fig. 4.4'''Axial Force at the Tower Base
  
 
An exhaustive comparison between different damper layouts shows that the worse conditions for both far-fault and near fault ground motions are obtained with case 3 followed close by cases 1 and 4. Best cases for an adequate control of the longitudinal and transverse displacements of the deck are obtained with cases 2 and 5. Similarly, axial forces of the tower and deck are better controlled with cases 2 and 5.
 
An exhaustive comparison between different damper layouts shows that the worse conditions for both far-fault and near fault ground motions are obtained with case 3 followed close by cases 1 and 4. Best cases for an adequate control of the longitudinal and transverse displacements of the deck are obtained with cases 2 and 5. Similarly, axial forces of the tower and deck are better controlled with cases 2 and 5.
Line 6,470: Line 6,353:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image290.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image290.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image291.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image291.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 6,479: Line 6,362:
  
  
==Fig. 4.5 Axial Force of the Deck at the Tower-Deck Connection==
+
'''Fig. 4.5''' Axial Force of the Deck at the Tower-Deck Connection
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image292.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image292.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image293.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image293.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 6,492: Line 6,375:
  
  
==Fig. 4.6 Deck-end Damper Forces==
+
'''Fig. 4.6''' Deck-end Damper Forces
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image294.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image294.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image295.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image295.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 6,505: Line 6,388:
  
  
==Fig. 4.7 Longitudinal Damper Forces at the Tower-Deck Connection ==
+
'''Fig. 4.7''' Longitudinal Damper Forces at the Tower-Deck Connection  
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image296.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image296.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-image297.png|312px]] '''</span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;">''' [[Image:draft_Samper_432909089-monograph-image297.png|312px]] '''</span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 6,518: Line 6,401:
  
  
==Fig. 4.8 Transverse Damper Forces at the Tower-Deck Connection==
+
'''Fig. 4.8''' Transverse Damper Forces at the Tower-Deck Connection
  
 
The analysis of the damper forces shows that for far-fault ground motion case 2 controls efficiently deck-end damper forces and longitudinal damper forces at the tower-deck connection. For near-fault ground motion, cases 2 and 5 adequately control deck-end damper forces, but for the longitudinal damper forces at the tower-deck connection, case 5 is the best choice and case 2 seems to be an unfavourable layout. Regarding the transverse damper forces at the tower-deck connection, time-history plots show that for both far-fault and near-fault ground motions, cases 4 and 5 are practically superimposed, experiencing the same behaviour. This implies that both cases are the same in terms of the control of the transverse damper forces.
 
The analysis of the damper forces shows that for far-fault ground motion case 2 controls efficiently deck-end damper forces and longitudinal damper forces at the tower-deck connection. For near-fault ground motion, cases 2 and 5 adequately control deck-end damper forces, but for the longitudinal damper forces at the tower-deck connection, case 5 is the best choice and case 2 seems to be an unfavourable layout. Regarding the transverse damper forces at the tower-deck connection, time-history plots show that for both far-fault and near-fault ground motions, cases 4 and 5 are practically superimposed, experiencing the same behaviour. This implies that both cases are the same in terms of the control of the transverse damper forces.
Line 6,526: Line 6,409:
 
As a complement of the time-history plots, Tables 4.2 to 4.5 show a summary of the maximum main responses for the five cases in terms of displacements, velocities, internal forces, damper forces and damper velocities respectively. ''Δ<sub>1-L</sub>'' is the maximum displacement of the tower-top in the longitudinal direction; ''Δ<sub>3-V</sub>'' is the maximum vertical displacement of the deck at the mid-span; ''Δ<sub>3-T</sub>'' is the maximum transverse displacement of the deck at the mid-span; and ''Δ<sub>4-L</sub>'' is the maximum longitudinal displacement of the deck. Analogously, velocities ''V'' at the same points for the control of displacements were defined, according to the nomenclature for the seismic responses applied in Chapter 3. Displacements and velocities are shown as absolute values for simplicity. Likewise, maximum internal forces on the structure are shown as absolute values for bending moments. Nomenclature for internal forces is the same considered in Chapter 3. In Table 4.5, maximum damper velocities (''V<sub>max</sub>'') and forces (''F<sub>max</sub>'') are shown for deck-end dampers, longitudinal tower dampers and transverse tower dampers respectively. Response of the dampers is shown in absolute values.
 
As a complement of the time-history plots, Tables 4.2 to 4.5 show a summary of the maximum main responses for the five cases in terms of displacements, velocities, internal forces, damper forces and damper velocities respectively. ''Δ<sub>1-L</sub>'' is the maximum displacement of the tower-top in the longitudinal direction; ''Δ<sub>3-V</sub>'' is the maximum vertical displacement of the deck at the mid-span; ''Δ<sub>3-T</sub>'' is the maximum transverse displacement of the deck at the mid-span; and ''Δ<sub>4-L</sub>'' is the maximum longitudinal displacement of the deck. Analogously, velocities ''V'' at the same points for the control of displacements were defined, according to the nomenclature for the seismic responses applied in Chapter 3. Displacements and velocities are shown as absolute values for simplicity. Likewise, maximum internal forces on the structure are shown as absolute values for bending moments. Nomenclature for internal forces is the same considered in Chapter 3. In Table 4.5, maximum damper velocities (''V<sub>max</sub>'') and forces (''F<sub>max</sub>'') are shown for deck-end dampers, longitudinal tower dampers and transverse tower dampers respectively. Response of the dampers is shown in absolute values.
  
==Table 4.2 Maximum Relative Displacements [cm] and Velocities [m/sec] in the Structure==
+
'''Table 4.2''' Maximum Relative Displacements [cm] and Velocities [m/sec] in the Structure
  
 
{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
Line 6,645: Line 6,528:
 
According to Table 4.2, similar results are obtained for the displacements in the presence of far-fault ground motion, for each measured response. Maximum differences are obtained for the longitudinal displacements of the deck (43%), with maximum value for case 3. Similar differences are obtained comparing velocities between the analyzed cases. For near-fault ground motion, more important differences are obtained, and especially for the vertical and transverse displacements of the deck. Analogue differences are obtained for velocities. Furthermore, for both near-fault and far-fault ground motions, maximum displacements and velocities are obtained for the transverse displacements of the deck, independent on the damper layout. Likewise, it is confirmed that the worse condition is obtained for case 3, and the best results are obtained for cases 1, 2 and 5, for both far-fault and near-fault earthquakes. In this sense, comparing cases 2 and 5, it is obvious that practically the same maximum responses are obtained in both situations
 
According to Table 4.2, similar results are obtained for the displacements in the presence of far-fault ground motion, for each measured response. Maximum differences are obtained for the longitudinal displacements of the deck (43%), with maximum value for case 3. Similar differences are obtained comparing velocities between the analyzed cases. For near-fault ground motion, more important differences are obtained, and especially for the vertical and transverse displacements of the deck. Analogue differences are obtained for velocities. Furthermore, for both near-fault and far-fault ground motions, maximum displacements and velocities are obtained for the transverse displacements of the deck, independent on the damper layout. Likewise, it is confirmed that the worse condition is obtained for case 3, and the best results are obtained for cases 1, 2 and 5, for both far-fault and near-fault earthquakes. In this sense, comparing cases 2 and 5, it is obvious that practically the same maximum responses are obtained in both situations
  
==Table 4.3 Maximum Main Forces on the Structure – Far-Fault Ground Motion==
+
'''Table 4.3''' Maximum Main Forces on the Structure – Far-Fault Ground Motion
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 6,699: Line 6,582:
  
  
==Table 4.4 Maximum Main Forces on the Structure – Near-Fault Ground Motion==
+
'''Table 4.4''' Maximum Main Forces on the Structure – Near-Fault Ground Motion
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 6,759: Line 6,642:
 
The analysis of the internal forces shows again the worse condition obtained with case 3. For far-fault ground motion, the best results are obtained with case 1, on the contrary of the near-fault ground motion, in which minimum responses are obtained with cases 2 and 4. Likewise, the analysis of the maximum internal forces shows more important differences than the analysis of displacements and velocities. On the other hand, comparing far-fault with near-fault ground motions, it is interesting to observe that maximum bending moments of the towers are obtained at the base in the longitudinal direction (in-plane) for the near-fault condition; on the contrary of the case of the far-fault condition, in which maximum bending moments are obtained in the transverse direction (out-of-plane). This implies that the selected dampers are more effective in reducing the in-plane bending moments of the towers for far-fault ground motion, independent on the damper layout, because according to the undamped analysis of the bridge models, for both far-fault and near-fault ground motions, maximum moments of the towers were always obtained in the longitudinal direction (in-plane). Likewise, as happens with the previous results, maximum responses are always obtained for near-fault ground motion.
 
The analysis of the internal forces shows again the worse condition obtained with case 3. For far-fault ground motion, the best results are obtained with case 1, on the contrary of the near-fault ground motion, in which minimum responses are obtained with cases 2 and 4. Likewise, the analysis of the maximum internal forces shows more important differences than the analysis of displacements and velocities. On the other hand, comparing far-fault with near-fault ground motions, it is interesting to observe that maximum bending moments of the towers are obtained at the base in the longitudinal direction (in-plane) for the near-fault condition; on the contrary of the case of the far-fault condition, in which maximum bending moments are obtained in the transverse direction (out-of-plane). This implies that the selected dampers are more effective in reducing the in-plane bending moments of the towers for far-fault ground motion, independent on the damper layout, because according to the undamped analysis of the bridge models, for both far-fault and near-fault ground motions, maximum moments of the towers were always obtained in the longitudinal direction (in-plane). Likewise, as happens with the previous results, maximum responses are always obtained for near-fault ground motion.
  
==Table 4.5 Maximum Damper Forces [kN] and Velocities [m/sec]==
+
'''Table 4.5''' Maximum Damper Forces [kN] and Velocities [m/sec]
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 6,879: Line 6,762:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image298.png|276px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image298.png|276px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Maximum displacements of the deck</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Maximum displacements of the deck</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image299.png|276px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image299.png|276px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Maximum bending moments at the tower base</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Maximum bending moments at the tower base</span>
Line 6,888: Line 6,771:
  
  
==Fig. 4.9 Comparison of Maximum Responses for Damped and Undamped Cases – AB4 Model==
+
'''Fig. 4.9''' Comparison of Maximum Responses for Damped and Undamped Cases – AB4 Model
  
 
The analysis was performed for ''AB4'' model, considering the event Gatos as input ground motion. For the damped analysis, layout of case 5 was applied, that is to say, both longitudinal and transverse protection of the bridge with fluid viscous dampers, according to Table 4.1.
 
The analysis was performed for ''AB4'' model, considering the event Gatos as input ground motion. For the damped analysis, layout of case 5 was applied, that is to say, both longitudinal and transverse protection of the bridge with fluid viscous dampers, according to Table 4.1.
Line 6,899: Line 6,782:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image300-c.png|600px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image300-c.png|600px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
:<span style="text-align: center; font-size: 75%;">(1) Longitudinal layout of supports and dampers</span></div>
+
:<span style="text-align: center; font-size: 75%;">'''(1)''' Longitudinal layout of supports and dampers</span></div>
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-image301-c.png|204px]] </span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-monograph-image301-c.png|204px]] </span>
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Detail A</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Detail A</span>
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-image302-c.png|228px]] </span>
+
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 75%;"> [[Image:draft_Samper_432909089-monograph-image302-c.png|228px]] </span>
  
 
<span style="text-align: center; font-size: 75%;">'''(c) '''Detail B</span>
 
<span style="text-align: center; font-size: 75%;">'''(c) '''Detail B</span>
Line 6,915: Line 6,798:
  
  
==Fig. 4.10 Optimal Layout of the Dampers==
+
'''Fig. 4.10''' Optimal Layout of the Dampers
  
 
Position of the damper at the tower-deck connection considers an oblique location. Of course, it is possible to locate those devices considering other proposals; however this layout is simple for repairing or maintenance. This configuration needs to be considered only as basic or schematic solution, because definitive position and the main details must be materialized according to the definitive design, manufacturer’s specifications and constructive issues.
 
Position of the damper at the tower-deck connection considers an oblique location. Of course, it is possible to locate those devices considering other proposals; however this layout is simple for repairing or maintenance. This configuration needs to be considered only as basic or schematic solution, because definitive position and the main details must be materialized according to the definitive design, manufacturer’s specifications and constructive issues.
Line 6,921: Line 6,804:
 
The exposed analysis represents the general tendency of cable-stayed bridges considering different layouts of supports and dampers. The analysis was performed using the worse conditions for both far-fault and near-fault ground motions according to the selected earthquake database, and of course, some variations may be experienced if different conditions are considered. Regarding the bridge models, similar conclusions may be obtained if ''AR4 ''bridge is analyzed, according to the results of Chapter 3.
 
The exposed analysis represents the general tendency of cable-stayed bridges considering different layouts of supports and dampers. The analysis was performed using the worse conditions for both far-fault and near-fault ground motions according to the selected earthquake database, and of course, some variations may be experienced if different conditions are considered. Regarding the bridge models, similar conclusions may be obtained if ''AR4 ''bridge is analyzed, according to the results of Chapter 3.
  
:<big>1.4 Modal Analysis Considering the Optimal Arrange of the Dampers</big>
+
:<big>4.4 Modal Analysis Considering the Optimal Arrange of the Dampers</big>
  
 
An exhaustive modal analysis was performed for the undamped bridge models in Chapter 3. That study left clear the importance of an adequate modal analysis as first step in the nonlinear seismic analysis of cable-stayed bridges.
 
An exhaustive modal analysis was performed for the undamped bridge models in Chapter 3. That study left clear the importance of an adequate modal analysis as first step in the nonlinear seismic analysis of cable-stayed bridges.
Line 6,931: Line 6,814:
 
Table 4.6 shows natural periods and nature of the modal shapes for the first 15 modes, in which some changes on the natural periods for both bridge models compared with the undamped cases are obtained. As was explained, those changes are in accordance with the change of the support conditions at the tower-deck connection, which enlarges the fundamental period for both structures. It is observed a higher fundamental period for ''AB4'' bridge (6.0 sec) compared with ''AR4'' bridge (3.24 sec). This implies for these new conditions, that ''AR4'' model is longitudinally stiffer than ''AB4'' model, aspect that can be explained because of the intrinsic additional stiffness provided by the shortest cables of the harp pattern in the presence of roller supports at the tower-deck connection. For fixed-hinge connections, as happens with the undamped cases, this additional stiffness is not obvious, as can be seen in Chapter 3. In this sense, it was demonstrated that the stay spacing was not decisive on the determination of the fundamental periods of cable-stayed bridges, with higher periods obtained for the harp pattern, if fixed-hinge connections at the tower-deck level are employed. In other words, a flexibility increase implies more incidence of the longitudinal stiffness provided by the stay cable layout.
 
Table 4.6 shows natural periods and nature of the modal shapes for the first 15 modes, in which some changes on the natural periods for both bridge models compared with the undamped cases are obtained. As was explained, those changes are in accordance with the change of the support conditions at the tower-deck connection, which enlarges the fundamental period for both structures. It is observed a higher fundamental period for ''AB4'' bridge (6.0 sec) compared with ''AR4'' bridge (3.24 sec). This implies for these new conditions, that ''AR4'' model is longitudinally stiffer than ''AB4'' model, aspect that can be explained because of the intrinsic additional stiffness provided by the shortest cables of the harp pattern in the presence of roller supports at the tower-deck connection. For fixed-hinge connections, as happens with the undamped cases, this additional stiffness is not obvious, as can be seen in Chapter 3. In this sense, it was demonstrated that the stay spacing was not decisive on the determination of the fundamental periods of cable-stayed bridges, with higher periods obtained for the harp pattern, if fixed-hinge connections at the tower-deck level are employed. In other words, a flexibility increase implies more incidence of the longitudinal stiffness provided by the stay cable layout.
  
==Table 4.6 Natural Periods and Modal Shapes for Damped Models==
+
'''Table 4.6''' Natural Periods and Modal Shapes for Damped Models
  
 
{| style="width: 88%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 88%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 7,044: Line 6,927:
 
{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
 
|-
 
|-
|  style="vertical-align: top;"|'''Table 4.7 '''Critical Damping Ratios – Damped Cases
+
|  style="vertical-align: top;width: 40%;"|'''Table 4.7 '''Critical Damping Ratios – Damped Cases
  
 
{| style="width: 50%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 50%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 7,069: Line 6,952:
  
  
|  style="vertical-align: top;"|Modal damping exposed in Table 4.7 was obtained applying the empirical formulation by Kawashima ''et al'' (1993). In general terms, critical damping ratios for the damped cases are different compared with the undamped cases, with the exception of damping associated with transverse bending vibrations.  
+
|  style="vertical-align: top;width: 60%;"|Modal damping exposed in Table 4.7 was obtained applying the empirical formulation by Kawashima ''et al'' (1993). In general terms, critical damping ratios for the damped cases are different compared with the undamped cases, with the exception of damping associated with transverse bending vibrations.  
 
|}
 
|}
  
Line 7,075: Line 6,958:
 
Damping associated to vertical bending vibrations for the damped models is lower than that obtained with the undamped cases, and especially ''AB4'' model. An opposite situation occurs with damping associated to torsional oscillations, in which damping for the undamped models are almost 50% the damping of the damped cases.
 
Damping associated to vertical bending vibrations for the damped models is lower than that obtained with the undamped cases, and especially ''AB4'' model. An opposite situation occurs with damping associated to torsional oscillations, in which damping for the undamped models are almost 50% the damping of the damped cases.
  
<span id='_GoBack'></span>'''<br/>'''
 
  
:<big>1.5 Optimal Damper Parameters</big>
+
:<big>4.5 Optimal Damper Parameters</big>
  
 
Point 4.3 demonstrated that the best damper layout corresponds to longitudinal dampers located at the deck-ends and at the tower-deck connection. However, the best option necessarily includes selection of the optimal damper parameters, considering that capacity of damping devices depends on the specific damping coefficient ''C'' and velocity exponent ''N''. An adequate selection of those parameters is not trivial, and for that reason the aim of this part is to select the best combination of ''C'' and ''N'' that minimize the seismic response of the structures as well as the response of the dampers for both far-fault and near-fault ground motions. It is known that high control of the seismic forces into the structure implies higher damper forces, which necessarily requires higher damper capacities. As a result, an adequate selection of the damper parameters is essential to avoid wrong designs with the subsequent uncertainty about the seismic behaviour.
 
Point 4.3 demonstrated that the best damper layout corresponds to longitudinal dampers located at the deck-ends and at the tower-deck connection. However, the best option necessarily includes selection of the optimal damper parameters, considering that capacity of damping devices depends on the specific damping coefficient ''C'' and velocity exponent ''N''. An adequate selection of those parameters is not trivial, and for that reason the aim of this part is to select the best combination of ''C'' and ''N'' that minimize the seismic response of the structures as well as the response of the dampers for both far-fault and near-fault ground motions. It is known that high control of the seismic forces into the structure implies higher damper forces, which necessarily requires higher damper capacities. As a result, an adequate selection of the damper parameters is essential to avoid wrong designs with the subsequent uncertainty about the seismic behaviour.
Line 7,085: Line 6,967:
 
The analyses are performed using ''AB4'' model, and considering that the seismic response of ''AR4 ''model is similar, according to Chapter 3. The optimal damper layout previously analyzed is considered here, taking into account the same specifications for all the dampers. The geometry, structural modelling, loads and combinations, materials and analysis hypotheses are the same considered before. All the analyses are performed using the code SAP2000, and considering all available nonlinearities. As seismic input, the worse conditions for both far-fault and near-fault ground motions are applied, meaning that the orthogonal three-component earthquakes of Event 5 (far-fault) and Gatos (near-fault) are used again. Nonlinear direct integration time history analysis is applied for all the analyses, using the Hilber-Hughes-Taylor-α step-by-step integration method to solve the equations of motion. Time integration parameters considered to reach an accurate convergence are 0.02 sec time-step size, -0.2 numerical damping, 0.02 sec maximum sub-step size, 0 sec minimum sub-step size, 70 and 140 maximum iterations per sub-step for far-fault and near-fault ground motions respectively, and 1x10<sup>-4</sup> and 1x10<sup>-3</sup> iteration convergence tolerance for far-fault and near-fault ground motions respectively. Damping mechanism is considered as Rayleigh’s type, according to the modal damping previously obtained.
 
The analyses are performed using ''AB4'' model, and considering that the seismic response of ''AR4 ''model is similar, according to Chapter 3. The optimal damper layout previously analyzed is considered here, taking into account the same specifications for all the dampers. The geometry, structural modelling, loads and combinations, materials and analysis hypotheses are the same considered before. All the analyses are performed using the code SAP2000, and considering all available nonlinearities. As seismic input, the worse conditions for both far-fault and near-fault ground motions are applied, meaning that the orthogonal three-component earthquakes of Event 5 (far-fault) and Gatos (near-fault) are used again. Nonlinear direct integration time history analysis is applied for all the analyses, using the Hilber-Hughes-Taylor-α step-by-step integration method to solve the equations of motion. Time integration parameters considered to reach an accurate convergence are 0.02 sec time-step size, -0.2 numerical damping, 0.02 sec maximum sub-step size, 0 sec minimum sub-step size, 70 and 140 maximum iterations per sub-step for far-fault and near-fault ground motions respectively, and 1x10<sup>-4</sup> and 1x10<sup>-3</sup> iteration convergence tolerance for far-fault and near-fault ground motions respectively. Damping mechanism is considered as Rayleigh’s type, according to the modal damping previously obtained.
  
:<big>1.5.1 Parametric Analysis</big>
+
:<big>4.5.1 Parametric Analysis</big>
  
 
In order to consider representative possibilities of linear and nonlinear viscous damping, damping coefficients between 5 and 50 MN/(m/sec)<sup>N</sup>, and velocity exponents between 0.015 and 1.0 are studied, implying commercial alternatives for the dampers, currently available according to some manufacturers. Those velocity exponents cover a wide-range, from linear to highly nonlinear dampers. More than 40 nonlinear analyses were performed, implying more than 120 hours of computer time.
 
In order to consider representative possibilities of linear and nonlinear viscous damping, damping coefficients between 5 and 50 MN/(m/sec)<sup>N</sup>, and velocity exponents between 0.015 and 1.0 are studied, implying commercial alternatives for the dampers, currently available according to some manufacturers. Those velocity exponents cover a wide-range, from linear to highly nonlinear dampers. More than 40 nonlinear analyses were performed, implying more than 120 hours of computer time.
Line 7,093: Line 6,975:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image303.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image303.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image304.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image304.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 7,107: Line 6,989:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image305.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image305.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image306.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image306.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 7,116: Line 6,998:
  
  
==Fig. 4.12 Maximum Vertical Displacements of the Deck at the Mid-Span==
+
'''Fig. 4.12''' Maximum Vertical Displacements of the Deck at the Mid-Span
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image307.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image307.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image308.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image308.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 7,129: Line 7,011:
  
  
==Fig. 4.13 Maximum Axial Forces at the Tower Base==
+
'''Fig. 4.13''' Maximum Axial Forces at the Tower Base
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image309.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image309.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image310.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image310.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 7,142: Line 7,024:
  
  
==Fig. 4.14 Maximum In-Plane Bending Moments at the Tower Base==
+
'''Fig. 4.14''' Maximum In-Plane Bending Moments at the Tower Base
  
 
Longitudinal displacements of the deck decrease as damping coefficient increases. For ''C'' > 30 MN/(m/sec)<sup>N </sup>displacements become independent on the damping coefficient, for both, far-fault and near-fault ground motion, which implies that control of longitudinal displacements of the deck cannot increase for damping coefficients higher than 30 MN/(m/sec)<sup>N</sup>. For far-fault ground motion, maximum displacements tend to 10 cm for high damping coefficients. In the case of near-fault ground motion, maximum displacements of the deck tend to 20 cm for high damping coefficients, independent on the velocity exponent ''N'', as happens with the far-fault condition. The analysis of the velocity exponent ''N'' shows a general tendency in which lower deck displacements are achieved with lower velocity exponents for far-fault ground motion. An opposite behaviour occurs with near-fault ground motion, especially for ''C'' ''' '''< 20 MN/(m/sec)<sup>N</sup>. This behaviour can be explained in the fact that velocities near to 1 m/sec are demanding the dampers for the near-fault condition, which implies that higher forces are demanding the dampers for high velocity exponents, implying lower deck response; on the contrary of the case of the far-fault ground motion.
 
Longitudinal displacements of the deck decrease as damping coefficient increases. For ''C'' > 30 MN/(m/sec)<sup>N </sup>displacements become independent on the damping coefficient, for both, far-fault and near-fault ground motion, which implies that control of longitudinal displacements of the deck cannot increase for damping coefficients higher than 30 MN/(m/sec)<sup>N</sup>. For far-fault ground motion, maximum displacements tend to 10 cm for high damping coefficients. In the case of near-fault ground motion, maximum displacements of the deck tend to 20 cm for high damping coefficients, independent on the velocity exponent ''N'', as happens with the far-fault condition. The analysis of the velocity exponent ''N'' shows a general tendency in which lower deck displacements are achieved with lower velocity exponents for far-fault ground motion. An opposite behaviour occurs with near-fault ground motion, especially for ''C'' ''' '''< 20 MN/(m/sec)<sup>N</sup>. This behaviour can be explained in the fact that velocities near to 1 m/sec are demanding the dampers for the near-fault condition, which implies that higher forces are demanding the dampers for high velocity exponents, implying lower deck response; on the contrary of the case of the far-fault ground motion.
Line 7,158: Line 7,040:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image311.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image311.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image312.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image312.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 7,167: Line 7,049:
  
  
==Fig. 4.15 Maximum Deck-End Damper Forces==
+
'''Fig. 4.15''' Maximum Deck-End Damper Forces
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image313.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image313.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image314.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image314.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 7,180: Line 7,062:
  
  
==Fig. 4.16 Maximum Damper Forces at the Tower-Deck Connection==
+
'''Fig. 4.16''' Maximum Damper Forces at the Tower-Deck Connection
  
 
The analysis of the damper velocities (Figs. 4.17 and 4.18) shows an opposite situation compared with the damper forces, that is to say, minimum velocities obtained for highly nonlinear dampers. In other words, minimum velocity responses imply maximum damper forces. Velocities tend to decrease as the damping coefficient increases, although for ''C'' > 30 MN/(m/sec)<sup>N</sup>, velocities are independent on the damping coefficient for both far-fault and near-fault ground motion. Likewise, it is clear that maximum damper velocities can be obtained for the near-fault condition.
 
The analysis of the damper velocities (Figs. 4.17 and 4.18) shows an opposite situation compared with the damper forces, that is to say, minimum velocities obtained for highly nonlinear dampers. In other words, minimum velocity responses imply maximum damper forces. Velocities tend to decrease as the damping coefficient increases, although for ''C'' > 30 MN/(m/sec)<sup>N</sup>, velocities are independent on the damping coefficient for both far-fault and near-fault ground motion. Likewise, it is clear that maximum damper velocities can be obtained for the near-fault condition.
Line 7,188: Line 7,070:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image315.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image315.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image316.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image316.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 7,197: Line 7,079:
  
  
== Fig. 4.17 Maximum Deck-End Damper Velocities==
+
'''Fig. 4.17''' Maximum Deck-End Damper Velocities
  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image317.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image317.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image318.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image318.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 7,210: Line 7,092:
  
  
==Fig. 4.18 Maximum Damper Velocities at the Tower-Deck Connection==
+
'''Fig. 4.18''' Maximum Damper Velocities at the Tower-Deck Connection
  
:<big>1.5.2 Selection of the Damper Parameters</big>
+
:<big>4.5.2 Selection of the Damper Parameters</big>
  
 
Selection of the optimal damper parameters necessarily consists in obtaining an efficient control of both the structural response and the damper response. In this sense, optimization techniques can be employed in this task. It seems to be that a reasonable approximation is to minimize the maximum longitudinal displacements of the deck as characteristic measure of the structural response, and to minimize the maximum damper forces as characteristic measure of the damper response; considering those maximum responses as absolute values for simplicity. By this way, we can define the longitudinal deck displacement matrix ''A'', in which ''a<sub>ij</sub>'' represents the maximum absolute longitudinal displacement of the deck [cm] for damping coefficient ''i'' and velocity exponent ''j''. Similarly, it is possible to define the damper force matrix ''B'', in which ''b<sub>ij</sub>'' represents the maximum absolute damper force [kN] associated to the damping coefficient ''i'' and velocity exponent ''j. ''Thus, defining matrix ''F = AB'', the task is to seek the minimum values of ''F''. This simple procedure can be applied separately for deck-end dampers as well as the dampers located at the tower-deck connection, for far-fault and near-fault ground motions.
 
Selection of the optimal damper parameters necessarily consists in obtaining an efficient control of both the structural response and the damper response. In this sense, optimization techniques can be employed in this task. It seems to be that a reasonable approximation is to minimize the maximum longitudinal displacements of the deck as characteristic measure of the structural response, and to minimize the maximum damper forces as characteristic measure of the damper response; considering those maximum responses as absolute values for simplicity. By this way, we can define the longitudinal deck displacement matrix ''A'', in which ''a<sub>ij</sub>'' represents the maximum absolute longitudinal displacement of the deck [cm] for damping coefficient ''i'' and velocity exponent ''j''. Similarly, it is possible to define the damper force matrix ''B'', in which ''b<sub>ij</sub>'' represents the maximum absolute damper force [kN] associated to the damping coefficient ''i'' and velocity exponent ''j. ''Thus, defining matrix ''F = AB'', the task is to seek the minimum values of ''F''. This simple procedure can be applied separately for deck-end dampers as well as the dampers located at the tower-deck connection, for far-fault and near-fault ground motions.
Line 7,218: Line 7,100:
 
Of course, this simplified approximation considers that the main parameters that affect both the seismic response of the structure and the seismic response of the dampers are conditioned by displacements of the deck and forces on the dampers. Approximations considering more sophisticated optimization techniques can also be applied, as well as energy approaches; however, for simplicity, the procedure here explained was used.
 
Of course, this simplified approximation considers that the main parameters that affect both the seismic response of the structure and the seismic response of the dampers are conditioned by displacements of the deck and forces on the dampers. Approximations considering more sophisticated optimization techniques can also be applied, as well as energy approaches; however, for simplicity, the procedure here explained was used.
  
:1.5.2.1 Far-fault ground motion
+
:4.5.2.1 Far-fault ground motion
  
 
:''(1) Deck-end dampers:''
 
:''(1) Deck-end dampers:''
Line 7,226: Line 7,108:
 
{|
 
{|
 
|-
 
|-
| <math display="inline"></math>
+
| [[Image:draft_Samper_432909089-monograph-image319.png|12px]]
| [[Image:draft_Samper_432909089-image320.png|center|600px]]
+
| [[Image:draft_Samper_432909089-monograph-image320.png|center|600px]]
 
|}
 
|}
  <math display="inline"></math> </div>
+
  [[Image:draft_Samper_432909089-monograph-image321.png|12px]] </div>
  
 
{| style="width: 68%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 68%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 7,284: Line 7,166:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image322.png|270px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image322.png|270px]] </div>
  
<math display="inline"></math>
+
[[Image:draft_Samper_432909089-monograph-image323.png|12px]]
  
 
{| style="width: 77%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 77%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 7,334: Line 7,216:
  
  
==In those analyses, units of C are [MN/(m/sec)<sup>N</sup>]. ==
+
In those analyses, units of C are [MN/(m/sec)<sup>N</sup>].
  
 
Values signed in green represent good candidates that minimize both maximum longitudinal displacements of the deck and maximum damper forces. ''C = 5 ''is not recommended because of the small energy dissipation associated. This implies that the optimal solution, for all damper locations, seems to be ''C = 10 ''or ''C = ''20. It is interesting to observe that the linear solution (''N = 1'') is not an optimal value, and for that reason, the best candidates for the velocity exponent are ''N = 0.1'' and ''N = 0.5.'' It is clear that ''C = 10''; ''N = 0.1'' is the same as ''C = 20; N = 0.5'' for both deck-end dampers and dampers located at the tower-deck connection. If we consider more structural response parameters, it is possible to achieve the desired optimal solution. Thus, employing results obtained from the parametric analysis, it is clear that the best option to control the vertical displacements of the deck is ''N = 0.5'' and ''C ≤ 30.'' For the tower moments, the best option is ''N = 0.5'' or ''N = 0.1'' and 10 ''≤ C ≤ 30.'' For deck-end damper forces, ''N = 0.5'' and ''C ≤ 30'' is a good solution. For deck-end damper velocities, an adequate control occurs if ''N = 0.1'' and ''10 ≤ C ≤ 50''; or ''N = 0.5 ''and ''C ≥ 20. ''The best solution for the damper forces at the tower-deck connection is ''N = 0.5 ''and ''C ≤ 30''. In the case of the damper velocities at the tower-deck connection'', N = 0.1'' and ''C ≥ 20'' is a good option. As a result, the best option involving all the considered aspects is to choose ''C = 20'' and ''N = 0.5.''
 
Values signed in green represent good candidates that minimize both maximum longitudinal displacements of the deck and maximum damper forces. ''C = 5 ''is not recommended because of the small energy dissipation associated. This implies that the optimal solution, for all damper locations, seems to be ''C = 10 ''or ''C = ''20. It is interesting to observe that the linear solution (''N = 1'') is not an optimal value, and for that reason, the best candidates for the velocity exponent are ''N = 0.1'' and ''N = 0.5.'' It is clear that ''C = 10''; ''N = 0.1'' is the same as ''C = 20; N = 0.5'' for both deck-end dampers and dampers located at the tower-deck connection. If we consider more structural response parameters, it is possible to achieve the desired optimal solution. Thus, employing results obtained from the parametric analysis, it is clear that the best option to control the vertical displacements of the deck is ''N = 0.5'' and ''C ≤ 30.'' For the tower moments, the best option is ''N = 0.5'' or ''N = 0.1'' and 10 ''≤ C ≤ 30.'' For deck-end damper forces, ''N = 0.5'' and ''C ≤ 30'' is a good solution. For deck-end damper velocities, an adequate control occurs if ''N = 0.1'' and ''10 ≤ C ≤ 50''; or ''N = 0.5 ''and ''C ≥ 20. ''The best solution for the damper forces at the tower-deck connection is ''N = 0.5 ''and ''C ≤ 30''. In the case of the damper velocities at the tower-deck connection'', N = 0.1'' and ''C ≥ 20'' is a good option. As a result, the best option involving all the considered aspects is to choose ''C = 20'' and ''N = 0.5.''
  
:1.5.2.2 Near-fault ground motion
+
:4.5.2.2 Near-fault ground motion
  
==The analysis of the near-fault condition is analogue to the far-fault analysis.==
+
The analysis of the near-fault condition is analogue to the far-fault analysis.
  
 
:''(1) Deck-end dampers:''
 
:''(1) Deck-end dampers:''
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  <math display="inline">A=\left[\begin{array}{ccccc}
+
  [[Image:draft_Samper_432909089-monograph-image324.png|600px]] </div>
C & N=0.015 & N=0.1 & N=0.5 & N=1\\
+
5 & 90.0 & 86.1 & 73.9 & 69.4\\
+
10 & 63.2 & 62.2 & 55.1 & 51.8\\
+
20 & 33.8 & 34.6 & 35.6 & 32.9\\
+
30 & 23.8 & 20.1 & 26.6 & 25.7\\
+
50 & 17.0 & 18.3 & 17.0 & 19.6
+
\end{array}\right];\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }B=</math><math>\left[\begin{array}{ccccc}
+
C & N=0.015 & N=0.1 & N=0.5 & N=1\\
+
5 & 4950 & 5230 & 6630 & 8470\\
+
10 & 9850 & 10160 & 12100 & 14200\\
+
20 & 19600 & 19700 & 20000 & 21300\\
+
30 & 29400 & 29100 & 27300 & 26500\\
+
50 & 49000 & 47800 & 38200 & 34200
+
\end{array}\right]</math> </div>
+
  
 
{| style="width: 63%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 63%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 7,410: Line 7,278:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  <math display="inline">B=\left[\begin{array}{ccccc}
+
  [[Image:draft_Samper_432909089-monograph-image325.png|276px]] </div>
C & N=0.015 & N=0.1 & N=0.5 & N=1\\
+
5 & 4900 & 5060 & 5640 & 6180\\
+
10 & 9800 & 9760 & 9500 & 8700\\
+
20 & 19500 & 18400 & 14400 & 13100\\
+
30 & 29100 & 26600 & 19600 & 16400\\
+
50 & 48300 & 44600 & 29000 & 20900
+
\end{array}\right]</math> </div>
+
  
 
{| style="width: 64%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
 
{| style="width: 64%;margin: 1em auto 0.1em auto;border-collapse: collapse;"  
Line 7,469: Line 7,330:
 
Summarizing, ''C = 20'' and ''N = 0.5'' is selected for far-fault ground motion; and ''C = 30'' and ''N = 0.015'' is applied for near-fault ground motion. Those values are used independent on the damper location.
 
Summarizing, ''C = 20'' and ''N = 0.5'' is selected for far-fault ground motion; and ''C = 30'' and ''N = 0.015'' is applied for near-fault ground motion. Those values are used independent on the damper location.
  
:<big>1.5.3 Influence of the Velocity Exponent and Damping Coefficient</big>
+
:<big>4.5.3 Influence of the Velocity Exponent and Damping Coefficient</big>
  
 
The analyses have shown the important incidence of the damper parameters on the seismic response of the structure and dampers. The velocity exponent of the dampers plays an important role on the seismic response of the dampers, in which linear dampers tend to minimize the damper forces although important damper velocities can be experienced mainly for low damping coefficients. However, it was demonstrated that the optimal solution, as a whole, involves the employ of nonlinear dampers, and especially in the presence of near-fault ground motions.
 
The analyses have shown the important incidence of the damper parameters on the seismic response of the structure and dampers. The velocity exponent of the dampers plays an important role on the seismic response of the dampers, in which linear dampers tend to minimize the damper forces although important damper velocities can be experienced mainly for low damping coefficients. However, it was demonstrated that the optimal solution, as a whole, involves the employ of nonlinear dampers, and especially in the presence of near-fault ground motions.
Line 7,477: Line 7,338:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image326.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image326.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) ''' ''C = 10 ''MN/(m/sec)<sup>N</sup></span>
 
<span style="text-align: center; font-size: 75%;">'''(a) ''' ''C = 10 ''MN/(m/sec)<sup>N</sup></span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image327.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image327.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) ''' ''C = 50 ''MN/(m/sec)<sup>N</sup></span>
 
<span style="text-align: center; font-size: 75%;">'''(b) ''' ''C = 50 ''MN/(m/sec)<sup>N</sup></span>
Line 7,495: Line 7,356:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image328.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image328.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a) ''' ''C = 10 ''MN/(m/sec)<sup>N</sup></span>
 
<span style="text-align: center; font-size: 75%;">'''(a) ''' ''C = 10 ''MN/(m/sec)<sup>N</sup></span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image329.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image329.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b) ''' ''C = 50 ''MN/(m/sec)<sup>N</sup></span>
 
<span style="text-align: center; font-size: 75%;">'''(b) ''' ''C = 50 ''MN/(m/sec)<sup>N</sup></span>
Line 7,513: Line 7,374:
 
As a summary, an important effect of the velocity exponent of the dampers is observed on the damper forces, as well as a not important effect on the structural response. The damping coefficient notably affects both the structural response and the damper response, with an increase of the structural response when the damping coefficient decreases, on the contrary of the damper response, in which an increase of the damping coefficient increases the damper forces.
 
As a summary, an important effect of the velocity exponent of the dampers is observed on the damper forces, as well as a not important effect on the structural response. The damping coefficient notably affects both the structural response and the damper response, with an increase of the structural response when the damping coefficient decreases, on the contrary of the damper response, in which an increase of the damping coefficient increases the damper forces.
  
:1.6 <big>Nonlinear Time-History Analysis </big>
+
:4.6 <big>Nonlinear Time-History Analysis </big>
  
 
The previous analyses were focused on the search of the optimal damper layout, optimal damper parameters and the importance of the damping coefficient and velocity exponent. This part analyzes the seismic response of both ''AB4'' and ''AR4 ''bridge models considering all the far-fault and near-fault events described in Chapter 3. The aim of this study is to obtain the nonlinear seismic response for the damped systems considering the optimal conditions before mentioned, in order to compare the responses between both structures, and with the undamped cases, considering the far-fault and near-fault conditions.
 
The previous analyses were focused on the search of the optimal damper layout, optimal damper parameters and the importance of the damping coefficient and velocity exponent. This part analyzes the seismic response of both ''AB4'' and ''AR4 ''bridge models considering all the far-fault and near-fault events described in Chapter 3. The aim of this study is to obtain the nonlinear seismic response for the damped systems considering the optimal conditions before mentioned, in order to compare the responses between both structures, and with the undamped cases, considering the far-fault and near-fault conditions.
Line 7,521: Line 7,382:
 
The study is divided into far-fault and near-fault ground motions, in which time histories and maximum responses are exposed for the structures and dampers. As occurs with the undamped cases, shear forces were not included in this analysis because the response of cable-stayed bridges (internal forces) is basically controlled by axial forces, bending moments and their interaction.
 
The study is divided into far-fault and near-fault ground motions, in which time histories and maximum responses are exposed for the structures and dampers. As occurs with the undamped cases, shear forces were not included in this analysis because the response of cable-stayed bridges (internal forces) is basically controlled by axial forces, bending moments and their interaction.
  
:<big>1.6.1 Far-Fault Ground Motion</big>
+
:<big>4.6.1 Far-Fault Ground Motion</big>
  
 
For the far-fault ground motion analysis, the same artificially generated earthquake events before employed are used here. Likewise, the same general considerations explained in Chapter 3 are valid here. In this sense, time-history plots shown in the following figures expose the response for the zone associated to the strong motion duration, obtained from the Arias Intensity of each event. Furthermore, at time equal to zero, the response is generally non-zero because it is obtained at the end of the nonlinear static analysis, and considered as starting point of the nonlinear direct integration time history analysis.
 
For the far-fault ground motion analysis, the same artificially generated earthquake events before employed are used here. Likewise, the same general considerations explained in Chapter 3 are valid here. In this sense, time-history plots shown in the following figures expose the response for the zone associated to the strong motion duration, obtained from the Arias Intensity of each event. Furthermore, at time equal to zero, the response is generally non-zero because it is obtained at the end of the nonlinear static analysis, and considered as starting point of the nonlinear direct integration time history analysis.
Line 7,529: Line 7,390:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image330.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image330.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image331.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image331.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,543: Line 7,404:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image332.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image332.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 51%;"| [[Image:draft_Samper_432909089-image333.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image333.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,559: Line 7,420:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image334.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image334.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image335.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image335.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,577: Line 7,438:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image336.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image336.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image337.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image337.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,593: Line 7,454:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image338.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image338.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 50%;"| [[Image:draft_Samper_432909089-image339.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image339.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,607: Line 7,468:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image340.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image340.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image341.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image341.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,623: Line 7,484:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image342.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image342.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image343.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image343.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,637: Line 7,498:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image344.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image344.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image345.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image345.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,655: Line 7,516:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image346.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image346.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image347.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image347.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,671: Line 7,532:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image348.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image348.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image349.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image349.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 7,963: Line 7,824:
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In-plane</span>
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In-plane</span>
  
<sup>b</sup> At the tower-deck connection <sup>d</sup> Out-of-plane
+
<span style="text-align: center; font-size: 75%;"><sup>b</sup> At the tower-deck connection <sup>d</sup> Out-of-plane</span>
  
 
The analysis of the maximum forces and velocities of the dampers can be appreciated in Table 4.11. For simplicity, response of the dampers is exposed in absolute values again. Almost the same results are observed for both structures, with insignificant differences between the seismic events. Important differences are observed comparing deck-end damper response with the tower damper response, in which average velocities of about 0.30 m/sec and 0.18 m/sec are obtained respectively. With regard to the average maximum damper forces, differences of about 20% are evaluated between both damper locations. Of course, maximum damper forces are obtained at the deck-ends, implying higher energy dissipation than that associated to the tower-deck connection.
 
The analysis of the maximum forces and velocities of the dampers can be appreciated in Table 4.11. For simplicity, response of the dampers is exposed in absolute values again. Almost the same results are observed for both structures, with insignificant differences between the seismic events. Important differences are observed comparing deck-end damper response with the tower damper response, in which average velocities of about 0.30 m/sec and 0.18 m/sec are obtained respectively. With regard to the average maximum damper forces, differences of about 20% are evaluated between both damper locations. Of course, maximum damper forces are obtained at the deck-ends, implying higher energy dissipation than that associated to the tower-deck connection.
Line 8,052: Line 7,913:
  
  
:<big>1.6.2 Near-Fault Ground Motion</big>
+
:<big>4.6.2 Near-Fault Ground Motion</big>
  
 
Addition of long-period velocity pulses on the seismic records involves a very different behaviour compared with results recently exposed. Of course, the main differences come from the real nature of the seismic events that now are analyzed, with evident lower frequency content, as can be seen in Chapter 3. Velocity pulses are the basic characteristic of the near-source effects, and the presence of those phenomena on long-period structures can be dramatic, with important response increases, as was demonstrated with the undamped bridges. Time histories are very different compared with those obtained during the far-fault analysis, and results show important variations from one event to another. Likewise, maximum responses are very different depending on the considered event; however, general tendencies and important observations can be proposed. The main observation is the important decrease of the seismic response when additional dampers are included, and especially the longitudinal response, as occurs with the far-fault analysis.
 
Addition of long-period velocity pulses on the seismic records involves a very different behaviour compared with results recently exposed. Of course, the main differences come from the real nature of the seismic events that now are analyzed, with evident lower frequency content, as can be seen in Chapter 3. Velocity pulses are the basic characteristic of the near-source effects, and the presence of those phenomena on long-period structures can be dramatic, with important response increases, as was demonstrated with the undamped bridges. Time histories are very different compared with those obtained during the far-fault analysis, and results show important variations from one event to another. Likewise, maximum responses are very different depending on the considered event; however, general tendencies and important observations can be proposed. The main observation is the important decrease of the seismic response when additional dampers are included, and especially the longitudinal response, as occurs with the far-fault analysis.
Line 8,062: Line 7,923:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image350.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image350.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image351.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image351.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,076: Line 7,937:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image352.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image352.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image353.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image353.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,096: Line 7,957:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image354.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image354.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image355.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image355.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,110: Line 7,971:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image356.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image356.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image357.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image357.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,124: Line 7,985:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image358.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image358.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image359.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image359.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,138: Line 7,999:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image360.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image360.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image361.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image361.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,152: Line 8,013:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image362.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image362.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image363.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image363.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,170: Line 8,031:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image364.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image364.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image365.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image365.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,186: Line 8,047:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image366.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image366.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image367.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image367.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,200: Line 8,061:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image368.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image368.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' ''AB4'' bridge</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image369.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image369.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' ''AR4'' bridge</span>
Line 8,496: Line 8,357:
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In-plane</span>
 
<span style="text-align: center; font-size: 75%;"><sup>a</sup> At the tower base <sup>c</sup> In-plane</span>
  
<sup>b</sup> At the tower-deck connection <sup>d</sup> Out-of-plane
+
<span style="text-align: center; font-size: 75%;"><sup>b</sup> At the tower-deck connection <sup>d</sup> Out-of-plane</span>
  
 
.Summarizing, the analysis considering near-fault ground motions shows that the largest seismic responses are obtained with events Gatos and Kobe. A similar situation occurs for the undamped bridges. Responses obtained for ''AB4 ''and ''AR4 ''models are similar, mainly for displacements and velocities. Most important differences are observed with the deck response for internal forces and cable forces, as long as the tower response is less sensitive. This behaviour is independent on the earthquake nature (far-fault – near-fault) or the damped or undamped condition of the bridges. As was concluded for the undamped condition, it is difficult to select the best stay cable layout in terms of the seismic response. There are not clear tendencies, and application of time-history analysis can be confused for this purpose.
 
.Summarizing, the analysis considering near-fault ground motions shows that the largest seismic responses are obtained with events Gatos and Kobe. A similar situation occurs for the undamped bridges. Responses obtained for ''AB4 ''and ''AR4 ''models are similar, mainly for displacements and velocities. Most important differences are observed with the deck response for internal forces and cable forces, as long as the tower response is less sensitive. This behaviour is independent on the earthquake nature (far-fault – near-fault) or the damped or undamped condition of the bridges. As was concluded for the undamped condition, it is difficult to select the best stay cable layout in terms of the seismic response. There are not clear tendencies, and application of time-history analysis can be confused for this purpose.
Line 8,585: Line 8,446:
  
  
:<big>1.6.3 Specifications of the Dampers</big>
+
:<big>4.6.3 Specifications of the Dampers</big>
  
 
Nonlinear time-history analysis of the bridge models considering the optimal dampers permits the specifications required for the dampers. This selection can be made in terms of the earthquake nature and locations of the dampers. Because of the same damper responses for both bridge models, the same dampers are specified. Table 4.15 summarizes the main damper specifications. The number of dampers corresponds to those required for each single damper location. Requirements for the dampers here exposed satisfy seismic requirements, and of course, additional damper parameters should be defined by the manufacturer, according to the design specifications.
 
Nonlinear time-history analysis of the bridge models considering the optimal dampers permits the specifications required for the dampers. This selection can be made in terms of the earthquake nature and locations of the dampers. Because of the same damper responses for both bridge models, the same dampers are specified. Table 4.15 summarizes the main damper specifications. The number of dampers corresponds to those required for each single damper location. Requirements for the dampers here exposed satisfy seismic requirements, and of course, additional damper parameters should be defined by the manufacturer, according to the design specifications.
Line 8,635: Line 8,496:
  
  
:<big>1.7 Comparative Results and Discussion</big>
+
:<big>4.7 Comparative Results and Discussion</big>
  
 
Results of the nonlinear time history analysis have demonstrated the direct incidence of the fluid viscous dampers on the seismic response of the bridge models. In Chapter 3, the dynamic characterization of cable-stayed bridges was obtained by means of the modal analysis. The response spectrum analysis gave a first approach of the seismic response of the bridge models, a general seismic characterization, and allowed the selection of the bridge models for the nonlinear time-history analysis. Time-history analysis gave an accurate seismic description for the considered bridge models, in which the responses where obtained and characterized in terms of stay cable layout and earthquake nature. Chapter 4 introduces the effect of viscous dampers as passive energy dissipation devices, in which selection of the damper layout, selection of the optimal damper parameters as well as the definitive seismic responses are obtained for both models in the presence of far-fault and near-fault ground motions.
 
Results of the nonlinear time history analysis have demonstrated the direct incidence of the fluid viscous dampers on the seismic response of the bridge models. In Chapter 3, the dynamic characterization of cable-stayed bridges was obtained by means of the modal analysis. The response spectrum analysis gave a first approach of the seismic response of the bridge models, a general seismic characterization, and allowed the selection of the bridge models for the nonlinear time-history analysis. Time-history analysis gave an accurate seismic description for the considered bridge models, in which the responses where obtained and characterized in terms of stay cable layout and earthquake nature. Chapter 4 introduces the effect of viscous dampers as passive energy dissipation devices, in which selection of the damper layout, selection of the optimal damper parameters as well as the definitive seismic responses are obtained for both models in the presence of far-fault and near-fault ground motions.
Line 8,641: Line 8,502:
 
The last part of this chapter compares the seismic responses in terms of the damped Vs undamped time histories for both bridge models and taking into account the earthquake nature. Comparative results on the average of the maximum responses are analyzed, and finally, results of the energy approach are exposed and discussed.
 
The last part of this chapter compares the seismic responses in terms of the damped Vs undamped time histories for both bridge models and taking into account the earthquake nature. Comparative results on the average of the maximum responses are analyzed, and finally, results of the energy approach are exposed and discussed.
  
:<big>1.7.1 Seismic Response Comparison</big>
+
:<big>4.7.1 Seismic Response Comparison</big>
  
 
With comparative purposes, Figs. 4.41 to 4.46 show average responses of longitudinal displacements of the deck, vertical displacements of the deck at the mid-span, in-plane bending moments at the tower base, in-plane bending moments of the deck at the mid-span, axial forces at the tower base and out-of-plane bending moments at the tower base respectively. Those average responses were obtained for each time step, considering all the analyzed events for both far-fault and near-fault ground motions; and selecting ''AB4'' bridge as the analyzed model. As was explained before, it is expected a similar response for ''AR4 ''bridge, and for that reason it is not considered here. The aim of this response comparison is to show the main differences between damped Vs undamped cases.
 
With comparative purposes, Figs. 4.41 to 4.46 show average responses of longitudinal displacements of the deck, vertical displacements of the deck at the mid-span, in-plane bending moments at the tower base, in-plane bending moments of the deck at the mid-span, axial forces at the tower base and out-of-plane bending moments at the tower base respectively. Those average responses were obtained for each time step, considering all the analyzed events for both far-fault and near-fault ground motions; and selecting ''AB4'' bridge as the analyzed model. As was explained before, it is expected a similar response for ''AR4 ''bridge, and for that reason it is not considered here. The aim of this response comparison is to show the main differences between damped Vs undamped cases.
Line 8,649: Line 8,510:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image370.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image370.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image371.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image371.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
Line 8,665: Line 8,526:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image372.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image372.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image373.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image373.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
Line 8,679: Line 8,540:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image374.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image374.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image375.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image375.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
Line 8,695: Line 8,556:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image376.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image376.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image377.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image377.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
Line 8,709: Line 8,570:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image378.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image378.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image379.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image379.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
Line 8,723: Line 8,584:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image380.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image380.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(a)''' Far-fault ground motion</span>
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image381.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image381.png|312px]]
  
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b)''' Near-fault ground motion</span>
Line 8,741: Line 8,602:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-image382.png|312px]] '''
+
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image382.png|312px]] '''
  
 
'''Fig. 4.47''' Average of the Maximum Longitudinal Displacements of the Deck
 
'''Fig. 4.47''' Average of the Maximum Longitudinal Displacements of the Deck
  
  
|  style="text-align: center;vertical-align: top;width: 55%;"| [[Image:draft_Samper_432909089-image383.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image383.png|312px]]
  
 
'''Fig. 4.48''' Average of the Maximum Vertical Displacements of the Deck at the Mid-Span
 
'''Fig. 4.48''' Average of the Maximum Vertical Displacements of the Deck at the Mid-Span
Line 8,752: Line 8,613:
  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-image384.png|312px]] '''
+
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image384.png|312px]] '''
  
 
'''Fig. 4.49''' Average of the Maximum Transverse Displacements of the Deck at the Mid-Span
 
'''Fig. 4.49''' Average of the Maximum Transverse Displacements of the Deck at the Mid-Span
  
  
|  style="text-align: center;vertical-align: top;width: 55%;"| [[Image:draft_Samper_432909089-image385.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image385.png|312px]]
  
 
'''Fig. 4.50''' Average of the Maximum Longitudinal Displacements at the Tower-Top
 
'''Fig. 4.50''' Average of the Maximum Longitudinal Displacements at the Tower-Top
Line 8,763: Line 8,624:
  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image386.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:draft_Samper_432909089-monograph-image386.png|312px]]
  
 
'''Fig. 4.51''' Average of the Maximum Transverse Displacements of the Tower-Top
 
'''Fig. 4.51''' Average of the Maximum Transverse Displacements of the Tower-Top
Line 8,776: Line 8,637:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image387.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image387.png|312px]]
  
 
'''Fig. 4.52 '''Average Compressive Forces at the Tower Base  
 
'''Fig. 4.52 '''Average Compressive Forces at the Tower Base  
|  style="text-align: center;vertical-align: top;width: 57%;"| [[Image:draft_Samper_432909089-image388.png|312px]]
+
|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:draft_Samper_432909089-monograph-image388.png|312px]]
  
 
'''Fig. 4.53 '''Average Compressive Forces of the Deck at the Tower-Deck Connection
 
'''Fig. 4.53 '''Average Compressive Forces of the Deck at the Tower-Deck Connection
Line 8,787: Line 8,648:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-image389.png|312px]] '''
+
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image389.png|312px]] '''
  
 
'''Fig. 4.54 '''Average Tension Forces of the Most Loaded Cables  
 
'''Fig. 4.54 '''Average Tension Forces of the Most Loaded Cables  
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-image390.png|312px]] '''
+
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image390.png|312px]] '''
  
 
'''Fig. 4.55 '''Average In-Plane Moments at the Tower Base
 
'''Fig. 4.55 '''Average In-Plane Moments at the Tower Base
Line 8,798: Line 8,659:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-image391.png|312px]] '''
+
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image391.png|312px]] '''
  
 
'''Fig. 4.56 '''Average Out-of-Plane Moments at the Tower Base  
 
'''Fig. 4.56 '''Average Out-of-Plane Moments at the Tower Base  
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-image392.png|312px]] '''
+
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image392.png|312px]] '''
  
 
'''Fig. 4.57 '''Average Moments of the Deck at the Mid-Span
 
'''Fig. 4.57 '''Average Moments of the Deck at the Mid-Span
Line 8,809: Line 8,670:
 
The analysis of the internal forces shows very similar results for the average axial forces of the towers (Fig. 4.52), independent on the bridge model, far-fault or near-fault condition and damped or undamped situation, as was observed with time-histories. This behaviour confirms the null effect of the viscous dampers on the axial forces of the towers. The effect of the additional dampers on the seismic response reduction is especially evident for compressive forces of the deck, tension forces of the cables, in-plane bending moments of the towers and moments of the deck, with average response reductions of 13%, 37%, 69% and 45% for far-fault ground motions respectively. For near-fault ground motions, average reductions of 12%, 35%, 54% and 32% are obtained respectively. This analysis demonstrates that additional viscous dampers are extremely efficient on the seismic response control of in-plane moments of the towers and deck. By this way, taking into account results obtained from this analysis and considering the displacement response comparison, it is possible to observe that the largest response reductions are generally obtained for the far-fault ground motions, although differences with the near-fault condition can be not very interesting. On the other hand, as happens with displacements, the comparison between far-fault and near-fault ground motions shows that the largest averages of the maximum responses are obtained for the near-fault condition, basically for in-plane responses, independent on the damped or undamped situation and bridge model. Likewise, comparing ''AB4'' with ''AR4'' model, for the damped and undamped conditions, it is confirmed that basically the same response of the towers is obtained, and important differences can be appreciated for the deck and cable responses. Surprisingly, average out-of-plane moments at the tower base expose an analogue behaviour compared with the transverse response of displacements before studied, confirming the negligible effect of the viscous dampers on the transverse response.
 
The analysis of the internal forces shows very similar results for the average axial forces of the towers (Fig. 4.52), independent on the bridge model, far-fault or near-fault condition and damped or undamped situation, as was observed with time-histories. This behaviour confirms the null effect of the viscous dampers on the axial forces of the towers. The effect of the additional dampers on the seismic response reduction is especially evident for compressive forces of the deck, tension forces of the cables, in-plane bending moments of the towers and moments of the deck, with average response reductions of 13%, 37%, 69% and 45% for far-fault ground motions respectively. For near-fault ground motions, average reductions of 12%, 35%, 54% and 32% are obtained respectively. This analysis demonstrates that additional viscous dampers are extremely efficient on the seismic response control of in-plane moments of the towers and deck. By this way, taking into account results obtained from this analysis and considering the displacement response comparison, it is possible to observe that the largest response reductions are generally obtained for the far-fault ground motions, although differences with the near-fault condition can be not very interesting. On the other hand, as happens with displacements, the comparison between far-fault and near-fault ground motions shows that the largest averages of the maximum responses are obtained for the near-fault condition, basically for in-plane responses, independent on the damped or undamped situation and bridge model. Likewise, comparing ''AB4'' with ''AR4'' model, for the damped and undamped conditions, it is confirmed that basically the same response of the towers is obtained, and important differences can be appreciated for the deck and cable responses. Surprisingly, average out-of-plane moments at the tower base expose an analogue behaviour compared with the transverse response of displacements before studied, confirming the negligible effect of the viscous dampers on the transverse response.
  
:<big>1.7.2 Energy Analysis</big>
+
:<big>4.7.2 Energy Analysis</big>
  
 
An adequate study of the energies involved is fundamental to understand the seismic response of the bridge models. In this stage, we are basically interested in the input energy provided by the ground motion to the structures, and the dissipated energy by additional viscous damping. A comparison between both energies gives an idea about the performance of the structures in terms of the absorbed and dissipated energy; and the efficiency of the additional viscous damping system.
 
An adequate study of the energies involved is fundamental to understand the seismic response of the bridge models. In this stage, we are basically interested in the input energy provided by the ground motion to the structures, and the dissipated energy by additional viscous damping. A comparison between both energies gives an idea about the performance of the structures in terms of the absorbed and dissipated energy; and the efficiency of the additional viscous damping system.
  
The input energy depends on the mass ''m<sub>s</sub>'' of the system, the input ground acceleration  <math display="inline">{\ddot{x}}_{\acute g}</math> and the relative velocity of the system <math display="inline">\dot{x}</math> . On the other hand, the dissipated energy by additional viscous damping depends on the damping coefficient ''c<sub>d</sub>'', the velocity exponent ''N'', and the velocity of the system  <math display="inline">\dot{x}</math> . However, in practical terms, the dissipated energy is strongly influenced by the damping coefficient, as was previously demonstrated (see Chapter 2). As example, Fig. 4.58 exposes the input energy and the dissipated energy by additional viscous damping considering events 5 and Kobe as input ground motions, applied to ''AR4 ''bridge. Both input records are characterized by the same duration.
+
The input energy depends on the mass ''m<sub>s</sub>'' of the system, the input ground acceleration  [[Image:draft_Samper_432909089-monograph-image393.png|18px]] and the relative velocity of the system [[Image:draft_Samper_432909089-monograph-image394.png|12px]] . On the other hand, the dissipated energy by additional viscous damping depends on the damping coefficient ''c<sub>d</sub>'', the velocity exponent ''N'', and the velocity of the system  [[Image:draft_Samper_432909089-monograph-image395.png|12px]] . However, in practical terms, the dissipated energy is strongly influenced by the damping coefficient, as was previously demonstrated (see Chapter 2). As example, Fig. 4.58 exposes the input energy and the dissipated energy by additional viscous damping considering events 5 and Kobe as input ground motions, applied to ''AR4 ''bridge. Both input records are characterized by the same duration.
  
 
Distribution of both input energy and dissipated energy are quite different comparing the seismic events. Event 5 represents a typical far-fault ground motion, in which both energies linearly vary with time. The input energy is gradually introduced, but also dissipated for all the event duration. The total input energy achieved is 100 MJ, and the total dissipated energy is 60 MJ. Kobe earthquake, JMA Station, is a near-fault ground motion, in which all the input energy is introduced in brief time (≈ 13 sec), and practically all the dissipated energy is achieved in 15 sec, taking into account that both input and dissipated energies start at time equal to 7 – 8 sec. For time over 20 sec, no additional input or dissipated energy is experienced. The total input energy introduced is 160 MJ, and the total dissipated energy is 100 MJ; higher than those obtained for Event 5.
 
Distribution of both input energy and dissipated energy are quite different comparing the seismic events. Event 5 represents a typical far-fault ground motion, in which both energies linearly vary with time. The input energy is gradually introduced, but also dissipated for all the event duration. The total input energy achieved is 100 MJ, and the total dissipated energy is 60 MJ. Kobe earthquake, JMA Station, is a near-fault ground motion, in which all the input energy is introduced in brief time (≈ 13 sec), and practically all the dissipated energy is achieved in 15 sec, taking into account that both input and dissipated energies start at time equal to 7 – 8 sec. For time over 20 sec, no additional input or dissipated energy is experienced. The total input energy introduced is 160 MJ, and the total dissipated energy is 100 MJ; higher than those obtained for Event 5.
Line 8,821: Line 8,682:
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-image396.png|312px]] '''
+
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image396.png|312px]] '''
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Event 5 </span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Event 5 </span>
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-image397.png|312px]] '''
+
|  style="vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image397.png|312px]] '''
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Kobe Earthquake, JMA Station</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Kobe Earthquake, JMA Station</span>
Line 8,834: Line 8,695:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<big>''' [[Image:draft_Samper_432909089-image398.png|408px]] '''</big></div>
+
<big>''' [[Image:draft_Samper_432909089-monograph-image398.png|408px]] '''</big></div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  [[Image:draft_Samper_432909089-image399.png|396px]] </div>
+
  [[Image:draft_Samper_432909089-monograph-image399.png|396px]] </div>
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
{| style="width: 100%;margin: 1em auto 0.1em auto;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-image400.png|318px]] '''
+
|  style="text-align: center;vertical-align: top;"|''' [[Image:draft_Samper_432909089-monograph-image400.png|318px]] '''
  
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion </span>
 
<span style="text-align: center; font-size: 75%;">'''(a) '''Far-fault ground motion </span>
|  style="vertical-align: top;"|'''4 [[Image:draft_Samper_432909089-image401.png|312px]] '''
+
|  style="vertical-align: top;"|'''4 [[Image:draft_Samper_432909089-monograph-image401.png|312px]] '''
  
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
 
<span style="text-align: center; font-size: 75%;">'''(b) '''Near-fault ground motion</span>
Line 8,869: Line 8,730:
 
As a general conclusion, very high energy dissipation rates are obtained, which confirms the important incidence of the addition of fluid viscous dampers as passive energy dissipation devices to reduce the seismic response of cable-stayed bridges.
 
As a general conclusion, very high energy dissipation rates are obtained, which confirms the important incidence of the addition of fluid viscous dampers as passive energy dissipation devices to reduce the seismic response of cable-stayed bridges.
  
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+
==Appendix A. Step-by-Step Nonlinear Time History Analysis==
<big>''' '''Appendix A</big></div>
+
 
+
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+
<big>Step-by-Step Nonlinear Time History Analysis </big></div>
+
  
 
<big>A.1 General Considerations</big>
 
<big>A.1 General Considerations</big>
Line 8,896: Line 8,753:
  
 
The α-method uses the Newmark method to solve the following modified equations of motion:
 
The α-method uses the Newmark method to solve the following modified equations of motion:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image402.png|456px]] [Eq. A.1]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
where ''M'' is the mass matrix, ''C'' is the viscous damping matrix, ''K'' is the stiffness matrix, ''F<sub>t</sub> ''is the vector of forces that depends on the time ''t'', α is the numerical damping parameter and ''u,  [[Image:draft_Samper_432909089-monograph-image403.png|12px]] ''and  [[Image:draft_Samper_432909089-monograph-image404.png|12px]] are the respective vectors of displacements, velocities and accelerations.
|-
+
| <math display="inline">M{\ddot{u}}_t+(1+\alpha )C{\dot{u}}_t+(1+\alpha )Ku_t=</math><math>(1+\alpha )F_t-\alpha F_t+\alpha C{\dot{u}}_{t-\Delta t}+</math><math>\alpha Ku_{t-\Delta t}</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. A.1]  
+
|}
+
where ''M'' is the mass matrix, ''C'' is the viscous damping matrix, ''K'' is the stiffness matrix, ''F<sub>t</sub> ''is the vector of forces that depends on the time ''t'', α is the numerical damping parameter and ''u,  <math display="inline">\dot{u}</math> ''and  <math display="inline">\ddot{u}</math> are the respective vectors of displacements, velocities and accelerations.
+
  
 
Equation A.1 can be expressed as:
 
Equation A.1 can be expressed as:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image405.png|222px]] [Eq. A.2]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">M{\ddot{q}}_{n+1}+(1+\alpha )F_{n+1}^q-\alpha F_n^q=</math><math>F_{n+\alpha }^t</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. A.2]  
+
|}
+
 
that represents the equilibrium equation for the instant ''t<sub>n+1</sub>'' expressed in generalized coordinates ''q''.
 
that represents the equilibrium equation for the instant ''t<sub>n+1</sub>'' expressed in generalized coordinates ''q''.
  
 
The following formulas are applied:
 
The following formulas are applied:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image406.png|366px]] [Eqs. A.3]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">\begin{array}{c}
+
{\ddot{q}}_{n+1}=\frac{1}{\gamma \Delta t}({\dot{q}}_{n+1}-{\dot{q}}_n)-\frac{1-\gamma }{\gamma }{\ddot{q}}_n\\
+
{\dot{q}}_{n+1}=\frac{\gamma }{\beta \Delta t}q_{n+1}-\Delta t\left(\frac{\gamma }{2\beta }-1\right){\ddot{q}}_n-\left(\frac{\gamma }{\beta }-1\right){\dot{q}}_n-\frac{\gamma }{\beta \Delta t}q_n\\
+
t_{n+\alpha }=t_{n+1}+\alpha (t_{n+1}-t_n)
+
\end{array}</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eqs. A.3]  
+
|}
+
 
where ''q'' is the vector of generalized coordinates, ''Δt'' is the time-step size and ''F<sup>q</sup>'' is part of the vector of forces only depending on the generalized coordinates and their derivatives.
 
where ''q'' is the vector of generalized coordinates, ''Δt'' is the time-step size and ''F<sup>q</sup>'' is part of the vector of forces only depending on the generalized coordinates and their derivatives.
  
Line 8,938: Line 8,773:
  
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
  <math display="inline">\beta =\frac{1}{4}{\left(1-\alpha \right)}^2;\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\gamma =</math><math>\frac{1}{2}-\alpha ;\mbox{​}\mbox{​}\mbox{​}\mbox{​}\mbox{​}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\alpha \in [-</math><math>\frac{1}{3},\mbox{ }0]</math> </div>
+
  [[Image:draft_Samper_432909089-monograph-image407.png|300px]] </div>
  
 
In the case of α=0, the method reduces to the Newmark`s constant or average acceleration method (trapezoid rule). Using α=0 offers the highest accuracy of the available methods, but may permit excessive vibrations in the higher frequency modes, i.e., those modes with periods of the same order as or less than the time-step size. For more negative values of alpha, the higher frequency modes are more severely damped. This is not physical damping, since it decreases as smaller time-steps are used. However, it is often necessary to use a negative value of alpha to encourage a nonlinear solution to converge. For best results, it is recommended the use of the smallest time-step practical, and select α as close to zero as possible, and trying with different values of α and time-step size to be sure that the solution is not too dependent upon these parameters [Computers & Structures, 2007]. For α=-1/3, maximum numerical dissipation is reached, aspect that can be dangerous because much high frequency content could be severely damped.
 
In the case of α=0, the method reduces to the Newmark`s constant or average acceleration method (trapezoid rule). Using α=0 offers the highest accuracy of the available methods, but may permit excessive vibrations in the higher frequency modes, i.e., those modes with periods of the same order as or less than the time-step size. For more negative values of alpha, the higher frequency modes are more severely damped. This is not physical damping, since it decreases as smaller time-steps are used. However, it is often necessary to use a negative value of alpha to encourage a nonlinear solution to converge. For best results, it is recommended the use of the smallest time-step practical, and select α as close to zero as possible, and trying with different values of α and time-step size to be sure that the solution is not too dependent upon these parameters [Computers & Structures, 2007]. For α=-1/3, maximum numerical dissipation is reached, aspect that can be dangerous because much high frequency content could be severely damped.
Line 8,950: Line 8,785:
 
{| style="width: 100%;border-collapse: collapse;"  
 
{| style="width: 100%;border-collapse: collapse;"  
 
|-
 
|-
|  style="text-align: center;vertical-align: top;width: 54%;"| [[Image:draft_Samper_432909089-image408-c.png|306px]]
+
|  style="text-align: center;vertical-align: top;width: 53%;"|[[Image:draft_Samper_432909089-monograph-image408-c.png|306px]]
  
 
'''Fig. A.1 '''Examples of Structures with Lumped Nonlinear Elements [Wilson, 2002]
 
'''Fig. A.1 '''Examples of Structures with Lumped Nonlinear Elements [Wilson, 2002]
Line 8,956: Line 8,791:
  
 
In this methodology, the dynamic equilibrium equations of a linear elastic structure with predefined nonlinear elements subjected to an arbitrary load can be written as:
 
In this methodology, the dynamic equilibrium equations of a linear elastic structure with predefined nonlinear elements subjected to an arbitrary load can be written as:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image409.png|252px]] [Eq. A.4]
|
+
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">M\ddot{u}(t)+C\dot{u}(t)+K_Lu(t)+r_N(t)=r(t)</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. A.4]  
+
|}
+
 
|}
 
|}
  
  
|}
+
where ''M'' is the diagonal mass matrix, ''C'' is the proportional damping matrix, ''K<sub>L</sub>'' is the stiffness matrix for the linear elastic elements (all elements except those with nonlinear behaviour), ''r<sub>N</sub>'' is the vector of forces from the nonlinear degrees-of-freedom in the link elements, ''r'' is the vector of applied loads and ''u,  [[Image:draft_Samper_432909089-monograph-image410.png|12px]] ''and  [[Image:draft_Samper_432909089-monograph-image411.png|12px]] are the respective displacement, velocity and acceleration vectors.
  
 +
If the computer model is unstable without the nonlinear elements, it is possible to add ''effective elastic elements'' (at the location of the nonlinear elements) of arbitrary stiffness. If these effective forces, ''K<sub>e</sub>u(t)'', are added to both sides of Eq. A.1, the exact equilibrium equations can be written as:
  
where ''M'' is the diagonal mass matrix, ''C'' is the proportional damping matrix, ''K<sub>L</sub>'' is the stiffness matrix for the linear elastic elements (all elements except those with nonlinear behaviour), ''r<sub>N</sub>'' is the vector of forces from the nonlinear degrees-of-freedom in the link elements, ''r'' is the vector of applied loads and ''u,  <math display="inline">\dot{u}</math> ''and  <math display="inline">\ddot{u}</math> are the respective displacement, velocity and acceleration vectors.
+
[[Image:draft_Samper_432909089-monograph-image412.png|264px]] [Eq. A.5]
  
If the computer model is unstable without the nonlinear elements, it is possible to add ''effective elastic elements'' (at the location of the nonlinear elements) of arbitrary stiffness. If these effective forces, ''K<sub>e</sub>u(t)'', are added to both sides of Eq. A.1, the exact equilibrium equations can be written as:
 
{| class='formulaSCP' style='width: 100%;'
 
|-
 
|
 
{| style='margin:auto;width: 100%; text-align:center;'
 
|-
 
| <math display="inline">M\ddot{u}+C\dot{u}+(K_L+K_e)u=r-r_N+K_eu</math>
 
|}
 
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. A.5]
 
|}
 
 
where ''K<sub>e</sub>'' is the effective stiffness of arbitrary value. Therefore, the ''exact'' dynamic equilibrium equations for the nonlinear model can be written as:
 
where ''K<sub>e</sub>'' is the effective stiffness of arbitrary value. Therefore, the ''exact'' dynamic equilibrium equations for the nonlinear model can be written as:
{| class='formulaSCP' style='width: 100%;'
+
 
|-
+
[[Image:draft_Samper_432909089-monograph-image413.png|126px]] [Eq. A.6]
|
+
 
{| style='margin:auto;width: 100%; text-align:center;'
+
|-
+
| <math display="inline">M\ddot{u}+C\dot{u}+\overline{K}u=\overline{R}</math>
+
|}
+
| style='width: 5px;text-align: right;white-space: nowrap;' | [Eq. A.6]  
+
|}
+
 
where
 
where
  
+
 
 
{|
 
{|
 
|-
 
|-
| <math display="inline">\overline{K}=K+K_e\mbox{ }\mbox{and}\mbox{ }\overline{R}=</math><math>r-r_N+K_eu</math>
+
| [[Image:draft_Samper_432909089-monograph-image414.png|222px]]
| <math display="inline">\overline{K}</math>
+
| [[Image:draft_Samper_432909089-monograph-image415.png|center|18px]]
 
|}
 
|}
is known, and the effective external load  <math display="inline">\overline{R}</math> must be evaluated by iteration. If a good estimate of the effective elastic stiffness can be made, the rate of convergence may be accelerated because the unknown term  <math display="inline">-r_N+K_eu</math> will be small [Wilson, 2002].
+
is known, and the effective external load  [[Image:draft_Samper_432909089-monograph-image416.png|12px]] must be evaluated by iteration. If a good estimate of the effective elastic stiffness can be made, the rate of convergence may be accelerated because the unknown term  [[Image:draft_Samper_432909089-monograph-image417.png|72px]] will be small [Wilson, 2002].
  
 
The solution for this integration methodology involves the application of the stiffness and mass orthogonal Load Dependent Ritz Vectors of the elastic structural system to reduce the size of the nonlinear system to be solved. For that reason, this methodology is also called ''Nonlinear Modal Time History Analysis, ''because in some sense, this is a hybrid between time history analysis and modal analysis''.'' The forces in the nonlinear elements are calculated by iteration at the end of each time or load step, and the uncoupled modal equations are solved exactly for each time increment. By iteration within each time-step, equilibrium, compatibility and all element force-deformation equations within each nonlinear element are identically satisfied.
 
The solution for this integration methodology involves the application of the stiffness and mass orthogonal Load Dependent Ritz Vectors of the elastic structural system to reduce the size of the nonlinear system to be solved. For that reason, this methodology is also called ''Nonlinear Modal Time History Analysis, ''because in some sense, this is a hybrid between time history analysis and modal analysis''.'' The forces in the nonlinear elements are calculated by iteration at the end of each time or load step, and the uncoupled modal equations are solved exactly for each time increment. By iteration within each time-step, equilibrium, compatibility and all element force-deformation equations within each nonlinear element are identically satisfied.
Line 9,213: Line 9,026:
 
For analysis of large structural systems, it is not possible to store all information within high-speed storage. If data needs to be obtained from low-speed disk storage, the effective speed of a computer can be reduced significantly. For that reason, it is recommended the use of computer codes with the transfer data to and from disk storage conducted in large blocks to minimize disk access time, also called ''paging operating systems''.
 
For analysis of large structural systems, it is not possible to store all information within high-speed storage. If data needs to be obtained from low-speed disk storage, the effective speed of a computer can be reduced significantly. For that reason, it is recommended the use of computer codes with the transfer data to and from disk storage conducted in large blocks to minimize disk access time, also called ''paging operating systems''.
  
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+
==References==
<big>''' '''References</big></div>
+
  
 
Abdel – Ghaffar, A.M. (1991), “Cable-Stayed Bridges under Seismic Action”, ''Proceedings of the Seminar Cable – Stayed Bridges: Recent Development and their Future'', Yokohama, Japan.
 
Abdel – Ghaffar, A.M. (1991), “Cable-Stayed Bridges under Seismic Action”, ''Proceedings of the Seminar Cable – Stayed Bridges: Recent Development and their Future'', Yokohama, Japan.

Latest revision as of 15:05, 29 May 2018


Acknowledgments

The authors wish to thank to the Department of Geotechnical Engineering and Geosciences and the Department of Construction Engineering at Technical University of Catalonia for their help and support during the doctorate years of Mr. Galo Valdebenito. This work is inspired in the basic result of that investigation. Likewise, thank to the Faculty of Engineering Sciences and the Department of Research and Development (DID) at Universidad Austral de Chile for their help and support in this publication.

Preface

Earthquakes can be really destructive. There is no doubt. Recent seismic events have demonstrated the important effects on structures, and especially on bridges. In this sense, cable-stayed bridges are not an exception, although their seismic performance during recent events has been satisfactory. Their inherent condition as part of life-lines makes the seismic design and retrofitting of such structures be seriously considered.

Traditionally, seismic protection strategies have been based on provide enough strength and ductility. In the case of buildings or bridges with adequate supports and degrees of redundancy, that approach can be satisfactory, however, in the case of structures with few degrees of redundancy, or questionable ductility, that scheme could be inadequate, and worse, dangerous, as usually happens with cable-stayed bridges. All traditional modern strategies to design seismic structures are focused on the adequate comprehension of the mechanisms involved, in which ductility can be provided by some elements specially designed for these purposes. In these sense, strategies such as performed-based design or displacement-based design consider that well-designed structures need to dissipate enough energy by hysteresis in order to obtain economic and safe structures.

The incorporation of additional energy dissipation and isolation devices, by means of passive, active, semi-active and hybrid strategies, constitutes without doubt efficient schemes to protect structures controlling or avoiding damage, in which the energy dissipation is guaranteed through the action of external elements specially designed for those purposes. By this way, now it is possible to provide enough strength and energy dissipation capacity at the same time, avoiding damage on important structural elements, with the subsequent guaranty of the functionality, very important on life-lines, even during strong ground motions.

The present work constitutes an approach to the seismic protection of cable-stayed bridges including the incorporation of fluid viscous dampers as additional energy dissipation devices. The idea of the authors is to provide an up-to-date vision of the problem taking into account that long-period structures such as those proposed here, need to be adequately protected against strong motions, and considering that, because of their importance, an elastic behaviour is desirable. Chapter 1 describes the object to study in general terms. Chapter 2 constitutes a state-of-the-art review regarding the seismic behaviour and performance of fluid viscous dampers as external energy dissipation devices. The mechanical behaviour and technological aspects are now introduced with an energetic point of view, in which some practical applications are exposed and discussed. Chapter 3 describes the seismic response of cable-stayed bridges without external seismic protection, considering a parametric analysis in order to study the effects of the stay cable layout, stay spacing and deck level. A complete modal characterization is exposed, followed by a response spectrum analysis for comparative purposes. The effect of variations of the stay forces is analyzed, and finally, a nonlinear step-by-step analysis is performed for the critical structures, considering the velocity dependence of such bridges and the effects of far-fault and near-fault ground motions. The last Chapter exposes the seismic analysis of the selected structures including the incorporation of fluid viscous dampers as passive additional energy dissipation devices. Because of the inherent nonlinear behaviour of the structures and external devices, a mandatory nonlinear direct integration time-history analysis is performed for all the cases, in which parametric analyses are carried out in order to select the best damper parameters, and for the case of both far-fault and near-fault ground motions. In this part, comparative results are exposed with the aim to propose some practical recommendations.

Galo E. Valdebenito
Ángel C. Aparicio
Llavaneras (Barcelona), October 2009.


Chapter 1. Introduction

1.1 Cable-Stayed Bridges and Seismic Protection

Bridges are without a doubt attractive civil engineering works from a structural point of view. But they are not only exciting as a structure: the project, construction, maintenance, operation as well as functional, aesthetic, economic and political aspects make them extremely interesting constituting a great social event [Maldonado et al, 1998].

Suspension bridges are very interesting and useful structures because they can be used for long-spans, solving many practical problems for which is necessary to cross large distances without intermediate supports. These kinds of structures are a challenge from all points of view, due to the constant increase of the main span length demand, constituting most of the times a human whim or that competitive and insatiable desire to break goals at any price. Cable supported bridges can be divided into suspended and cable-stayed bridges, as can be appreciated in Fig. 1.1.

From a structural point of view, both types of bridges are completely different, since contrary to suspended bridges, in cable-stayed bridges the cables are prestressed. Keeping in mind functional and economical aspects, suspension bridges permit longer spans with more economical results than cable-stayed bridges [Podolny and Scalzi, 1986]. Actually, the longer main spans in cable-stayed bridges reach 900 m, although recent investigations show the feasibility and possibility of building bridges of this kind with main spans exceeding 1000 m. These studies are based on the current high standard technologies and the lightness of superstructures that use orthotropic slabs [Aschrafi, 1998; Nagai et al, 1998].

Draft Samper 432909089-monograph-image1-c.png
(a) Cable-Stayed bridge
Draft Samper 432909089-monograph-image2-c.png
(b) Suspended bridge
Fig. 1.1 Cable-Supported Bridges

In spite of the relative simplicity of bridges, the recent earthquake events of San Fernando (1971), Loma Prieta (1989), Northridge (1994), Kobe (1995) and Taiwan (1999) have shown that these systems are very vulnerable, mainly those of reinforced concrete. For that reason, is a high-priority to improve the comprehension of this phenomenon, learning from the recent earthquake lessons [Priestley et al, 1996]. These structural systems expose a few degrees of redundancy, and the collapse mechanisms should be known in detail to reach an appropriate performance. Some aspects that should be considered are: degree of redundancy of the system, soil-structure interaction, spatial variability effects, near source effects, geological faults and geotechnical aspects, bridge length effects, vertical component of motion and damping [Valdebenito, 2005]. All these aspects are explained in the references of Ghasemi (1999), Kawashima (2000); Cheung et al (2000) and Calvi (2004).

The structural analysis of a bridge depends undoubtedly on the structural modelling. Therefore, a well-done modelling is reflected in the degree of accuracy of the results. The vertiginous development of high-performance computers permits to solve more complex and large structures, testing a lot of conditions in a relatively short time. Thus, computing time will depend on the modelling used and the required accuracy for the results. Because of almost all the seismic isolators or energy dissipators experience non-linear behaviour, consideration of non-linear aspects in the analysis of the bridge – energy dissipation system is advisable. In spite of the current computer capacity and better non-linear structural analysis software, it is clear that the time and knowledge level of the designer are two serious limitations of the extensive application of these methodologies [Jara and Casas, 2002]. In fact, sometimes is preferable the use of simplified methods that show sufficiently accurate results in short time. In the case of long-span cable-stayed bridges, the problem is more complex, maybe due to the high non-linear behaviour of those structures, and hence, non-linear analysis becomes an indispensable condition, leaving aside the classical response spectrum analysis or the equivalent static analysis. Thus, a relatively complex structure can be solved by the iterative definition of the stiffness and equivalent damping.

Traditional seismic control strategies are based on the modification of stiffness, mass or geometric properties of the structure, reducing inertial forces and displacements caused by an earthquake. Thus, in the current design is necessary to permit controlled structural damage by the ductility provided, with the aim of avoiding too conservative designs and expensive costs. In other words, in the current philosophy, a structure with energy dissipation capacity is required, more than a resistant structure against all events. Although it is certain that traditional strategies for the seismic protection of bridges have progressed in the last years, for appropriate bridge strength and to assure a satisfactory behaviour for different intensity levels, development of special vibration control devices has given origin to a new path in seismic engineering. In general terms, instead of provide more strength, is more attractive to reduce internal forces and displacements through special isolation systems or energy dissipation devices. This energy distribution means that the seismic energy proceeding from the subsoil is distributed to different structural components and thus significant energy accumulation is avoided.

Amongst the existent control systems on bridges, passive strategies are well accepted because of their low comparative price, simple installation and maintenance as well as their great reliability and better theoretical and technological development [Jara and Casas, 2002]. Active, semi-active and hybrid systems seems to be an excellent strategy for the seismic control of structures, however, a lack of regulations and uncertainty regarding their real performance under strong ground motions are important limitations for their application. Without a doubt, there is a very promising future, mainly with semi-active and hybrid systems because of their incomparable advantages, although now their use is very limited, not been properly tested on real structures with real earthquakes. Thus, the general approach reducing the seismic demand of structures, more than trying to increase their strength or deformation capacity with appropriate criteria, is without a doubt an advantageous seismic protection system. These new seismic control strategies are conceived for the reduction of the seismic demand, and the appropriate application of this approach leads to systems that behave elastically during great earthquakes, on the contrary of a traditional design, where high energy dissipation capacity by controlled damage is needed. Passive control systems convert the kinetic energy of the system into heat, transferring it among different vibration modes. They do not require additional external energy for their operation, constituting their main advantage. In general terms, these systems operate elastically during great earthquakes, permitting structural functionality conditions after the event. Because of their low cost, high efficiency and low maintenance, they are additional seismic protection systems widely used in the world. Passive control systems can be classified as follows (Table 1.1):

Table 1.1 Passive Seismic Control Systems [Adapted from Valdebenito and Aparicio, 2005]
Base Isolation Energy Dissipators Seismic Connectors Resonant Dampers
1. Rubber Bearings (RB) 1. Metallic Yield Dampers (MD) 1. Shock transmission Units TU) 1. Tuned Mass dampers (TMD)
2. High Damping Rubber Bearings (HDR) 2. Friction Dampers (FD) 2. Displacement Control Devices (DCD) 2. Tuned Liquid Dampers (TLD)
3. Lead Rubber Bearings (LRB) 3. Viscoelastic Dampers (VE) 3. Rigid Connection Devices (RCD)
4. Rubber Bearings with Additional Energy Dissipation 4. Fluid Viscous Dampers (VF)
5. Sliding Bearings (SB) 5. Lead Extrusion Dampers (LED)
6. Shape Memory Alloy (SMA)


Base isolation and dissipation result in decreasing the energy applied to the system and the transformation from energy to heat. This is also designated as energy approach, which especially takes into account the energy character of the seismic event. In the seismic isolation, the structure is separated from the subsoil, automatically limiting the energy that affects the structure, which is considerably reduced. As a result, the natural period is increased, which causes a considerable reduction of the structural acceleration during seismic events. Depending on the installed type of isolator, they do not only guarantee the vertical load transmission but also the restoring capacity during and after a seismic event.

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Fig. 1.2 (a) Energy Dissipation of a Traditional Bridge, (b) Energy Dissipation of a Seismic Isolated Bridge [Adapted from Jara and Casas, 2002]


Fig. 1.2 (a) exposes a traditionally designed bridge, in which the seismic energy is dissipated by damage at the plastic zones (plastic hinges). For the above-mentioned, an adequate ductility to dissipate the earthquake energy is required. Fig. 1.2 (b) shows the case of an isolated bridge with rubber bearings. In this situation, inertial forces on the pylon are reduced, and the inelastic energy dissipation during severe earthquakes is achieved by hysteretic deformation of the supports [Jara and Casas, 2002].


Base isolation systems and seismic connectors applied to bridges have been properly tested and used for more than 20 years, and there is a lot of documentation and experience regarding to this. In relation to energy dissipation systems, the use of fluid viscous dampers can be the future for the application to large structures such as long-span cable-stayed bridges, mainly due to their high capacity, robustness, and good results of recent investigations.

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Fig. 1.3 Minimized Seismic Energy Penetration by Seismic Isolation and Energy Dissipation

It seems that additional damping devices are clearly adequate considering the current high standards and technology, and in conjunction with isolation, produce the best possible seismic protection, mainly if the structural system is not velocity-dependent. On one hand, isolation reduces the spectral acceleration (demand), and on the other hand, fluid viscous dampers dissipate input energy avoiding structural damage (Fig. 1.3). A good state-of-the-art in relation to supplemental energy dissipation can be found in the work of Soong and Spencer (2002).


In the case of cable-stayed bridges, their seismic behaviour has been, in general terms, very satisfactory, maybe due to their great flexibility. In spite of the above-mentioned, comprehension of their behaviour is very complex being appropriate and promising to consider special systems of additional seismic protection. On those structures, these additional systems have been applied basically to control vibrations on cables due to the effect of the wind and rain (rain - wind vibration), to solve aerodynamic problems on unstable and complex structures and for the seismic retrofit of existing bridges. Now, application of these devices for the control of seismic actions begins to be used with more frequency; not only on the cables to mitigate the cable-deck interaction [Macdonald and Georgakis, 2002] but also to isolate the superstructure, as can be appreciated in the recently inaugurated Rion-Antirion Bridge (Fig. 1.4), in the Gulf of Corinthian, Greece [Infanti et al, 2004].

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Fig. 1.4 Rion-Antirion Bridge, Greece [from [1] www.aecom.com]]

Design of almost all cable-stayed bridges located at moderate-to-high seismicity zones is more complex than design of conventional bridges. Bridge design regulations and modern previsions have been developed in general terms and for standard bridges, in order to provide safe and economical structures. As general design philosophy, it is accepted the important request of having structural damage but permitting emergency communications for a not frequent severe earthquake. For the new cable-stayed bridges, code previsions cannot be applicable, being necessary the urgent improvement of regulations and general recommendations for the seismic design of these bridges, based on numeric, experimental or full-scale testing investigations. Also, the lack of information about the real performance of these bridges during strong earthquakes increases the uncertainty in terms of an appropriate design [Abdel-Ghaffar, 1991]. In fact, according to Eurocode 8 Part 2 [CEN, 1998b], cable-stayed bridges are classified as special bridges, aspect that implies that these regulations need to be considered only as general recommendations. At the moment, existent regulations with regard to passive systems are limited to seismic isolation and energy dissipation devices, without the incorporation of hybrid, active or semi-active systems. Design specifications for bridges with LRB systems, published by the New Zealand Ministry of Works and Development in 1983, were the first regulations about bridges with special seismic protection based on isolation and energy dissipation systems. Later, in the 90s, official recommendations for the first time in USA [1991, 2000], Italy (1991), Japan (1996), and Europe through Eurocode 8 [CEN, 1998a, 1998b] were published.

1.2 Historical Background

The early stayed bridges used chains or bars for the stays. The advent of various types of structural cables, with their inherent high carrying capacity and ease of installation, led engineers and contractors to replace the chains and bars [Podolny and Scalzi, 1986].

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Fig. 1.5 Löscher-type Timber Bridge [Courtesy of the British Constructional Steelwork Association, Ltd]


The concept of a bridge partially suspended only by inclined stays is credited to C.J. Löscher, a carpenter from Fribourg, Switzerland who built a completely timber bridge including stays and tower in 1784, with a span of 32 m. (Fig. 1.5).



Cable-stayed bridges might have become a conventional form of construction had it not been for the bad publicity that followed the collapses of two bridges: the 79 m pedestrian bridge crossing the Tweed River near Dryburgh-Abbey (England) in 1818; and the 78 m long bridge over the Saale River near Nienburg, Germany, in 1824 [Podolny and Scalzi, 1986]. The famous French engineer, Navier, discussed these failures with his colleagues, and his adverse comments are assumed to have condemned the stay-bridge concept to relative obscurity. Whatever the reason, engineers turned to the suspension bridge, which was also emerging, as the preferred type of bridge for river crossings.

The principle of using stays to support a bridge superstructure returned with the works of John Roebling. The Niagara Falls Bridge (Fig. 1.6), the Old St. Clair Bridge in Pittsburgh (USA), the Cincinnati Bridge over the Ohio River (USA) and the Brooklyn Bridge (Fig. 1.7) in New York (USA) are good examples.

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Fig. 1.6 Niagara Falls Bridge [Courtesy of the Niagara Falls Bridge Commission]

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Fig. 1.7 Brooklyn Bridge [from [2] www.elclubdigital.com]]


It should be noted that the stays used by Roebling in his suspension bridges were used as an addition to the classical suspension bridge with the main catenary cable and its suspenders. During Roebling’s time the suspension bridge concept was suffering with failures resulting from wind forces. He knew that by incorporating the diagonal stays he could minimize the susceptibility of his structures to adverse wind loading. However, it is not clear whether he used the two suspension systems compositely.

Towards the end of the 19th century, the success of these hybrid structures – part suspension, part stayed – resulted in a slowing down of the use of structures supported exclusively by inclined rods. However, it was not until 1899 that the French engineer A. Gislard further advanced the development of stayed bridges by the introduction of a new system of hangers, at the same time economic and sufficiently rigid [Walter, 1999]. The system was characterized by the addition of cables intended to take up the horizontal components of the forces set up by the stays. This arrangement cancels out any compressive forces in the deck and thus avoids deck instability.

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Fig. 1.8 The Bridge over the Donzère Canal, France [photo: J. Kerisel]

Surprisingly, the first “modern” cable-stayed bridges were built in concrete by Eduardo Torroja in the 1920s (Tampul aqueduct) and by Albert Caquot in 1952 (Donzère Canal Bridge, Fig. 1.8).

However, the real development came from Germany with papers published by Franz Dischinger and with the famous series of steel bridges crossing the river Rhine, as the Oberkassel Bridge, in Düsseldorf, Germany (Fig. 1.9).

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Fig. 1.9 Oberkassel Bridge, Düsseldorf, Germany

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Fig. 1.10 Maracaibo Bridge, Venezuela [from en.structurae.de]


The increasing popularity of this new type of structure with German engineers slowly extended to other countries. Thus, the Italian architect R. Morandi designed several cable-stayed bridges in reinforced and prestressed concrete. His most outstanding work is the bridge on Lake Maracaibo, Venezuela, built in 1962 (Fig. 1.10).

The international development of this bridge type began in the 1970s, but a very big step forward took place in the 1990s, when cable-stayed bridges entered the domain of very long spans which was previously reserved for suspension bridges. As examples, the Barrios de Luna Bridge – also called the Fernandez Casado Bridge – in Spain (430 m, 1983, Fig. 1.11); the Yang Pu Bridge in Shangai, China (602 m, 1993, Fig. 1.12); the Normandie Bridge in Le Havre, France (856 m, 1994, Fig. 1.13) and the Tatara Bridge in Japan (890 m, 1998, Fig. 1.14). It is extremely interesting to analyse the progress in the world record for cable-stayed bridges, since it provides keys to understand the evolution of their design (Fig. 1.15).

The recently inaugurated Millau Bridge in the Tarn Valley, France, is one of the world’s famous multi-span cable-stayed bridge, with 342 m main span length and 343 m height for the highest pylon. This also called “bridge over the clouds” is one of the more interesting French engineering works at the present (Fig. 1.16). In the same way, the new Sutong Bridge, in Nantong, China (inaugurated in 2008), is considered the longest cable-stayed bridge of the world, with a main span length of 1088 m, and surpassing the Japanese record reached with the Tatara Bridge (Fig. 1.17).

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Fig. 1.11 Barrios de Luna Bridge, Spain [from en.structurae.de]

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Fig. 1.12 Yang Pu Bridge, China [photo: M. Virlogeux]

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Fig. 1.13 Normandie Bridge, France [from fr.structurae.de]

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Fig. 1.14 Tatara Bridge, Japan [from [3] www.answers.com]]



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Fig. 1.15 Evolution of Record Spans for Cable-Stayed Bridges [Virlogeux, 1999]
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Fig. 1.16 Millau Bridge, France

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Fig. 1.17 Sutong Bridge, Nantong, China


Although the use of energy dissipation devices began as an attempt to control the cable vibration on cable-stayed and suspension bridges, very common on those structures due to the inherent low damping of the cable system, the inclusion of additional seismic protection, with the introduction of passive and active energy dissipation devices, has just begun. In this sense, the use of fluid viscous dampers in the recently inaugurated Rion-Antirion Bridge (Greece) is an exceptional opportunity to test in situ, with a real structure in a high-seismicity zone, those devices (Fig. 1.18). The deck of this multi-span cable-stayed bridge is continuous and fully suspended from four pylons (total length of 2252 meters). Its approach viaducts comprise 228m of concrete deck on the Antirion side and 986m of steel composite deck on the Rion side. The Main Bridge seismic protection system comprises fuse restraints and viscous dampers of dimensions heretofore never built. The same act in parallel, connecting the deck to the pylons. The restrainers of the Rion Antirion Bridge were designed as a rigid link intended to withstand high wind loads up to a pre-determined force. Under the reaction of the design earthquake, fuse restrainers will fail and leave the dampers free to dissipate the earthquake-induced energy acting upon the structure. The Approach Viaducts were seismically isolated utilizing elastomeric isolators and viscous dampers [Infanti et al, 2004].

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Fig. 1.18 Rion-Antirion Viscous Dampers [Courtesy of FIP Industriale, Italy]

Another interesting application of passive/active devices is to retrofit existent bridges. After important earthquake events, or adjusting the seismic behaviour of existent structures in accordance with new codes and specifications, many bridges need to be retrofitted. For cable-stayed bridges, it seems to be impractical to reinforce structural members, and it will be more simple and efficient to conduct the bridge retrofit by using isolation systems if the system is proved to be feasible [Lai et al, 2004].


The recent application of active (i.e. hybrid, semi-active) systems on cable-stayed bridges is very limited. Actually, a benchmark structural problem for cable-stayed bridges was defined in order to provide a test bed for the development of strategies for the seismic control of those structures. The problem is based on the new cable-stayed bridge that spans the Mississippi River: the Bill Emerson Memorial Bridge, in Cape Girardeau, Missouri, USA. [Dyke et al, 2002].

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Fig. 1.19 Dongting Lake Bridge, China

Real applications of active systems to cable-stayed bridges are limited only to aerodynamic structural control of the stays. In this sense, the recent application of Magnetorheological Dampers on the Dongting Lake Bridge over the Yangtze River in the southern central China (Fig. 1.19) is the first known application of those devices to control the rain-wind vibration. The installation finished in June 2002 [Chen et al, 2003].


Chapter 2. Fluid Viscous Damping Technology

2.1 General Overview

Structures situated on seismic areas must be designed to resist earthquake ground motions. A fundamental rule regarding the seismic design of structures, express that higher damping implies lower induced seismic forces. For conventional constructions, the induced earthquake energy is dissipated by the structural components of the system designed to resist gravity loads. It is well known that damping level during the elastic seismic behaviour is generally very low, which implies not much dissipated energy. During strong ground motion, energy dissipation can be reached through damage of important structural elements, and considering only the resulting response forces within the structure due to an earthquake leads to massive structural dimensions, stiff structures with enormous local energy accumulation and plastic hinges. This strengthening method combined with usual bearing arrangements permits plastic deformations by way of leading to yield stress and cracks. In this sense, structural repair after an important seismic event is generally very expensive, the structure is set temporarily out of service and sometimes a lot of damaged structures must be demolished [Alvarez, 2004].

General concepts for appropriate protection of structures against earthquakes do not exist, as every structure is quite unique and requires individual considerations. Earthquakes are often interpreted in terms of deformations and acting forces induced upon the structure. As a consequence, there is a tendency to think only about increasing the strength of the structure. Actually, forces and displacements are nothing but a mere manifestation of seismic attacks and do not in fact represent their very essence. An earthquake is actually an energy phenomenon and the forces causing stresses in the structure are the final effect of that event.

In recent years, other strategies have been developed to reduce the seismic response of the structures using additional passive devices. A passive control system may be defined as a system which does not require an external power source for operation and utilizes the motion of the structure to develop the control forces, as a function of the response of the structure at the location of the passive control system, according to Fig. 2.1.

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Fig. 2.1 Block Diagram of Passive Control System [Symans and Constantinou, 1999]

A passive control system may be used to increase the energy dissipation capacity of a structure through localized discrete energy dissipation devices located either within a seismic isolation system or over the height of the structure. Such systems may be referred to as supplemental energy dissipation systems [Symans and Constantinou, 1999]


Passive supplemental damping strategies, including base isolation systems, viscoelastic dampers and tuned mass dampers are well understood and are widely accepted by the engineering community as a means for mitigating the effects of dynamic loading on structures. In this sense, energy dissipation systems can be considered as an important passive strategy in which the objective of these devices is to absorb a significant amount of the seismic input energy, thus reducing the demand on the structure by means of the relative motion within the passive devices which, in turn, dissipate energy. In general terms, these devices are not part of the structural system that resists gravity loads, constituting an external system that can be easily replaced after a strong earthquake. Of course, in this case the structural functionality is not affected as well as the stability of the structure, with a low replacement cost of such devices compared with repair or service interruption costs.

Additional damping devices dissipate energy by means of yielding, friction, Viscoelastic action or fluid flow through orifices [Soong and Dargush, 1997; Constantinou, 2003]. In this sense, fluid viscous dampers constitutes one of the well accepted energy dissipation systems by the scientific and engineering community, being considered as additional damping system in this work, as was previously explained and justified. These systems are capable of dissipate an important amount of energy during strong ground motions as well as to control long period displacements. These dampers are basically comprised of a cylinder filled with silicone fluid (oil or paste) and a piston that divides it into two chambers and is free to move in both directions. In case of sudden movements, due to earthquakes or other dynamic actions like braking, wind, etc., lamination of silicone fluid occur through an appropriate hydraulic circuit and leads to energy dissipation. In case of slow displacements, due to structure thermal expansion, such flow is obstructed, so that during normal service the behaviour is substantially rigid, acting like a shock absorber. Because of those advantages, utilization of this technology permits to take full advantage of the strength of structural elements, because it is possible to maximize energy dissipation reaching the maximum level of force that the structure can sustain, without exceeding it. As a consequence, structural elements remain in the elastic field also during high intensity earthquakes.

Actually, manufacture of fluid viscous dampers permits to design such devices for a wide range of specific requirements of velocity and force, constituting a good choice for implementation on new and existing facilities. Those devices are properly tested at specific laboratories, especially when they are applied on important structures or they are required for special conditions. In this sense, manufacturers such as Alga s.P.a. (Milano, Italy); FIP Industriale s.P.a. (Selvazzano, Italy), Taylor Devices, Inc. (New York, USA), Maurer Söhne (München, Germany), Mageba (Bülach, Switzerland) or Nanjing Damper Technology Engineering Co. Ltd (Nanjing, China) design and manufacture a wide variety of such systems.

Today an increasing number of applications of energy dissipation devices on bridges for the control of seismic displacements and energy dissipation is taking place. The more common solution is, probably, the use of linear / non-linear viscous dampers, permitting an adequate control of the displacements avoiding an increase of the structural internal forces and the increase of stiffness for piers and abutments [Jerónimo and Guerreiro, 2002].

The new tendencies regarding the seismic analysis and design of fluid viscous dampers capture the frequency dependence of such devices [Singh et al, 2003]; the earthquake response of non-linear fluid viscous dampers [Peckan et al, 1999; Lin and Chopra, 2002]; the seismic performance and behaviour of these devices during near-field ground motion [Tan et al, 2005; Xu et al, 2007] and the performance-based design of viscous dampers [Kim et al, 2003; Li and Liang, 2007]. A state-of-the-art review can be found in the works of Lee and Taylor (2001) and Symans et al (2008).

2.2 Technological Aspects
2.2.1 Historical Background

As with many other types of engineered components, the requirements, needs and available funds from the military allowed rapid design evolution of fluid dampers to satisfy the needs of armed forces. Early fluid damping devices operated by viscous effects, where the operating medium was sheared by vanes or plates within the damper. Designs of this type were mere laboratory curiosities, since the maximum pressure available from shearing a fluid is limited by the onset of cavitation, which generally occurs at between 0.06 N/mm2 and 0.1 N/mm2, depending on the viscosity of the fluid. This operating pressure was so low that for any given output level, a viscous damper was much larger and more costly than other types [Taylor, 1996].

In the late 1800`s, applications for dampers arose in the field of artillery, where a high performance device was needed to attenuate the recoil of large cannons. After extensive experimentation, the French Army incorporated a unique (and “top-secret”) fluid damper into the design of their 75 mm gun. These first fluid damper designs used inertial flows, where oil was forced through small orifices at high speeds, in turn generating high damping forces. This allowed the damper to operate at relatively high operating pressures, in the 20 N/mm2 range. The output of those devices was not affected by viscosity changes of the fluid, but rather by the specific mass of the fluid, which changes only slightly with temperature. Thus, the technology of fluid inertial dampers became widespread within the armies and navies of most countries in the 1900 – 1945 period.

During the World War II, the emergence of radar and similar electronic systems required the development of specialized shock isolation techniques. During the Cold War period, the guided missile became the weapon of choice for the military, and the fluid inertial damper was again turned to by the military as the most cost effective way of protecting missiles against both conventional and nuclear weapon detonation. In these cases, transient shock from a miss near weapons detonation can contain free field velocities of 3 to 12 m/s, displacements of up to 2000 mm, and accelerations up to 1000 times gravity. For that reason, extremely high damping forces were needed to attenuate these transient pulses on large structures, and fluid inertial dampers became a preferred solution to these problems [Taylor, 1996].

With the end of the Cold War in the late 80`s, much of this fully developed defence technology became available for civilian applications. In this context, demonstration of the benefits of damping technology on structures could take place immediately, using existing dampers and the seismic test facilities available at U.S. university research centres. In this sense, application of fluid viscous dampers as part of seismic energy dissipation systems was experimentally and analytically studied, being validated by extensive testing on one-sixth to one-half scale building and bridge models in the period 1990 – 1993 at the Multidisciplinary Centre for Earthquake Engineering Research (MCEER), located on the campus of the State University of New York at Buffalo in USA. Thus, implementation of fluid viscous damping technology began relatively swiftly, with wind protection usage beginning in 1993, and seismic protection usage beginning in 1995 [Taylor and Duflot, 2002].

2.2.2 General Behaviour

Fluid viscous dampers operate on the principle of fluid flow through orifices. A stainless steel piston travels through chambers that are filled with silicone oil. The silicone oil is inert, non flammable, non toxic and stable for extremely long periods of time. The pressure difference between the two chambers cause silicone oil to flow through an orifice in the piston head and seismic energy is transformed into heat, which dissipates into the atmosphere. This associated temperature increase can be significant, particularly when the damper is subjected to long-duration or large-amplitude motions [Makris 1998; Makris et al, 1998]. Mechanisms are available to compensate for the temperature rise such that the influence on the damper behaviour is relatively minor [Soong and Dargush, 1997]. However, the increase in temperature may be of concern due to the potential for heat-induced damage to the damper seals. In this case, the temperature rise can be reduced by reducing the pressure differential across the piston head (e.g., by employing a damper with a larger piston head) [Makris et al, 1998]. Interestingly, although the damper is called a fluid viscous damper, the fluid typically has a relatively low viscosity (e.g., silicone oil with a kinematic viscosity on the order of 0.001 m2 /s at 20°C). The term fluid viscous damper is associated with the macroscopic behaviour of the damper which is essentially the same as that of an ideal linear or nonlinear viscous dashpot (i.e., the resisting force is directly related to the velocity). Generally, the fluid damper includes a double-ended piston rod (i.e., the piston rod projects outward from both sides of the piston head and exits the damper at both ends of the main cylinder). Such configurations are useful for minimizing the development of restoring forces (stiffness) due to fluid compression [Symans et al, 2008]. The force/velocity relationship for this kind of damper can be characterized as F = C.Vα where F is the output force, V the relative velocity across the damper; C is the damping coefficient and α is a constant exponent which is usually a value between 0.1 and 1.0 for earthquake protection, although at the present time some manufactures begin to apply dampers with very low damping coefficients, typically in the order of 0.02. Fluid viscous dampers can operate over temperature fluctuations ranging from –40°C to +70°C, and they have the unique ability to simultaneously reduce both stress and deflection within a structure subjected to a transient. This is because a fluid viscous damper varies its force only with velocity, which provides a response that is inherently out-of-phase with stresses due to flexing of the structure [Taylor and Duflot, 2002].

Fluid velocity is very high in the piston head so the upstream pressure energy converts almost entirely to kinetic energy. When the fluid subsequently expands into the full volume on the other side of the piston head it slows down and loses its kinetic energy into turbulence. There is very little pressure on the downstream side of the piston head compared with the full pressure on the upstream side of the piston head. This difference in pressures produces a large force that resists the motion of the damper. Viscous dampers, when correctly designed and fabricated, have zero leakage and require no accumulator or external liquid storage device to keep them full of fluid. They have nearly perfect sealing. In a correctly designed and fabricated viscous damper there is nothing to wear out or deteriorate over time so there is no practical limit on expected life. Warranty periods of 35 years are common [Lee and Taylor, 2001]. Fig. 2.2 shows a general view of a fluid viscous damper, and Fig. 2.3 shows fluid viscous dampers for a high-speed railway bridge in Spain.

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Fig 2.2 General view of a Fluid Viscous Damper [Courtesy of FIP Industriale s.P.a., Italy]

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Fig. 2.3 Fluid Viscous Dampers for De Las Piedras-High Speed Railway Bridge, Spain [Courtesy of Maurer Sönhe, Germany]


Fig. 2.4 exposes a schematic of a typical fluid viscous damper showing its elements, which are described next.

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Fig. 2.4 Typical Viscous Damper [Lee and Taylor, 2001]

The piston rod is machined from high alloy steel stainless steel and then highly polished. This high polish provides long life for the seal. The piston rod is designed for rigidity as it must resist compression buckling and must not flex under load, which would injure the seal.


The cylinder contains the working fluid and must withstand the pressure loading when the damper operates. Cylinders are usually made from seamless steel tubing and are sometimes machined from steel bars. Proof pressure is generally 1 - 5 times expected internal pressure for the maximum credible seismic event.

Structural applications require a fluid that is fire-resistant, non-toxic, thermally stable and that will not degrade with age. Under current OSHA (Occupational Safety & Health) guidelines this means a flash point of at least 200°F. Silicone fluid is often used as it has a flash point over 650°F and is cosmetically inert, completely non-toxic and one of the most thermally stable fluids available.

The seal must provide a service life of at least 35 years without replacement. As dampers often sit for long periods without use, the seal must not exhibit long-term sticking or allow fluid seepage. The dynamic seal is made from high-strength structural polymer to eliminate sticking or compression set during long periods of inactivity. Acceptable materials include Teflon®, stabilized nylon and members of the acetyl resin family. Dynamic seals made from structural polymers do not age, degrade or cold flow over time.

The piston head attaches to the piston rod and effectively divides the cylinder into two separate pressure chambers. This space between the outside diameter of the piston and in the inside diameter of the cylinder forms the orifice. Very often the piston head is made from a different material than the cylinder to provide thermal compensation. As the temperature rises the annulus between the piston head and the cylinder shrinks to compensate for thinning of the fluid.

The damper shown in Fig. 2.4 uses an internal accumulator to make up for the change in volume as the rod strokes. This accumulator is either a block of closed-cell plastic foam or a movable pressurized piston, or a rubber bladder. The accumulator also accommodates thermal expansion of the silicone fluid.

Viscous dampers add energy dissipation to a structure, which significantly reduces response to earthquakes, blasts, wind gusts and other shock and vibration inputs. A value of 30% of the critical damping ratio is a practical upper limit for combined viscous and structural damping. Around 25% of this is viscous damping and the remaining 5% is structural damping [Lee and Taylor, 2001]. This provides a 50% reduction in structural response compared with the same structure without viscous dampers. Note that the addition of viscous dampers does not change the period of the structure. This is because viscous damping is 90 degrees out of phase with the structural forces. Fig. 2.5 shows a typical plot of base shear against interstorey drift, taken from a laboratory test, according to Lee and Taylor (2001).

Draft Samper 432909089-monograph-image25.png

Fig. 2.5 Typical Plot of Base Shear Against Interstorey Drift [Lee and Taylor, 2001]

Draft Samper 432909089-monograph-image26.png

Fig. 2.6 Base Shear Against Interstorey Drift with Added Dampers [Lee and Taylor, 2001]


Note that the hysteresis loop is very flat and thin as there is only 5% damping. Figure 2.6 shows a plot of the same structure with the same input only this time with added viscous damping. Note that interstorey drift is 50% less and that the hysteresis curve is much fuller. In this case, 20% of added linear damping to the structure increased its earthquake resistance compared to that of the same structure without added damping. The area inside the hysteresis loop is the same as in Fig. 2.5. It is theoretically possible to provide enough viscous damping to completely prevent plastic hinging. This provides a totally linear structure. Economically, it is best to retain some plastic hinging as this results in the least overall cost. Viscous dampers still limit interstorey drift sufficiently to provide immediate occupancy after a worst-case event. They also limit and control the degree of plastic hinging and greatly reduce base shear and interstorey shear [Lee and Taylor, 2001]. Only as comparative purpose Table 2.1 shows equivalent damping coefficients for different structures and components. It is clear that an enormous amount of energy can be dissipated with the implementation of seismic dampers, reaching the largest values of dissipated energy. Of course, with those quantities, structural damping in the case of cable-stayed bridges may represent no more than 3% of the additional damping provided by the dampers, that is to say, a negligible amount.

Table 2.1 Comparison of Equivalent Damping Coefficients ξ of Different Structures and Components [Courtesy of Maurer Sönhe, Germany]
Structural Component Damping ratio (ξ)
Steel bridge 0.02
Concrete bridge 0.05
Elastomeric bearing 0.05 – 0.06
High damping rubber bearing 0.16 – 0.19
Lead rubber bearing and friction pendulum 0.30 – 0.40
Fluid viscous dampers Up to 0.60


In terms of the efficiency, the damping coefficient ξ relates to the efficiency η according to:

Draft Samper 432909089-monograph-image27.png [Eq. 2.1]

This ends up in a maximum efficiency η = 96% for fluid viscous dampers.


As a summary, the overall characteristics of fluid viscous dampers include:

  • During service conditions the device is not pre-tensioned and the fluid is under insignificant pressure
  • An extra-low damping exponent, such as those proposed from some manufacturers, provides maximum and well-defined force to a certain limit. No structural damages due to higher damping forces occur even in case the vibration frequency exceeds the expected value.
  • With the current technology, velocity ranges from 0.1 mm/sec to 1500 mm/sec or even more can be reached for fluid viscous dampers, which implies wide-variety of applications.
  • Maximum response force is given within tenths of second, so structural displacements and vibrations can be more effectively minimized.
  • Automatic volume compensation of the fluid caused by temperature changes without pressure increase inside the devices. Any compensation containers are located inside.
  • No maintenance works necessarily. Visual inspection can be recommended during the period bridge inspections. Depending on the accumulated displacements and displacement velocities the service life can be reach up to 40 years.
  • With the current development, the devices are not prone to leaking
  • Range of operating temperatures varies from -40ºC to +70ºC.
  • Non-toxic, not inflammable and not ageing fluids are applied.
2.2.3 Application to Bridges

Decks for viaducts and long-span bridges require adequate expansion joints for large displacements under service conditions to absorb the effects of creep and thermal expansion. A common structural layout used in Europe, consists of continuous deck supported by POT devices [Priestley et al, 1996]. By this way, the idea of employing devices with an insignificant response under long-period displacements and at the same time, capable of dissipating much induced seismic energy, was developed.

Some manufacturers differentiate the type of damper according to the motion of the device in the presence of slow displacements. In this case, as for example when thermal expansion occurs, in the OTP type the fluid flows from one chamber to the other with minimum opposition (normally smaller than 10% of the maximum force), while in the OP type such a flow is obstructed, so that during normal service the behaviour is substantially rigid [see the scheme of the typical application of viscous dampers on bridges in Fig. 2.7].

Application of fluid viscous dampers to bridges have been used since middle 90`s. Although these devices may be applied to any kind of structures, their application is easier and more effective in bridges. One of the problems in the use of such devices is that the analysis of the dynamic behaviour becomes more elaborated and difficult than the analysis of a bridge with its seismic resistance based on the ductile capacity of the piers [Virtuoso et al, 2000]. Figs. 2.8 and 2.9 show some examples of application of fluid viscous dampers to bridges.

An important aspect to consider is that, if there is some available stiffness and resistance in the connection between the deck and the piers/towers or abutments, it is possible to obtain optimised solutions without inducing significant forces in the structure. That stiffness as the advantage of guaranteeing recentering capability after an earthquake can be used to improve the structure behaviour under other actions [Virtuoso et al, 2000].

Draft Samper 432909089-monograph-image28-c.png
Fig. 2.7 Typical Application of Viscous Dampers in Bridges [Courtesy of FIP Industriale s.P.a., Italy]
Draft Samper 432909089-monograph-image29.jpeg

Fig. 2.8 Fluid Viscous Dampers at G4-Egnatia Motorway Bridge, Greece [Courtesy of Maurer Sönhe, Germany]

Draft Samper 432909089-monograph-image30.png

Fig. 2.9 850 kN Capacity Damper for the Chun-Su Bridge, South Korea [Courtesy of FIP Industriale s.P.a., Italy]


2.3 Mechanical Behaviour
2.3.1 Energy Approach

An earthquake is an energy phenomenon and therefore this energy character should be considered to achieve the best possible seismic protection for the structure. Without seismic protection system, the seismic energy is entering the structure very concentrated at the fixed axis. By means of shock transmission units the entering energy is distributed to several spots within the structure. In this case the energy input into the structure is still in same magnitude like without those devices, but now the energy is spread over the entire structure in more portions. By implementing additional energy dissipation capability, less energy is entering the structure, with the consequent response mitigation.

The principles of physics that govern the effects of dissipation on the control of dynamic phenomena were studied more than two centuries ago [D`Alembert, Traité de Dynamique, 1743]. Nonetheless, their practical application has come about much later and within a much different time-frame in several sectors of engineering. As was previously exposed, the sector that was the first to adopt such damping technology was the military [France, 1897], followed by the automobile industry. In 1956 Housner already suggested an energy-based design of structures. Kato and Akiyama (1975) and Uang and Bertero (1990) made a valuable contribution to the development of the aspects of an energy-based approach, which presently meets with great concensus.

The dynamic equation of a single-degree-of-freedom structure with mass ms damping coefficient cs, stiffness ks and control force u, subject to ground acceleration Draft Samper 432909089-monograph-image31.png is:

Draft Samper 432909089-monograph-image32.png [Eq. 2.2]

where Draft Samper 432909089-monograph-image33.png , Draft Samper 432909089-monograph-image34.png and Draft Samper 432909089-monograph-image35.png are the displacement, velocity and acceleration responses respectively. The involved parameters are clearly explained in Fig. 2.10, which shows a simplified scheme for a single-degree-of-freedom system. Of course, each term in Eq. 2.2 is a force.

Draft Samper 432909089-monograph-image36.png

Fig. 2.10 Complex Bridge Structure Explained with a Simplified Single Oscillation Mass

Integrating Eq. 2.2 with respect to x:

Draft Samper 432909089-monograph-image37.png

where each term is now an energy component. Thus, we can define:

Draft Samper 432909089-monograph-image38.png [Eq. 2.3]

Draft Samper 432909089-monograph-image39.png [Eq. 2.4]


Draft Samper 432909089-monograph-image40.png [Eq. 2.5]

Draft Samper 432909089-monograph-image41.png [Eq. 2.6]

Draft Samper 432909089-monograph-image42.png [Eq. 2.7]

An energy balance equation can be proposed in terms of the above defined:

Draft Samper 432909089-monograph-image43.png [Eq. 2.8]

where:

Ek: Kinetic energy

Ev: Dissipated energy by inherent damping

Ee: Elastic strain energy

Eh: Dissipated energy by additional damping devices

Ei: Induced energy in the structure.

The concept of energy approach (Fig. 2.11) easily explains the energy terms involved in Eq. 2.8. The amount of structural stored energy (Es) has to be as low as possible to avoid damages. Therefore the value of the dissipated energy (Ed) must be great. In the term Eh energy dissipated by hysteretic or plastic deformation may be included; however this part must be kept low, as this way of energy dissipation causes structural yielding and cracks. For that reason, the drastic increase of the value of the energy of additional damping devices is the final opportunity to control the energy balance of the structure.

Draft Samper 432909089-monograph-image44.png Fig. 2.11 Concept of Energy Approach Considering the Energy Exchange Between Structure and Environment


Draft Samper 432909089-monograph-image45.png : Stored energy within structure

Draft Samper 432909089-monograph-image46.png : Dissipated energy within structure

Thus: