Dynamic Analysis of Hypar Membrane Structures Subjected to Seismic Excitations

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==Dynamic Analysis of Hypar Membrane Structures Subjected to Seismic Excitations   ==

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==Abstract==

The seismic response of tensile membrane structures (TMS)  is investigated. To the authors knowledge, this is the  first study on TMSs subjected to a seismic record  reported in the literature.  A type of membrane structures usually considered as a  reference in other works are employed in the present study.  The selected ''hyperbolic-paraboloid'' fabric structures,  also referred as ''hypar'' TMSs, are subjected to an earthquake  accelerogram from a relatively large earthquake recorded  at Norcia, Italy.  To obtain the TMSs seismic response,  a finite element formulation reported in a previous study,  and which accounts for wrinkling phenomena, orthotropic material  modeling, and geometrical nonlinearity, is employed.  The analyses are performed in two stages; first for the  prestressed case and then the seismic loading is added.  It is found that the seismic response of TMSs should not  be disregarded by designers beforehand, since important  increments in the dynamic response of the displacements produce an incremet of around 9% for stresses. However, a very important increment of around 80%  for support reaction forces is computed,  when compared with the static case.  It is also found that the orientation of the  frame-supporting structure has a significant impact  on the computed seismic reactions.

'''Keywords:''' ''tensile membrane structures, seismic excitations, dynamic analysis, finite element method''

==1 Introduction==

The structural analysis and design of tensile membrane  structures (TMS) is a subject that has become important  in recent years and some methodologies and recommendations  have arisen to cope with the complex behavior of these  light weight structures with impressive designs.  Noticeably, the TMS analysis and design concerns are  focused on certain types of loads which are considered  critical for fabric structures. Recent studies focus  primarily in demands on TMS due to wind and snow loads  to analyze the structures, to compute their reliability  and to review other aspects  (e.g., <span id='citeF-1'></span>[[#cite-1|[1]]], <span id='citeF-2'></span>[[#cite-2|[2]]], <span id='citeF-3'></span>[[#cite-3|[3]]], <span id='citeF-4'></span>[[#cite-4|[4]]], <span id='citeF-5'></span>[[#cite-5|[5]]], <span id='citeF-6'></span>[[#cite-6|[6]]], <span id='citeF-7'></span>[[#cite-7|[7]]] and <span id='citeF-8'></span>[[#cite-8|[8]]]).

The seismic response of TMSs is virtually absent in the  literature, and recommended guidelines  (e.g., <span id='citeF-9'></span>[[#cite-9|[9]]])  and formal code regulations are still under development  in many countries (e.g., <span id='citeF-10'></span>[[#cite-10|[10]]]).  Even though the wind and snow loads seem to capture  the attention of more designers and studies on TMSs,  it will be seen in the present work that consideration  of the earthquake forces may be relevant for some specific  issues regarding the seismic response of TMSs and  their supporting structures.

Recent literature dealing with the analysis of TMSs is based  on the finite element method (FEM), however detailed  information on the employed programs is not always reported  or commercial software is used  (<span id='citeF-1'></span>[[#cite-1|[1]]], <span id='citeF-11'></span>[[#cite-11|[11]]], <span id='citeF-5'></span>[[#cite-5|[5]]]);  this is important, since there is a wide variability of  results when using FEM to analyze TMSs (<span id='citeF-5'></span>[[#cite-5|[5]]]).  In the present study an on-purpose own developed code which  accounts for wrinkling phenomena, orthotropic material modeling,  and geometrical non-linearity is used, which formulation and  discretization is established in detail in (<span id='citeF-12'></span>[[#cite-12|[12]]])  and references within.  <span id='citeF-13'></span>[[#cite-13|[13]]] and  <span id='citeF-14'></span>[[#cite-14|[14]]] used a similar formulation.

Although there are TMSs with different shapes reported in  the literature, the hyperbolic paraboloid, nicknamed as ''hypar'',  has become some sort of benchmark for TMS studies and is used  by several authors in recent works  (e.g., <span id='citeF-1'></span>[[#cite-1|[1]]], <span id='citeF-4'></span>[[#cite-4|[4]]], <span id='citeF-6'></span>[[#cite-6|[6]]],  <span id='citeF-7'></span>[[#cite-7|[7]]], <span id='citeF-8'></span>[[#cite-8|[8]]]).  Hypar TMSs with different supporting sub structures are  considered in the present study.    Considering the preceding paragraphs, the main objective  of this study is to assess the seismic response of hypar  TMSs and their frame-supporting structure subjected to  earthquake loads using our referred FEM code.

==2 Finite Element Formulation==

The finite element analysis formulation to be used  for the dynamic analyses of membrane structures is based  on a previous study <span id='citeF-12'></span>[[#cite-12|[12]]],  which is used for the structures under analysis in this study. The curvilinear coordinate system used to formulate the membrane finite element is shown in Fig. [[#img-1|1]].

<div id='img-1'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Fig_Mem3_1.png|240px|Membrane coordinate system]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' Membrane coordinate system
|}

From <span id='citeF-12'></span>[[#cite-12|[12]]], the internal forces given in curvilinear coordinates for membrane structures are

<span id="eq-1"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\textrm{f}^{\, \mathit{int}}_{iI} = \int _{\Omega _0}        \textrm{B}^{cur}_{\alpha \beta iI} \, \textrm{S}^{\alpha \beta } d \Omega _0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

where: <math display="inline">i</math> is the global degree of freedom (dof), <math display="inline">I</math> is the local element node, <math display="inline">\Omega _0</math> is the reference configuration domain, and the Greek indices <math display="inline">\alpha </math>, <math display="inline">\beta </math> on the membrane  mid-surface take on values of 1 and 2 in a plane stress  state in the Euclidean space. Besides, <math display="inline">\textrm{S}</math> is the <math display="inline">2^{nd}</math> Piola-Kirchhoff stress tensor, and the fourth-order strain-displacement tensor in curvilinear coordinates is given by

<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\textrm{B}^{cur}_{\alpha \beta iI} =        \frac{1}{2}\left(                    N_{I,\alpha } x^h_{i,\beta } + N_{I,\beta } x^h_{i,\alpha }                  \right) </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}

with

<span id="eq-3"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>x^h_{i,\alpha } = \sum ^{n_\textrm{node}}_{J=1} N_{J,\alpha } x_{iJ} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

where <math display="inline">N_I</math> denotes the shape function of node <math display="inline">I</math> of the finite element. Using voigt notation to transform internal forces from tensorial to matrix notation, equation ([[#eq-1|1]]) yields,

<span id="eq-4"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{f}^{\, \mathit{int}}_{I} = \int _{\Omega _0}        \left[\mathbf{B}^T_{I}\right]^{cur} \, \left\{\mathbf{S} \right\}^{cur} d \Omega _0 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}

where the strain matrix <math display="inline">\mathbf{B}_{I}^{cur}</math> is given by

<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{B}_{I}^{cur} =    \begin{bmatrix}\frac{\partial N_I}{\partial \xi ^1} \frac{\partial x_1^h}{\partial \xi ^1}   &       \frac{\partial N_I}{\partial \xi ^1} \frac{\partial x_2^h}{\partial \xi ^1}   &       \frac{\partial N_I}{\partial \xi ^1} \frac{\partial x_3^h}{\partial \xi ^1}   \\  \\       \frac{\partial N_I}{\partial \xi ^2} \frac{\partial x_1^h}{\partial \xi ^2}   &       \frac{\partial N_I}{\partial \xi ^2} \frac{\partial x_2^h}{\partial \xi ^2}   &       \frac{\partial N_I}{\partial \xi ^2} \frac{\partial x_3^h}{\partial \xi ^2}   \\  \\       \frac{\partial N_I}{\partial \xi ^1} \frac{\partial x_1^h}{\partial \xi ^2}   +       \frac{\partial N_I}{\partial \xi ^2} \frac{\partial x_1^h}{\partial \xi ^1}   &       \frac{\partial N_I}{\partial \xi ^1} \frac{\partial x_2^h}{\partial \xi ^2}   +       \frac{\partial N_I}{\partial \xi ^2} \frac{\partial x_2^h}{\partial \xi ^1}   &       \frac{\partial N_I}{\partial \xi ^1} \frac{\partial x_3^h}{\partial \xi ^2}   +       \frac{\partial N_I}{\partial \xi ^2} \frac{\partial x_3^h}{\partial \xi ^1}          \end{bmatrix} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}

Following the details given in <span id='citeF-12'></span>[[#cite-12|[12]]],  the internal forces can be computed as indicated ahead to take into account the orthotropic material behavior.

It is important to mention that fabric manufacturers usually  do not provide in their catalogs the thickness of their  membrane products, and they report only the tensile stiffness.  This last parameter is enough to perform a static or  quasi-static analysis. If the tensile stiffness is given  and the formulation given in <span id='citeF-12'></span>[[#cite-12|[12]]] is going  to be used, some changes must be carried out. Because both  alternatives could be of interest for researchers, and because  the formulation given by <span id='citeF-12'></span>[[#cite-12|[12]]] is versatile  enough to implement any of the alternatives, a brief  description is given in the following when no thickness  is directly available.

First, it is noted that in <span id='citeF-12'></span>[[#cite-12|[12]]] the  internal forces are obtained by means of

<span id="eq-6"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{f}^{\, \mathit{int}} = A_0 \, t \, \mathbf{B}^T \, \mathbf{Q}^T \,       \mathbf{T}_{\sigma } \, \mathbf{S} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

where: <math display="inline">A_0</math> is the element surface in reference configuration, <math display="inline">t</math> is the membrane thickness, <math display="inline">\mathbf{B}</math> is the nonlinear  strain-displacement matrix, <math display="inline">\mathbf{Q}</math> is the transformation matrix  from a curvilinear system to a rectangular one, <math display="inline">\mathbf{T}_{\sigma }</math> is the  rotation matrix from fiber axes to local axes,  and '''S''' is the <math display="inline">2^{nd}</math> Piola-Kirchhoff stress vector  (force per unit area). In equation ([[#eq-6|6]]), the stresses are computed as

<span id="eq-7"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{S} = \mathbf{C} \cdot \mathbf{E} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}

where: <math display="inline">\mathbf{E}</math> is the Green-Lagrange strain vector,  and the constitutive equation <math display="inline">\mathbf{C}</math> is expressed as

<span id="eq-8"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{C} = \frac{E}{1-\nu ^2}         \begin{bmatrix}1    &   \nu  &   0   \\  \\          \nu  &   1    &   0   \\  \\          0    &   0    &  \dfrac{1-\nu }{2}        \end{bmatrix} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}

In equation ([[#eq-8|8]]), <math display="inline">E</math> is the Young modulus and <math display="inline">\nu </math> is the Poisson coefficient.  Since material properties given in most manufacturers'  catalogs neither give a value for the thickness <math display="inline">t</math> nor for  the Young modulus <math display="inline">E</math>, then the internal forces for  membrane analysis must be evaluated using

<span id="eq-9"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{f}^{\, \mathit{int}} = A_0 \, t \, \mathbf{B}^T       \, \mathbf{Q}^T \, \mathbf{T}_{\sigma } \, \mathbf{S}^{l} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}

where: <math display="inline">\mathbf{S}^l</math> is the <math display="inline">2^{nd}</math> Piola-Kirchhoff  stress vector (force per unit length), and

<span id="eq-10"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{S}^{l} = \mathbf{D} \cdot \mathbf{E} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}

with

<span id="eq-11"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{D} = \frac{E_{ts}}{1-\nu ^2}         \begin{bmatrix}1    &   \nu  &   0   \\  \\          \nu  &   1    &   0   \\  \\          0    &   0    &  \dfrac{1-\nu }{2}        \end{bmatrix} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|}

where: <math display="inline">E_{ts}</math> is the tensile stiffness of the material,  given in units of force per unit length  (as usual in manufacturers of membrane materials).  It is remarked that the tensile stiffness of the material  is numerically equivalent to the Young modulus times the thickness, <math display="inline">E_{ts} = E \, t</math>, therefore the thickness is no longer  required as stated in the original formulation as  given in <span id='citeF-12'></span>[[#cite-12|[12]]].

For the case when an orthotropic material is modelled,  two directions of the tensile stiffness must be given,  which are <math display="inline">E_w</math> for the warp direction and <math display="inline">E_f</math> for the fill direction.  Consequently, in the orthotropic plane stress constitutive equation, <math display="inline">E_x</math> must be replaced by <math display="inline">E_w</math>, and <math display="inline">E_y</math> must be replaced by <math display="inline">E_f</math>.  If desired, these changes would allow us to solve other problems,  for instance the numeric examples given in <span id='citeF-5'></span>[[#cite-5|[5]]].

Unfortunately, the previous simplification is only valid  for static or quasi-static analyses.  For a dynamic seismic analysis to explicitly determine  the thickness, <math display="inline">t</math>, is necessary.  Consequently, the original formulation in  <span id='citeF-12'></span>[[#cite-12|[12]]] is used, so that the right structure  mass is accounted for and a consistent analysis assessing  the internal forces given by equation ([[#eq-6|6]]) can be performed,  which allows to complete the formulation with

<span id="eq-12"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{f}^{\, int} + \mathbf{M}\mathbf{a} = \mathbf{f}^{\, ext} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}

here, the external forces <math display="inline">\mathbf{f}^{\, ext}</math> are given by  the <math display="inline">2^{nd}</math> Newton's law, by considering the seismic record  as the acceleration for the studied site which is defined later.  It is noteworthy that the formulation in <span id='citeF-12'></span>[[#cite-12|[12]]] is of  a wide applicability, and since it was originally developed to  cope with dynamic analyses, it can be used to analyze  the TMS under seismic excitations given below.

The algorithm used to solve equation ([[#eq-12|12]]) from a known solution time at step <math display="inline">n</math> (e.g., displacement <math display="inline">\mathbf{u}_n</math>, velocity <math display="inline">\dot{\mathbf{u}}_n</math> and acceleration <math display="inline">\ddot{\mathbf{u}}_n</math>), take into account the generalised<math display="inline">-\alpha </math> time integration scheme leading to

<span id="eq-13"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{f}^{\, int} ( \mathbf{u}_{n+\alpha _f} ) +       \frac{\alpha _m}{\beta \, \Delta t^2} \mathbf{M} \mathbf{u}_{n+1}       - \mathbf{f}^{\, ext}  </math>
|-
| style="text-align: center;" | <math>       = \mathbf{M} \left[\frac{\alpha _m}{\beta \, \Delta t^2} \mathbf{u}_n                            + \frac{\alpha _m}{\beta \, \Delta t} \dot{\mathbf{u}}_n                            + \left(\frac{\alpha _m}{2\beta - 1} \ddot{\mathbf{u}}_n \right)                     \right]    </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}

where

<span id="eq-14"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\mathbf{u}_{n+\alpha _f} = ( 1 - \alpha _f) \mathbf{u}_n + \alpha _f \mathbf{u}_{n+1} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}

Low-frequency dissipation is optimal with

<span id="eq-15"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\rho _{\infty } \in \left[0, 1\right],         \alpha _f = \frac{1}{1+\rho _{\infty }},        \alpha _m = \frac{2-\rho _{\infty }}{1+\rho _{\infty }} </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
|}

and when

<span id="eq-16"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\beta = \frac{1}{4}(1 + \alpha _m - \alpha _f )^2 </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
|}

the method is second-order accurate and posses  high frequency dissipation.

==3 Examples==

===3.1 Structure 1===

In this study a tensile-structure supported on a sub-structure  is investigated.  The geometry of the tensile-structure, as well as the surrounding  cable along the perimeter, are shown in figure [[#img-2|2]].

<div id='img-2'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Geom_1.png|210px|Membrane geometry and surrounding cable]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' Membrane geometry and surrounding cable
|}

In table [[#table-1|1]] the values of the geometry of the tensile-structure  and the surrounding cable are listed.


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
|+ style="font-size: 75%;" |<span id='table-1'></span>Table. 1 Membrane geometry
|- style="border-top: 2px solid;"
|         Node    
| x-Coord    
| y-Coord    
| z-Coord   
|- style="border-top: 2px solid;"
| [0.5ex]  <math display="inline">1</math>     
| <math>0.0</math>
| <math>0.0</math>
| <math>0.0</math>
|-
| <math>2</math>
| <math>3.0</math>
| <math>0.3</math>
| <math>5.0</math>
|-
| <math>3</math>
| <math>6.0</math>
| <math>0.0</math>
| <math>6.0</math>
|-
| <math>4</math>
| <math>5.7</math>
| <math>3.0</math>
| <math>5.0</math>
|-
| <math>5</math>
| <math>6.0</math>
| <math>6.0</math>
| <math>4.0</math>
|-
| <math>6</math>
| <math>3.0</math>
| <math>5.7</math>
| <math>5.0</math>
|-
| <math>7</math>
| <math>0.0</math>
| <math>6.0</math>
| <math>6.0</math>
|-
| <math>8</math>
| <math>0.3</math>
| <math>3.0</math>
| <math>5.0</math>
|-
| <math>9</math>
| <math>-0.30945</math>
| <math>-0.30945</math>
| <math>3.8299</math>
|-
| <math>10</math>
| <math>6.30945</math>
| <math>-0.30945</math>
| <math>6.1701</math>
|-
| <math>11</math>
| <math>6.30945</math>
| <math>6.30945</math>
| <math>3.8299</math>
|-
| <math>12</math>
| <math>-0.30945</math>
| <math>6.30945</math>
| <math>6.1701</math>
|}

From the coordinates listed in table [[#table-1|1]],  it can be observed that the highest points for the membrane  correspond to nodes 3 and 7, while the highest points for the  cables correspond to nodes 10 and 12.  The direction along the highest points is coincident with the  principal direction of the fiber reinforcement of the membrane,  i.e. the warp direction, also indicated in figure [[#img-2|2]].

In figure [[#img-3|3]] the nodes of the supporting sub-structure are shown.

<div id='img-3'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Geom_2.png|300px|Nodes and plan view of the supporting sub-structure]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 3:''' Nodes and plan view of the supporting sub-structure
|}

Coordinates for the geometry of the supporting sub-structure  shown in figure [[#img-3|3]] are listed in table [[#table-2|2]].


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
|+ style="font-size: 75%;" |<span id='table-2'></span>Table. 2 Sub-structure geometry
|- style="border-top: 2px solid;"
|         Node    
| x-Coord    
| y-Coord    
| z-Coord   
|- style="border-top: 2px solid;"
| [0.5ex]  <math display="inline">13</math>    
| <math>1.1048</math>
| <math>1.1048</math>
| <math>0.0</math>
|-
| <math>14</math>
| <math>4.8952</math>
| <math>1.1048</math>
| <math>0.0</math>
|-
| <math>15</math>
| <math>4.8952</math>
| <math>4.8952</math>
| <math>0.0</math>
|-
| <math>16</math>
| <math>1.1048</math>
| <math>4.8952</math>
| <math>0.0</math>
|-
| <math>17</math>
| <math>-2.30945</math>
| <math>-0.30945</math>
| <math>0.0</math>
|-
| <math>18</math>
| <math>-0.30945</math>
| <math>-2.30945</math>
| <math>0.0</math>
|-
| <math>19</math>
| <math>6.3095</math>
| <math>-2.3095</math>
| <math>0.0</math>
|-
| <math>20</math>
| <math>8.3095</math>
| <math>-0.30945</math>
| <math>0.0</math>
|-
| <math>21</math>
| <math>8.3095</math>
| <math>6.3095</math>
| <math>0.0</math>
|-
| <math>22</math>
| <math>6.3095</math>
| <math>8.3095</math>
| <math>0.0</math>
|-
| <math>23</math>
| <math>-0.30945</math>
| <math>8.3095</math>
| <math>0.0</math>
|-
| <math>24</math>
| <math>-2.3094</math>
| <math>6.3094</math>
| <math>0.0</math>
|}

The resulting mesh by using the previous listed membrane  and support coordinates are depicted in figure [[#img-4|4]] and,  as it can be observed in the figure, consists of linear triangles  with a <math display="inline">20 \times 20</math> mesh for the membrane. The cables and posts are  defined by a two-noded unidimensional element. Both mentioned  types of elements are Total Lagrangian non-linear geometric elements.  Also, in figure [[#img-4|4]] the control points  '''A''', '''B''' and '''C''' used  for obtaining the displacements and stresses to be discussed later are shown.

<div id='img-4'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Mesh_Iso_605.png|420px|Mesh and reference points for the studied structure]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' Mesh and reference points for the studied structure
|}

The material properties of the membrane,  after <span id='citeF-5'></span>[[#cite-5|[5]]],  are listed in table [[#table-3|3]],  where a thickness for the membrane has been added.


{|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
|+ style="font-size: 75%;" |<span id='table-3'></span>Table. 3 Membrane properties
|- style="border-top: 2px solid;"
|        Tensile stiffness warp direction, <math display="inline">E_w</math>      
| 600.0 <math display="inline">kN/m</math>   
|-
| Tensile stiffness fill direction, <math display="inline">E_f</math>      
| 600.0 <math display="inline">kN/m</math>   
|-
| Shear stiffness, <math display="inline">G</math>                         
| 30.0 <math display="inline">kN/m</math>   
|-
| Poisson, <math display="inline">v_{wf} = v_{fw}</math>                   
| 0.1          
|-
| Thickness, <math display="inline">t</math>                               
| 1.0 <math display="inline">mm</math>     
|-
| Density, <math display="inline">\rho </math>                              
| 1800.0 <math display="inline">kg/m^3</math>    
|}

The cable, tensors and posts material properties can be found  in table [[#table-4|4]]. Note that we denote post as a  compression member in the supporting sub-structure  (purple elements in figure [[#img-4|4]]), and tensor as  a tension member of the supporting sub-structure  (cyan elements in figure [[#img-4|4]]).


{|  class="floating_tableSCP wikitable" style="text-align: center; margin: 1em auto;min-width:50%;"
|+ style="font-size: 75%;" |<span id='table-4'></span>Table. 4 Sub-structure properties
|- style="border-top: 2px solid;"
|                
| Young modulus  
| Cross section                 
| Density     
|- style="border-top: 2px solid;"
| [0.5ex]             Cable   
| 210 <math display="inline">GPa</math>      
| <math>0.127 \times 10^{-3}</math><math display="inline">m^2</math>  
| 7800 <math display="inline">kg/m^3</math>   
|-
| Tensor  
| 210 <math display="inline">GPa</math>      
| <math>0.127 \times 10^{-3}</math><math display="inline">m^2</math>  
| 7800 <math display="inline">kg/m^3</math>   
|-
| Post    
| 210 <math display="inline">GPa</math>      
| <math>4.748 \times 10^{-3}</math><math display="inline">m^2</math>  
| 8500 <math display="inline">kg/m^3</math>      
|}

The structure is subjected to two load stages,  and the wrinkling model described in <span id='citeF-12'></span>[[#cite-12|[12]]] is used  in both phases. The first stage corresponds to prestress,  which is applied as per the values indicated in table [[#table-5|5]].


{|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
|+ style="font-size: 75%;" |<span id='table-5'></span>Table. 5 Prestress properties
|- style="border-top: 2px solid;"
|        Membrane prestress: warp=fill      
| 3.0 <math display="inline">kN/m</math>   
|-
| Cable prestress                    
| 2.36 <math display="inline">MN/m^2</math> 
|-
| Tensor prestress                   
| 1.18 <math display="inline">MN/m^2</math>      
|}

The prestress stage is explained in what follows:

* Prestress.- A quasi-static analysis is carried out by applying the               prestress linearly with a step time increment <math display="inline">\Delta t = 0.01</math>.

The second load stage corresponds to the seismic loads,  which are applied by using as input the accelerations of  the 6.6 magnitude (Richter scale) earthquake recorded at  Northern Norcia, Italy, on October 30, 2016 and shown  in figure [[#img-5|5]],<span id='citeF-15'></span>[[#cite-15|[15]]].

<div id='img-5'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Italy.png|480px|Accelerogram form the Norcia, Italy earthquake]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 5:''' Accelerogram form the Norcia, Italy earthquake
|}

The seismic loading is added to the resulting prestressing  state in the second stage as follows:

* Seismic load.- A seismic undamped dynamic analysis with a time              step increment <math display="inline">\Delta t = 0.005</math> <math display="inline">s</math> is performed              up to a total of 8000 time steps equivalent to a              duration of <math display="inline">40</math> <math display="inline">s</math> of the earthquake record.              In particular, the global ''X'' direction              was analyzed by considering that the 100%              of the earthquake acted along this direction

It is pointed out that for the case of a geometric non-linear  dynamic analysis, Newmark's dynamic method does not provide  a right solution; therefore, <math display="inline">\alpha{-}</math>methods like the  HHT <span id='citeF-16'></span>[[#cite-16|[16]]],  WBZ <span id='citeF-17'></span>[[#cite-17|[17]]] or  Generalized-<math display="inline">\alpha </math> <span id='citeF-18'></span>[[#cite-18|[18]]]  are required.

The results from the dynamic analysis highlight important issues  to be considered for these kinds of structures.  To elaborate on this, first consider the largest displacement  component at control point A depicted in figure [[#img-6|6]].  At this point the displacement due to prestress leads to a descend  of the structure to a maximum of 2.9 mm, and from there the  maximum absolute displacement during the earthquake further  increases to a maximum of 10.9 mm. This is equivalent to a 375% increase,  if the prestress displacement is used as reference.  Even though such increase is significant in percentage,  is not critical from a serviceability point of view for the structure,  since the lowest point in the membrane is located at a 4 m height.

<div id='img-6'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_A_Disp.png|480px|Displacement in the Z direction for control point A]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 6:''' Displacement in the Z direction for control point A
|}

To further elaborate on the discussion, the largest displacement  component for control point B is shown in figure [[#img-7|7]],  where it can be observed that the structure raises to a maximum  of 1.4 mm due to the prestress, and from there it further reaches  a maximum absolute displacement of 5.9 mm during the earthquake.  This is equivalent to a 421% increase,  is the prestress displacement is used as reference.

<div id='img-7'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_B_Disp.png|480px|Displacement in the Z direction for control point B]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 7:''' Displacement in the Z direction for control point B
|}

It is noted that by employing the same scale as in  the previous two figures, the largest displacement  component for control point C is in the X direction;  however, for a consistent comparison (i.e. in the same  direction as in the previous two figures) the  displacement in the Z direction for control point C  is shown in figure [[#img-8|8]].  At this point the displacement due to prestress leads  to a descend of the structure to a maximum of 5.3 mm;  during the earthquake the absolute maximum displacement is 4.5 mm.

<div id='img-8'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_C_Disp.png|480px|Displacement in the Z direction for control point C]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 8:''' Displacement in the Z direction for control point C
|}

The displacement in the X direction for control  point C is observed in figure [[#img-9|9]];  it is shown that the displacement due to prestress  is practically zero (exactly 7.3x10-5 mm),  and it reaches an absolute maximum of 5.5 mm during the earthquake.

<div id='img-9'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_Cx_Disp.png|480px|Displacement in the X direction for control point C]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 9:''' Displacement in the X direction for control point C
|}

From the previously described results it could be  natural to expect similar increments in the  membrane stresses. However, the stress increments  in the membrane are not as significant;  this is evidenced in figures [[#img-10|10]] and [[#img-11|11]].  For the warp stress, the maximum increment due to  the dynamic loads is located at point C and is  equal to a static value of 1.76 kN/m,  while the maximum dynamic value is equal to 1.92 kN/m.  This means that the stress increase is in the  order of 9.0%, considering the static value as reference.  For the fill stress similar trends are found,  being the maximum increase of the order of 8.5% located  at point C (with static and dynamic values of 1.64 kN/m  and 1.78 kN/m, respectively).  In figures [[#img-10|10]] and [[#img-11|11]]  the stresses at the other control  points (A and B) are also indicated;  similar trends can be observed but they are not included  since their increments are lower than those of point C.

<div id='img-10'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_S_warp.png|480px|Stresses in the warp direction]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 10:''' Stresses in the warp direction
|}

<div id='img-11'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_S_fill.png|480px|Stresses in the fill direction ]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 11:''' Stresses in the fill direction 
|}

With the results discussed so far, one may  think that the quantitative values of stresses  and displacements due to earthquake loads are  not that relevant for design purposes when membrane  structures are considered (as opposed to the  usually more critical values resulting from wind loading).  However, a different story could be found if the  design is focused on the foundation supports  (anchorage) as will be discussed shortly after,  and consequently care should be exercised by  the designer in such a case.

In figures [[#img-12|12]],  [[#img-13|13]] and [[#img-14|14]]  the most critical behavior at the structure  foundation supports, corresponding respectively  to nodes 16, 23 and 24 in figure [[#img-3|3]], is shown.  In figure [[#img-12|12]],  the static reaction in the Z direction is 57.8 kN,  and the corresponding dynamic value is 63.1 kN;  it means an increment of 9.1% in the  compression force on the support.

<div id='img-12'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_R_PMay.png|480px|Reactions at major post]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 12:''' Reactions at major post
|}

<div id='img-13'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_R_TMay1.png|480px|Reactions at major tensor 1]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 13:''' Reactions at major tensor 1
|}

<div id='img-14'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_R_TMay2.png|480px|Reactions at major tensor 2]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 14:''' Reactions at major tensor 2
|}

In figure [[#img-13|13]] the maximum response  in the Z direction is exhibit, which is a tension  force and is given by a negative value in the reaction;  the static value of such response is -22.3 kN,  while maximum absolute dynamic response is -31.5 kN,  meaning that an increase of tension of 41% is found.  Likewise, figure [[#img-14|14]] shows that  the maximum response is, one more time, a tensile  force in the Z direction; this time the static value  of such response is -22.3 kN, while maximum absolute  dynamic response is -31.2 kN (i.e., a 40% increment).

Figure [[#img-15|15]] shows the norm of the major  reactions on the structure, which corresponds to  node 23 in figure [[#img-3|3]].  It can be observed that the static response  value is 23.5 kN, while the maximum dynamic  response is 33.1 kN; a 40.9% increment.  This increase is significant, since the usual  design load factors do not cover such a large increment in terms of force.

<div id='img-15'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_i_Italy_R_TMay1_N.png|480px|Norm of reactions at major tensor 1]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 15:''' Norm of reactions at major tensor 1
|}

===3.2 Structure 2===

The increment in the reaction at the foundation supports  during an earthquake may be generated by the horizontal component  of the compression post due to the transmission of seismic  horizontal forces to the structure.  To investigate further this issue, the structure response  is obtained by considering that the posts are not inclined,  but oriented in the vertical direction. The new geometry with the changed nodes is shown in  figure [[#img-16|16]],  and they are the nodes 13, 14, 15 and 16.

<div id='img-16'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Geom_3.png|300px|Updated geometry of the membrane and surrounding cable]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 16:''' Updated geometry of the membrane and surrounding cable
|}

For a better visual appreciation, the new structure  can be observed in 3D in figure [[#img-17|17]],  where it can be readily observed that the posts are now vertical.

<div id='img-17'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Mesh_iso_606.png|480px|Updated geometry of the membrane and surrounding cable]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 17:''' Updated geometry of the membrane and surrounding cable
|}

In figure [[#img-18|18]] the norm of the major  reactions is shown for the new structure.  It corresponds to node 23 in  figure [[#img-16|16]] and it can be observed  that the static response has a value of 24.3 kN,  while maximum absolute dynamic response is 44.6 kN,  which represents a surprising 83.5% increment with  respect to the static response.  This value is enormous, and it is considered  that no common factor in codified design accounts  for such a large increment.

<div id='img-18'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_729493725672-Sis_08_r_Italy_R_TMay1_N.png|480px|Norm of reactions at major tensor 1 for new geometry]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 18:''' Norm of reactions at major tensor 1 for new geometry
|}

It can be concluded that the increase in the  reactions of the membrane foundation supports  is very significant, this should be considered  when designing the supports and posts  orientations in seismic prone regions.

From the previous description of displacements  and stresses and the discussion along,  it can be concluded that although the seismic  response of TMSs seems not to be considered  important in the literature, care should be exercised  in seismic regions when designing these kind  of structures, because the displacements and  support reactions increments can be relevant,  and this highlights the necessity to verify that  critical cases due to earthquake load are not reached.  It is also concluded that the orientation of the  elements on the supporting structure (e.g., the posts)  can have an impact on the obtained forces at the foundation.  Further research to investigate other geometries  (in the TMS as well as in the supporting structure)  and other seismic loads, as well as a comparison with  other types of loading (e.g., wind loading)  is strongly recommended.

==4 Conclusions==

Although the structural analysis and design of tensile  membrane structures (TMS) is a subject that has become  relevant lately, it was found that recent studies are mainly  focus on demands imposed to TMSs by wind and snow loads,  and that the seismic response of TMSs is virtually absent  in the literature. To the authors best knowledge,  this is the first study dealing with seismic demands on TMSs  using the described finite element formulation.  The findings can be important, especially if it is considered  that formal code regulations are still under development,  and that the conclusions given in this section could be  useful to code developers and practitioners, among others.

More specifically,  the seismic response of hypar TMSs under seismic  excitations is computed. A FEM which accounts  for wrinkling phenomena, orthotropic material modeling,  and geometrical non-linearity is employed for  the dynamic analysis.  Displacements, stresses and reactions are described  for selected nodes. A record of a relatively large  earthquake recorded at Northern Norcia, Italy, was used  for the analyses. It is pointed out once more that the seismic  response of TMSs is virtually absent in the  specialized literature and that codes and standards  are still under developments for these structures; therefore, it was decided to use methods and formulations  known in the structural engineering field to perform  the dynamic analysis with the described FEM  and accelerogram.

It was found that important increments are obtained  in terms of displacements when the dynamic loads are  considered, compared to the prestressed static loads.  Although stress increments are not as large,  the supports reactions exhibit a significant increment  of the order of 40% (always compared versus the static case).  Moreover, if the geometry of the supporting structure  orientations is varied (e.g., inclined posts versus  vertical posts) the increase at the reactions  can be even higher (   over 80% ). This means that special attention should be exercised when  the supports are to be designed, since they are critical  parts of the structures and if they do not withstand the  seismic demands or do not perform adequately under seismic  excitations, the whole structure could fail or  not being serviceable anymore.

It is highlighted that the seismic response of  TMSs should be checked in seismic regions,  because the displacements and support reactions  increments can be significant, and possible  critical cases due to earthquake load should  be inspected.  The readers could think of a case when the wind and  snow loadings are not that critical for a TMSs and  the seismic loading could be significant. For instance,  consider a hypothetical or real case of a TMS in a  seismic-prone region, while at the same time located  inside another larger structure preventing wind or  snow loading (or simply an open location where snow  in not present and wind velocities are not significant);  previsions for such possibilities should be incorporated  in codes and guidelines for TMSs, and the conclusions  referred here can be an aid for such a purpose.

Finally, it is also pointed out that the  orientation of the elements on the supporting  structure (e.g., the posts) can have an important  impact on the obtained forces at the foundation, as discussed in this study.  Therefore, further studies considering other  geometries in the TMS and in the supporting  structure, other seismic loads and comparisons  with other types of loading are strongly recommended to advance the knowledge of the response of TMSs under seismic excitations.

==5 Acknowledgements==

We are grateful to ''Universidad de Guanajuato'' for its support and to the ''International Centre for Numerical Methods in Engineering''  (CIMNE) for providing us with the pre and post processor  ''GiD'' <span id='citeF-19'></span>[[#cite-19|[19]]] in our CIMNE-Classroom at the Civil Engineering Department.

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