Valdes-Vazquez 2018a - Revision history

Rimni: /* 3. Examples */

2021-08-30T12:17:09Z

‎3. Examples

Rimni: /* 3. Examples */

2021-08-30T12:11:51Z

‎3. Examples

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Rimni: /* 2. Finite element formulation */

2021-08-30T12:02:14Z

‎2. Finite element formulation

Rimni at 14:44, 7 May 2020

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Rimni at 14:34, 7 May 2020

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-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  [[#img-5|Figure 5]],<span id='citeF-15'></span>[[#cite-15|[15]]].
+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  [[#img-5|Figure 5]], <span id='citeF-15'></span>[[#cite-15|[15]]].
 <div id='img-5'></div>
@@ Line 606: / Line 606: @@
 In  [[#img-13|Figure 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,  [[#img-14|Figure 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).
- [[#img-15|Figure 15]] shows the norm of the major  reactions on the structure, which corresponds to  node 23 in  [[#img-3|Figure 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.
+[[#img-15|Figure 15]] shows the norm of the major  reactions on the structure, which corresponds to  node 23 in  [[#img-3|Figure 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>

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 ==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 Figure [[#img-1|1]].
+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  [[#img-1|Figure 1]].
 <div id='img-1'></div>

@@ Line 201: / Line 201: @@
 {| 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{f}^{\, int} ( \mathbf{u}_{n+\alpha _f} ) +       \frac{\alpha _m}{\beta \, \Delta t^2} \mathbf{M} \mathbf{u}_{n+1}       - \mathbf{f}^{\, ext}  </math> <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>
 <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)

@@ Line 202: / Line 202: @@
 |-
 | 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>
-|-
+<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="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)

@@ Line 328: / Line 328: @@
 | <math>6.30945</math>
 | <math>3.8299</math>
-|-
+|-style="border-bottom: 2px solid;"
 | <math>12</math>
 | <math>-0.30945</math>
@@ Line 411: / Line 411: @@
 | <math>8.3095</math>
 | <math>0.0</math>
-|-
+|- style="border-bottom: 2px solid;"
 | <math>24</math>
 | <math>-2.3094</math>

@@ Line 496: / Line 496: @@
 * 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]]].
@@ Line 509: / Line 510: @@
 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
+* 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.

@@ Line 480: / Line 480: @@
 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]].
+<div class="center" style="font-size: 75%;">'''Table 5'''. Prestress properties</div>
 {|  class="floating_tableSCP wikitable" style="text-align: left; margin: 1em auto;min-width:50%;"
 |- style="border-top: 2px solid;"
 |        Membrane prestress: warp=fill

@@ Line 263: / Line 263: @@
 |}
-In table [[#table-1|1]] the values of the geometry of the tensile-structure  and the surrounding cable are listed.
+In Table [[#table-1|1]] the values of the geometry of the tensile-structure  and the surrounding cable are listed.
 <div class="center" style="font-size: 75%;">'''Table 1'''. Membrane geometry</div>
@@ Line 478: / Line 478: @@
 |}
-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]].
+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]].

@@ Line 253: / Line 253: @@
 ===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]].
+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>