Scipediacontent at 14:33, 13 April 2017

2017-04-13T14:33:17Z

Scipediacontent at 14:32, 13 April 2017

2017-04-13T14:32:13Z

Scipediacontent: Scipediacontent moved page Draft Content 624005357 to Manzoli 2011a

2017-03-31T11:33:31Z

Scipediacontent moved page Draft Content 624005357 to Manzoli 2011a

Scipediacontent: Created page with "====Resumo==== Estudam-se os aspectos referentes à qualidade e objetividade das respostas obtidas com as diferentes formulações (simétrica e não-simétrica) de elementos..."

2017-03-31T10:54:37Z

Created page with "====Resumo==== Estudam-se os aspectos referentes à qualidade e objetividade das respostas obtidas com as diferentes formulações (simétrica e não-simétrica) de elementos..."

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@@ Line 609: / Line 609: @@
 |-
 | <math>\left\{\begin{array}{c}
-\openbracketleft {u\openbracketright }_x\\
+\mbox{〚 }{u\mbox{〛}}_x\\
-\openbracketleft {u\openbracketright }_y
+\mbox{〚 }{u\mbox{〛}}_y
 \end{array}\right\}=l_e\left\{\begin{array}{c}
 {\gamma }_{xy}^p\\

@@ Line 64: / Line 64: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>{\overset{\mbox{ˆ}}{\boldsymbol{\mbox{d}}}}_1=\openbracketleft \boldsymbol{\mbox{u}}\openbracketright </math>
+| <math>{\overset{\mbox{ˆ}}{\boldsymbol{\mbox{d}}}}_1=\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}</math>
 |-
 |<math>{\overset{\mbox{ˆ}}{\boldsymbol{\mbox{d}}}}_2=0</math>
@@ Line 85: / Line 85: @@
 | <math>\boldsymbol{\epsilon }=\sum_{i=1}^3\boldsymbol{\mbox{B}}_i\quad \left(\boldsymbol{\mbox{d}}_i-\right. </math><math>\left. {\overset{\mbox{ˆ}}{\boldsymbol{\mbox{d}}}}_i\right)</math>
 |-
-|<math>=\sum_{i=1}^3\boldsymbol{\mbox{B}}_i\quad \boldsymbol{\mbox{d}}_i-</math><math>\boldsymbol{\mbox{B}}_1\openbracketleft \boldsymbol{\mbox{u}}\openbracketright </math>
+|<math>=\sum_{i=1}^3\boldsymbol{\mbox{B}}_i\quad \boldsymbol{\mbox{d}}_i-</math><math>\boldsymbol{\mbox{B}}_1\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}</math>
 |-
 |<math>=\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-</math><math>\overset{\mbox{ˆ}}{\boldsymbol{\epsilon }}</math>
@@ Line 92: / Line 92: @@
 |}
-onde a matriz '''B'''  agrupa as matrizes '''B'''<sub>''i''</sub>  (deformação-deslocamento) convencionais do M.E.F., o vetor '''d'''  agrupa os vetores de deslocamentos nodais de todos os nós, '''d'''<sub>''i''</sub> , e <math display="inline">\overset{\mbox{ˆ}}{\boldsymbol{\epsilon }}=</math><math>\boldsymbol{\mbox{B}}_1\openbracketleft \boldsymbol{\mbox{u}}\openbracketright </math>  corresponde à parte das deformações associadas ao deslocamento de corpo rígido proveniente da descontinuidade. Segundo o sistema de referência local adotado, tem-se:
+onde a matriz '''B'''  agrupa as matrizes '''B'''<sub>''i''</sub>  (deformação-deslocamento) convencionais do M.E.F., o vetor '''d'''  agrupa os vetores de deslocamentos nodais de todos os nós, '''d'''<sub>''i''</sub> , e <math display="inline">\overset{\mbox{ˆ}}{\boldsymbol{\epsilon }}=\boldsymbol{\mbox{B}}_1\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}</math>  corresponde à parte das deformações associadas ao deslocamento de corpo rígido proveniente da descontinuidade. Segundo o sistema de referência local adotado, tem-se:
 <span id='eq0015'></span>
@@ Line 109: / Line 109: @@
 \frac{\partial N_1}{\partial y}\quad  & \frac{\partial N_1}{\partial x}
 \end{array}\right]\left\{\begin{array}{c}
-\openbracketleft {u\openbracketright }_x\\
+\mbox{〚 }{u\mbox{〛}}_x\\
-\openbracketleft {u\openbracketright }_y
+\mbox{〚 }{u\mbox{〛}}_y
 \end{array}\right\}=\frac{1}{l_e}\left\{\begin{array}{c}
 \\
-\openbracketleft {u\openbracketright }_y\\
+\mbox{〚 }{u\mbox{〛}}_y\\
-\openbracketleft {u\openbracketright }_x
+\mbox{〚 }{u\mbox{〛}}_x
 \end{array}\right\}</math>
 |}
@@ Line 187: / Line 187: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\boldsymbol{\mbox{t}}\left(\openbracketleft \boldsymbol{\mbox{u}}\openbracketright \right)-</math><math>\boldsymbol{\mbox{N}}^T\quad \boldsymbol{\sigma }=</math><math>\boldsymbol{0}</math>
+| <math>\boldsymbol{\mbox{t}}\left(\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}\right)-</math><math>\boldsymbol{\mbox{N}}^T\quad \boldsymbol{\sigma }=</math><math>\boldsymbol{0}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | ( 8)
@@ Line 200: / Line 200: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\boldsymbol{\mbox{f}}_{int}=\boldsymbol{\mbox{B}}^T\quad \boldsymbol{\mbox{D}}\quad \left(\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{M}}}{l_e}\openbracketleft \boldsymbol{\mbox{u}}\openbracketright \right)\quad A_e</math>
+| <math>\boldsymbol{\mbox{f}}_{int}=\boldsymbol{\mbox{B}}^T\quad \boldsymbol{\mbox{D}}\quad \left(\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{M}}}{l_e}\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}\right)\quad A_e</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | ( 9)
@@ Line 211: / Line 211: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\boldsymbol{\mbox{t}}\left(\openbracketleft \boldsymbol{\mbox{u}}\openbracketright \right)-</math><math>\boldsymbol{\mbox{N}}^T\quad \boldsymbol{\mbox{D}}\quad \left(\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{M}}}{l_e}\openbracketleft \boldsymbol{\mbox{u}}\openbracketright \right)=</math><math>\boldsymbol{0}</math>
+| <math>\boldsymbol{\mbox{t}}\left(\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}\right)-</math><math>\boldsymbol{\mbox{N}}^T\quad \boldsymbol{\mbox{D}}\quad \left(\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{M}}}{l_e}\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}\right)=</math><math>\boldsymbol{0}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | ( 10)
@@ Line 244: / Line 244: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\openbracketleft \dot{\boldsymbol{\mbox{u}}}\openbracketright =</math><math>{\left(\boldsymbol{\mbox{K}}^{{'}}+\boldsymbol{\mbox{N}}^T\boldsymbol{\mbox{DM}}/l_e\right)}^{-1}\boldsymbol{\mbox{N}}^T\boldsymbol{\mbox{DB}}\quad \dot{\boldsymbol{\mbox{d}}}</math>
+| <math>\mbox{〚 }\dot{\boldsymbol{\mbox{u}}}\mbox{〛}={\left(\boldsymbol{\mbox{K}}^{{'}}+\boldsymbol{\mbox{N}}^T\boldsymbol{\mbox{DM}}/l_e\right)}^{-1}\boldsymbol{\mbox{N}}^T\boldsymbol{\mbox{DB}}\quad \dot{\boldsymbol{\mbox{d}}}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | ( 12)
@@ Line 258: / Line 258: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>{\dot{\boldsymbol{\mbox{f}}}}_{inf}=\boldsymbol{\mbox{B}}^T\boldsymbol{\mbox{D}}\left(\boldsymbol{\mbox{B}}\dot{\boldsymbol{\mbox{d}}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{M}}}{l_e}\openbracketleft \dot{\boldsymbol{\mbox{u}}}\openbracketright \right)A_e</math>
+| <math>{\dot{\boldsymbol{\mbox{f}}}}_{inf}=\boldsymbol{\mbox{B}}^T\boldsymbol{\mbox{D}}\left(\boldsymbol{\mbox{B}}\dot{\boldsymbol{\mbox{d}}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{M}}}{l_e}\mbox{〚 }\dot{\boldsymbol{\mbox{u}}}\mbox{〛}\right)A_e</math>
 |-
 |<math>=\left[\boldsymbol{\mbox{B}}^T\boldsymbol{\mbox{DB}}-\right. </math><math>\left. \frac{1}{l_e}\boldsymbol{\mbox{B}}^T\boldsymbol{\mbox{DM}}\quad {\left(\boldsymbol{\mbox{K}}^{{'}}+\boldsymbol{\mbox{N}}^T\boldsymbol{\mbox{DM}}/l_e\right)}^{-1}\boldsymbol{\mbox{N}}^T\boldsymbol{\mbox{DB}}\right]A_e\quad \dot{\boldsymbol{\mbox{d}}}</math>
@@ Line 295: / Line 295: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\overset{\mbox{ˆ}}{\boldsymbol{\epsilon }}=\frac{\boldsymbol{\mbox{M}}}{l_e}\openbracketleft \boldsymbol{\mbox{u}}\openbracketright </math>
+| <math>\overset{\mbox{ˆ}}{\boldsymbol{\epsilon }}=\frac{\boldsymbol{\mbox{M}}}{l_e}\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | ( 15)
@@ Line 307: / Line 307: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\boldsymbol{\mbox{t}}\left(\openbracketleft \boldsymbol{\mbox{u}}\openbracketright \right)-</math><math>\boldsymbol{\mbox{M}}^T\quad \boldsymbol{\mbox{D}}\quad \left(\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{M}}}{l_e}\openbracketleft \boldsymbol{\mbox{u}}\openbracketright \right)=</math><math>\boldsymbol{0}</math>
+| <math>\boldsymbol{\mbox{t}}\left(\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}\right)-</math><math>\boldsymbol{\mbox{M}}^T\quad \boldsymbol{\mbox{D}}\quad \left(\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{M}}}{l_e}\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}\right)=</math><math>\boldsymbol{0}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | ( 16)
@@ Line 321: / Line 321: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\overset{\mbox{ˆ}}{\boldsymbol{\epsilon }}=\frac{\boldsymbol{\mbox{N}}}{l_e}\openbracketleft \boldsymbol{\mbox{u}}\openbracketright </math>
+| <math>\overset{\mbox{ˆ}}{\boldsymbol{\epsilon }}=\frac{\boldsymbol{\mbox{N}}}{l_e}\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | ( 17)
@@ Line 331: / Line 331: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>\boldsymbol{\mbox{t}}\left(\openbracketleft \boldsymbol{\mbox{u}}\openbracketright \right)-</math><math>\boldsymbol{\mbox{N}}^T\quad \boldsymbol{\mbox{D}}\quad \left(\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{N}}}{l_e}\openbracketleft \boldsymbol{\mbox{u}}\openbracketright \right)=</math><math>\boldsymbol{0}</math>
+| <math>\boldsymbol{\mbox{t}}\left(\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}\right)-</math><math>\boldsymbol{\mbox{N}}^T\quad \boldsymbol{\mbox{D}}\quad \left(\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-\right. </math><math>\left. \frac{\boldsymbol{\mbox{N}}}{l_e}\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}\right)=</math><math>\boldsymbol{0}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | ( 18)
@@ Line 349: / Line 349: @@
 {| style="text-align: center; margin:auto;"
 |-
-| <math>{\boldsymbol{\epsilon }}_S=\boldsymbol{\epsilon }+</math><math>\frac{\boldsymbol{\mbox{N}}}{k}\openbracketleft \boldsymbol{\mbox{u}}\openbracketright </math>
+| <math>{\boldsymbol{\epsilon }}_S=\boldsymbol{\epsilon }+</math><math>\frac{\boldsymbol{\mbox{N}}}{k}\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}</math>
 |-
-|<math>=\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-</math><math>\frac{\boldsymbol{\mbox{M}}}{l_e}\openbracketleft \boldsymbol{\mbox{u}}\openbracketright +</math><math>\frac{\boldsymbol{\mbox{N}}}{k}\openbracketleft \boldsymbol{\mbox{u}}\openbracketright </math>
+|<math>=\boldsymbol{\mbox{B}}\quad \boldsymbol{\mbox{d}}-</math><math>\frac{\boldsymbol{\mbox{M}}}{l_e}\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}+</math><math>\frac{\boldsymbol{\mbox{N}}}{k}\mbox{〚 }\boldsymbol{\mbox{u}}\mbox{〛}</math>
 |}
 | style="width: 5px;text-align: right;white-space: nowrap;" | ( 19)

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Manzoli 2011a - Revision history

Scipediacontent at 14:33, 13 April 2017

Scipediacontent at 14:32, 13 April 2017

Scipediacontent: Scipediacontent moved page Draft Content 624005357 to Manzoli 2011a

Scipediacontent: Created page with "====Resumo==== Estudam-se os aspectos referentes à qualidade e objetividade das respostas obtidas com as diferentes formulações (simétrica e não-simétrica) de elementos..."