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==Resumo==
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As superfícies de interação em resultantes de tensão podem ser de grande utilidade nos processos de análise estrutural, mas sua obtenção para pórticos planos ou espaciais, geralmente, é em esforços combinados de momentos fletores e normais. A literatura apresenta as superfícies em formas planas, quadriculares, complexas ou mistas para análise não linear de estruturas que apresentam os problemas de instabilidade local e global na execução. O modelo de regressão linear múltipla é um método que permite a obtenção de superfícies de interação em resultantes de tensão a partir de análises de elementos sólidos 3D. As análises elastoplásticas de pórticos planos ou espaciais usando estas superfícies facilitam os processos de análise estrutural para a execução de projetos com melhor segurança estrutural. Neste trabalho, a abordagem será para estruturas metálicas com superfícies em resultantes de tensão obtidas por análises não lineares do modelo de dano de vigas de Timoshenko 3D.
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'''Palavras-chave:''' Superfícies de interação; vigas de Timoshenko 3D; regressão linear múltipla; resultantes de tensões, pórticos metálicos; análise elastoplástica.
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==Abstract==
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The interaction surfaces in stress resultants can be of great use in the structural analysis processes, but their obtaining for plane or space frames, generally, is in combined efforts of bending moments and normal. The literature presents the surfaces in plane, quadric, complex or mixed forms for nonlinear analysis of structures that have the problems of local and global instability in the executing. The multiple linear regression model is a method that permit obtain interaction surfaces in the stress resultants from 3D solid element analysis. The plane and space frames elastoplastic analysis using these surfaces facilitate the structural analysis processes for the execution of projects with better structural safety. In this work, the approach will be for metallic structures with stress resultants surfaces obtained by Timoshenko 3D beams damage model non-linear analysis.
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'''Keywords: '''interaction curves, 3D Timoshenko beams, multiple linear regression, stress resultants, steel frame, elastoplastic analysis.
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<span id='_Ref457256930'></span>
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==1 Introdução==
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A análise elastoplástica com pórticos espaciais ou planos necessita da função de escoamento que controla o término da fase elástica e o estado plástico da estrutura. Usar superfícies de interação em resultantes de tensões é de mais fácil entendimento para os projetistas porque geralmente os esforços seccionais são apresentados nestas resultantes, a saber, momentos, cortantes e axial. O modelo de dano em vigas de Timoshenko em vigas 3D permite obter os esforços em resultantes de tensões com a versatilidade de poder usá-lo para estruturas de concreto armado ou aço, de acordo com os parâmetros adotados. No trabalho de Vieira [1] faz-se a aplicação da tese doutoral de Hanganu [2] para o caso de estruturas de aço com a definição do limite de dano quando o valor da função de endurecimento (<math display="inline">k\left( d\right) )</math> for igual a máxima resistência ao cortante octaédrica (<math display="inline">{\tau }_{oct}^{m\acute{a}x}(d)</math>). As verificações realizadas demonstraram que os limites plásticos foram atendidos. Este trabalho fará a aplicação das superfícies de interação em pórticos planos e espaciais, de modo que seja verificado se a regressão linear múltipla consegue demostrar se a função é de boa utilidade ou não na análise elastoplástica.
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==2 Métodos==
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A pesquisa foi desenvolvida com as formulações apresentadas nas seções <span id='cite-_Ref4583336'></span>[[#_Ref4583336|2.1]] a <span id='cite-_Ref4583347'></span>[[#_Ref4583347|2.6]] e 3 (três) estudos de casos na seção <span id='cite-_Ref477377056'></span>[[#_Ref477377056|2.7]].
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<span id='_Ref4583336'></span>
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===2.1 Funções de Escoamento===
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Na literatura, as funções de escoamento de uma seção retangular para combinações de esforços seccionais de momento fletor, axial, cortante e torção para pórticos planos e espaciais são apresentadas por Lubliner [3], Mrázik [4] e Crisfield [5] de forma resumida como:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>f={n}^{2}+{m}_{i}-1=0\, com\, i=x,y</math>
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|}
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|  style="width: 5px;text-align: right;white-space: nowrap;"|( 1 )
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|}
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Onde:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math display="inline">n=\frac{N}{{N}_{xp}}</math>; <math display="inline">{m}_{i}=</math><math>\frac{{M}_{i}}{{M}_{ip}}</math>
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|}
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|  style="width: 5px;text-align: right;white-space: nowrap;"|( 2 )
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|}
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com
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<math display="inline">N</math> = esforço normal atuante;
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<math display="inline">{N}_{xp}</math> = esforço axial de plastificação;
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<math display="inline">{M}_{i}</math> = esforço de momento atuante;
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<math display="inline">{M}_{ip}</math> = esforço de momento de plastificação;
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<math display="inline">x\, e\, y</math> = direções dos esforços no sistema de referência.
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Outras funções apresentam as interações de esforços seccionais como segue:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>f={n}^{2}+\frac{s}{\sqrt{3}}{m}_{i}n+{m}_{i}^{2}-1=0\,</math>
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|}
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|  style="width: 5px;text-align: right;white-space: nowrap;"|( 3 )
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|}
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>f={n}^{2}+3s{m}_{i}n+\frac{9}{4}{m}_{i}^{2}-1=0\,</math>
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|}
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|  style="width: 5px;text-align: right;white-space: nowrap;"|( 4 )
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|}
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com
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>s=\frac{{M}_{i}}{\left| {M}_{ip}\right| }\,</math>
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|}
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|  style="width: 5px;text-align: right;white-space: nowrap;"|( 5 )
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>f={m}_{z}+\frac{3}{4}{m}_{y}^{2}-1=0</math>
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|}
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|  style="width: 5px;text-align: right;white-space: nowrap;"|( 6 )
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|}
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>f={m}_{i}^{2}+{f}_{i}^{2}-1=0</math>
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|}
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|  style="width: 5px;text-align: right;white-space: nowrap;"|( 7 )
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|}
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{| class="formulaSCP" style="width: 100%; text-align: center;" 
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|-
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| <math>{f}_{i}=\frac{{F}_{i}}{{F}_{ip}}</math>
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|}
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<math display="inline">{F}_{i}</math> = força cortante atuante;
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<math display="inline">{F}_{ip}</math> = força cortante de plastificação.
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Existem dificuldades para a obtenção das funções de escoamento por ensaios experimentais, assim como por modelos computacionais porque as mesmas dependem da geometria da seção transversal e das propriedades do material. A abordagem baseada no modelo de dano em vigas de Timoshenko 3D para a obtenção das superfícies com regressão linear múltipla é apresentada em Vieira e Silva [6] e com mais detalhes em Vieira [1].
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===2.2 Modelo de dano ===
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Hanganu [2] desenvolve o modelo de dano isotrópico para problemas termicamente estáveis, na configuração material lagrangiana com pequenas deformações e deslocamentos com a descrição do dano pela variável <math display="inline">d</math> em função de uma superfície elementar com um volume de material degradado como na <span id='cite-_Ref529774485'></span>[[#_Ref529774485|'''Figura 1''']]:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>d=\frac{{S}_{n}-{\overline{S}}_{n}}{{S}_{n}}=1-\frac{{\overline{S}}_{n}}{{S}_{n}}</math>
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529712878'></span>( 8 )
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|}
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Onde
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<math display="inline">{S}_{n}</math> = área total da seção;
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<math display="inline">{\overline{S}}_{n}</math>= área resistente efetiva;
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<math display="inline">{S}_{n}-{\overline{S}}_{n}\,</math>  = área ocupada pelas aberturas.
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{| style="width: 100%;border-collapse: collapse;" 
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|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Vieira_908925676-image1.png|174px]] 
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|}
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<span id='_Ref529774485'></span>'''Figura 1'''. Superfície com dano.
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A relação de equilíbrio entre a tensão de Cauchy <math display="inline">\mathit{\boldsymbol{\sigma }}</math>''' '''e a tensão efetiva <math display="inline">\overline{\mathit{\boldsymbol{\sigma }}}</math>''' '''é mostrada pela equação <span id='cite-_Ref529712885'></span>[[#_Ref529712885|( 9 )]] e a <span id='cite-_Ref529775476'></span>[[#_Ref529775476|'''Figura 2''']]:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>\sigma S=\, \overline{\sigma }\boldsymbol{\, }\overline{S}</math>
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529712885'></span>( 9 )
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|}
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{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
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|-
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|  style="text-align: center;vertical-align: top;width: 49%;"|[[Image:Draft_Vieira_908925676-image2.png|258px]] 
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|  style="text-align: center;vertical-align: top;width: 50%;"|[[Image:Draft_Vieira_908925676-image3.png|258px]] 
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|-
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|  style="vertical-align: top;"|1) Região real com dano.
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|  style="text-align: center;vertical-align: top;"|2) Região equivalente sem dano.
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|}
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<span id='_Ref529775476'></span>'''Figura 2'''. Tensão de Cauchy <math display="inline">\sigma</math>  e tensão efetiva <math display="inline">\overline{\sigma }</math>.
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Fazendo as relações entre as equações <span id='cite-_Ref529712878'></span>[[#_Ref529712878|( 8 )]] e <span id='cite-_Ref529712885'></span>[[#_Ref529712885|( 9 )]] obtém-se:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math display="inline">\mathit{\boldsymbol{\sigma }}=\, \overline{\mathit{\boldsymbol{\sigma }}}\boldsymbol{\, }\left( 1-\right. </math><math>\left. d\right) =\left( 1-d\right) \mathit{\boldsymbol{E}}\epsilon</math> 
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 10 )
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|}
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Onde:
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<math display="inline">E</math> = módulo de elasticidade do material;
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<math display="inline">\epsilon</math>  = deformação do material.
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Para problemas termicamente estáveis é válida a inequação de Clasius-Planck para representar a dissipação ( <math display="inline">{\Xi }_{m}</math>), sempre crescente, com a potência dissipativa <math display="inline">\overset{\cdot}{{\Xi }_{m}}</math> sendo positiva em um ponto para a forma lagrangiana seguinte:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>\overset{\cdot}{{\Xi }_{m}}=\frac{1}{{m}_{0}}\, {\mathit{\boldsymbol{\sigma }}}^{T}\overset{\cdot}{\mathit{\boldsymbol{\epsilon }}}-</math><math>\overset{\cdot}{\boldsymbol{\Psi }}=\left( \frac{1}{{m}_{o}}{\mathit{\boldsymbol{\sigma }}}^{T}-\right. </math><math>\left. \frac{\partial \boldsymbol{\Psi }}{\partial \mathit{\boldsymbol{\epsilon }}}\right) \overset{\cdot}{\mathit{\boldsymbol{\epsilon }}}-</math><math>\frac{\partial \boldsymbol{\Psi }}{\partial d}\overset{\cdot}{d}\geq 0</math>
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529714747'></span>( 11 )
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|}
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Com
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<math display="inline">{\Psi }_{0}</math> = energia livre elástica de Helmholtz do material sem danos;
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<math display="inline">\Psi</math>  = energia livre de Helmholtz para um modelo com dano isotérmico;
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<math display="inline">{m}_{0}</math>= densidade na configuração material.
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O termo <math display="inline">\left( \frac{1}{{m}_{o}}{\mathit{\boldsymbol{\sigma }}}^{T}-\right. </math><math>\left. \frac{\partial \boldsymbol{\Psi }}{\partial \mathit{\boldsymbol{\epsilon }}}\right) \overset{\cdot}{\mathit{\boldsymbol{\epsilon }}}-</math><math>\frac{\partial \boldsymbol{\Psi }}{\partial d}\overset{\cdot}{d}\geq 0</math> necessita cumprir-se em qualquer variação temporal arbitrária da variável independente <math display="inline">\epsilon</math> . Assim, <math display="inline">\overset{\cdot}{\epsilon }</math> pode ser “zero” ou
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>\left( \frac{1}{{m}_{o}}{\mathit{\boldsymbol{\sigma }}}^{T}-\frac{\partial \boldsymbol{\Psi }}{\partial \mathit{\boldsymbol{\epsilon }}}\right) =</math><math>0</math>
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529778988'></span>( 12 )
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|}
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Desenvolvendo a equação <span id='cite-_Ref529778988'></span>[[#_Ref529778988|( 12 )]] chega-se a:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>\mathit{\boldsymbol{\sigma }}={m}_{o}{\left( \frac{\partial \boldsymbol{\Psi }}{\partial \epsilon }\right) }^{T}=</math><math>\left( 1-d\right) {\mathit{\boldsymbol{C}}}^{0}\mathit{\boldsymbol{\epsilon }}={\mathit{\boldsymbol{C}}}^{s}\mathit{\boldsymbol{\epsilon }}</math>
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 13 )
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|}
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Onde
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<math display="inline">{\mathit{\boldsymbol{C}}}^{s}</math>= <math display="inline">\left( 1-\right. </math><math>\left. d\right) {\mathit{\boldsymbol{C}}}^{0}</math> é a matriz constitutiva secante do material com dano.
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Por consequência o termo restante da equação <span id='cite-_Ref529714747'></span>[[#_Ref529714747|( 11 )]] torna-se em
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>\overset{\cdot}{{\Xi }_{m}}={-\frac{\partial \boldsymbol{\Psi }}{\partial d}\overset{\cdot}{d}\boldsymbol{=\Psi }}_{0}\overset{\cdot}{d}\geq 0</math>
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529779568'></span>( 14 )
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|}
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Pelas equações <span id='cite-_Ref529714747'></span>[[#_Ref529714747|( 11 )]] e <span id='cite-_Ref529779568'></span>[[#_Ref529779568|( 14 )]] o dano nunca pode diminuir, ou seja,  <math display="inline">\overset{\cdot}{d}\geq 0</math>.
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A função equivalente utilizada no modelo de Hanganu [2] é mostrada na <span id='cite-_Ref529715855'></span>[[#_Ref529715855|'''Figura 3''']] com <math display="inline">\acute{{f}_{t}}</math> e <math display="inline">\acute{{f}_{c}}</math> como resistências de tração e compressão, respectivamente.
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O termo <math display="inline">n</math> é
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>n=\frac{{f}_{c}}{{f}_{t}}</math>
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 15 )
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|}
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{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
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|  style="text-align: center;vertical-align: top;width: 100%;"|[[Image:Draft_Vieira_908925676-image4.png|402px]]
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<span id='_Ref529715855'></span>'''Figura 3'''. Função limite de dano no plano principal <math display="inline">{\sigma }_{1}-</math><math>{\sigma }_{2}</math>.
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|}
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A equação que a representa é a seguinte:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math display="inline">\overline{F}=\, G\left( \overline{\sigma }\right) -G({f}_{c})\leq 0</math> 
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 16 )
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|}
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Onde
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<math display="inline">G\left( \chi \right)</math>  = função escalar, inversível, positiva e derivada positiva, a determinar.
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A função de evolução do limite de dano, Hanganu [2], é mostrada na <span id='cite-_Ref529718486'></span>[[#_Ref529718486|'''Figura 4''']]:
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{| style="width: 100%;border-collapse: collapse;" 
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|-
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|  style="text-align: center;vertical-align: top;width: 100%;"|[[Image:Draft_Vieira_908925676-image5.png|432px]] 
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|}
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<span id='_Ref529718486'></span>'''Figura 4'''. Representação da função <math display="inline">G\left( \overline{\sigma }\right)</math>
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O presente trabalho foi focado em estruturas de aço. Desta feita, adotou-se o critério de von Mises que depende de somente um parâmetro, ou seja, a máxima resistência ao cortante octaédrica <math display="inline">{\tau }_{oct}^{m\acute{a}x}</math>, considerando somente o 2º invariante do tensor desviador de tensões <math display="inline">\, {J}_{2}</math>, desprezando a influência do 1º invariante do tensor de tensões e do 3º invariante do tensor desviador de tensões <math display="inline">{J}_{3}</math>. De acordo com este critério, se alcança
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o limite do dano quando o valor da função de endurecimento <math display="inline">\kappa \, (d)</math> alcança a máxima resistência ao cortante octaédrico <math display="inline">{\tau }_{oct}^{m\acute{a}x}\, (d)</math>.
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>\kappa \, (d)={\tau }_{oct}^{m\acute{a}x}(d)</math>
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 17 )
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|}
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Este critério é representado na equação  <span id='cite-_Ref529821017'></span>[[#_Ref529821017|( 18 )]]:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>F\left( {\mathit{\boldsymbol{J}}}_{\mathit{\boldsymbol{2}}};\overline{\sigma }\right) =</math><math>f\left( {J}_{2}\right) -\overline{\sigma }d=\sqrt{3{\mathit{\boldsymbol{J}}}_{\mathit{\boldsymbol{2}}}}-</math><math>\overline{\sigma }d=0</math>
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529821017'></span>( 18 )
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|}
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===2.3 Superfícies de interação por regressão linear múltipla===
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Na obtenção das superfícies foram feitas várias combinações de carregamentos de forma a ter um grupo de pontos para gerar a superfície proposta, ou seja, pontos que tenham alcançado a superfície de escoamento. Para um dado carregamento, obtém-se um ponto, como por exemplo o ponto 1 da <span id='cite-_Ref529720062'></span>[[#_Ref529720062|'''Figura 5''']], cujas coordenadas (n1, m1) são o esforço axial e momento fletor respectivamente. Mais detalhes sobre os processos de obtenção das superfícies podem ser lidos em Vieira [1].
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{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
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|-
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|  style="text-align: center;vertical-align: top;"|''' [[Image:Draft_Vieira_908925676-image6.png|420px]] '''
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|}
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<span id='_Ref529720062'></span>'''Figura 5'''. Pontos gerados para criar a função de escoamento (caso uniaxial).
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A superfície para o caso da <span id='cite-_Ref529720062'></span>[[#_Ref529720062|'''Figura 5''']] tem a seguinte descrição no formato do modelo de regressão linear múltipla:
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>f={\beta }_{1}{n}^{2}+{\beta }_{2}m-1=0</math>
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|}
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|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 19 )
384
|}
385
386
387
Onde
388
389
<math display="inline">n\, e\, m</math> = esforços normal e fletor adimensionais, respectivamente;
390
391
<math display="inline">{\beta }_{1}\, e\, {\beta }_{2}</math> = coeficientes obtidos pela regressão linear múltipla.
392
393
Muitas aplicações da análise de regressão envolvem situações em que há mais de uma variável de regressão. Um modelo de regressão que contém mais de um regressor recebe o nome de modelo de regressão múltipla como por exemplo em Montgomery [7].
394
395
O modelo desenvolvido para a formulação pretendida tem a seguinte forma:
396
397
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
398
|-
399
| 
400
{| style="text-align: center; margin:auto;width: 100%;"
401
|-
402
| <math>f={\beta }_{1}{{\overline{n}}_{x}}^{{\beta }_{13}}+{\beta }_{2}{{\overline{f}}_{y}}^{{\beta }_{14}}+</math><math>{\beta }_{3}{{\overline{f}}_{z}}^{{\beta }_{15}}+{\beta }_{4}{{\overline{m}}_{x}}^{{\beta }_{16}}+</math><math>{\beta }_{5}{{\overline{m}}_{y}}^{{\beta }_{17}}+{\beta }_{6}{{\overline{m}}_{z}}^{{\beta }_{18}}+</math><math>{\beta }_{7}{{{\overline{n}}_{x}}^{{\beta }_{19}}\, {\overline{m}}_{x}}^{{\beta }_{20}}+</math><math>{\beta }_{8}{{{\overline{n}}_{x}}^{{\beta }_{21}}{\overline{m}}_{y}}^{{\beta }_{22}}+</math><math>{\beta }_{9}{{{\overline{n}}_{x}}^{{\beta }_{23}}{\overline{m}}_{z}}^{{\beta }_{24}}+</math><math>{\beta }_{10}{\, {\overline{m}}_{x}}^{{\beta }_{25}}{{\overline{m}}_{y}}^{{\beta }_{26}}+</math><math>{\beta }_{11}{\, {\overline{m}}_{x}}^{{\beta }_{27}}{{\overline{m}}_{z}}^{{\beta }_{28}}+</math><math>{\beta }_{12}{{\overline{m}}_{y}}^{{\beta }_{29}}{{\overline{m}}_{z}}^{{\beta }_{30}}-1=</math><math>0</math>
403
404
405
|}
406
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529721691'></span>( 20 )
407
|}
408
409
410
Onde:
411
412
<math display="inline">{\overline{n}}_{x}=\frac{{n}_{x}}{{n}_{xp}}</math> com <math display="inline">{n}_{x}</math> e <math display="inline">{n}_{xp}</math> como o esforço axial atuante e plástico;
413
414
<math display="inline">{\overline{f}}_{y}=\frac{{f}_{y}}{{f}_{yp}}</math> com <math display="inline">{f}_{y}</math> e <math display="inline">{f}_{yp}</math> como o esforço cortante atuante e plástico;
415
416
<math display="inline">{\overline{f}}_{z}=\frac{{f}_{z}}{{f}_{zp}}</math> com <math display="inline">{f}_{z}</math> e <math display="inline">{f}_{zp}</math> como o esforço cortante atuante e plástico;
417
418
<math display="inline">{\overline{m}}_{x}=\frac{{m}_{x}}{{m}_{xp}}</math> com <math display="inline">{m}_{x}</math> e <math display="inline">{m}_{xp}</math> como o momento torçor atuante e plástico;
419
420
<math display="inline">{\overline{m}}_{y}=\frac{{m}_{y}}{{m}_{yp}}</math>  com <math display="inline">{m}_{y}</math> e <math display="inline">{m}_{yp}</math> como o momento fletor atuante e plástico;
421
422
<math display="inline">{\overline{m}}_{z}=\frac{{m}_{z}}{{m}_{zp}}</math> com <math display="inline">{m}_{z}</math> e <math display="inline">{m}_{zp}</math> como o momento fletor atuante e plástico.
423
424
Na regressão as observações da equação <span id='cite-_Ref529721691'></span>[[#_Ref529721691|( 20 )]] podem ser apresentadas como
425
426
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
427
|-
428
| 
429
{| style="text-align: center; margin:auto;width: 100%;"
430
|-
431
| <math>Y={\beta }_{0}+{\beta }_{1}{x}_{i1}+{\beta }_{2}{x}_{i2}+\cdots +{\beta }_{k}{x}_{ik}+</math><math>{\epsilon }_{i}=0</math>
432
|}
433
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 21 )
434
|}
435
436
437
Com <math display="inline">i=1,2,\, \cdots ,n</math>
438
439
Onde
440
441
<math display="inline">n</math> = número de observações (ensaios);
442
443
<math display="inline">{\beta }_{k}</math> = coeficientes de regressão da resposta <math display="inline">Y</math>;
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<math display="inline">k\,</math> = variáveis independes (regressores): <math display="inline">{\overline{f}}_{x}</math>; <math display="inline">{\overline{f}}_{y}</math>; <math display="inline">{\overline{f}}_{z}</math>; <math display="inline">\, {\overline{m}}_{x}</math>; <math display="inline">\, {\overline{m}}_{y}</math>; <math display="inline">{\overline{m}}_{z}</math> e suas combinações;
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447
<math display="inline">{\epsilon }_{i}</math> = erros do modelo.
448
449
O enfoque matricial da formulação é mostrado como segue:
450
451
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
452
|-
453
| 
454
{| style="text-align: center; margin:auto;width: 100%;"
455
|-
456
| <math>\mathit{\boldsymbol{Y}}=\mathit{\boldsymbol{X\beta }}+\mathit{\boldsymbol{\epsilon }}=</math><math>0</math>
457
|}
458
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 22 )
459
|}
460
461
462
Com
463
464
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
465
|-
466
| 
467
{| style="text-align: center; margin:auto;width: 100%;"
468
|-
469
| <math>\mathit{\boldsymbol{Y=\, }}\left[ \begin{matrix}\begin{matrix}{y}_{1}\\{y}_{2}\\\vdots \end{matrix}\\{y}_{n}\end{matrix}\right] ;\mathit{\boldsymbol{\, X}}=</math><math>\left[ \begin{matrix}1\\1\\\begin{matrix}\vdots \\1\end{matrix}\end{matrix}\begin{matrix}{x}_{11}\\{x}_{21}\\\begin{matrix}\vdots \\{x}_{n1}\end{matrix}\end{matrix}\begin{matrix}{x}_{12}\\{x}_{22}\\\begin{matrix}\vdots \\{x}_{n2}\end{matrix}\end{matrix}\begin{matrix}\cdots \\\cdots \\\begin{matrix}\vdots \\\cdots \end{matrix}\end{matrix}\begin{matrix}{x}_{1k}\\{x}_{2k}\\\begin{matrix}\vdots \\{x}_{nk}\end{matrix}\end{matrix}\right]</math> 
470
|}
471
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 23 )
472
|}
473
474
475
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
476
|-
477
| 
478
{| style="text-align: center; margin:auto;width: 100%;"
479
|-
480
| <math>\mathit{\boldsymbol{\beta =\, }}\left[ \begin{matrix}\begin{matrix}{\beta }_{0}\\{\beta }_{1}\\\vdots \end{matrix}\\{\beta }_{k}\end{matrix}\right] ;\mathit{\boldsymbol{\epsilon }}=</math><math>\left[ \begin{matrix}{\epsilon }_{1}\\{\epsilon }_{2}\\\begin{matrix}\vdots \\{\epsilon }_{n}\end{matrix}\end{matrix}\right]</math> 
481
|}
482
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 24 )
483
|}
484
485
486
Onde
487
488
<math display="inline">\mathit{\boldsymbol{Y}}</math> = é o vetor de observações de dimensão <math display="inline">(n\times 1)</math>;
489
490
<math display="inline">\mathit{\boldsymbol{X}}</math>''' =''' é o tensor (matriz) de dimensão <math display="inline">(n\times p)</math> dos níveis das variáveis independentes;
491
492
<math display="inline">\mathit{\boldsymbol{\beta }}</math> = é o vetor dos coeficientes de regressão de dimensão <math display="inline">(p\times 1)</math>;
493
494
<math display="inline">\mathit{\boldsymbol{\epsilon }}</math> = é o vetor dos erros aleatórios de dimensão <math display="inline">(n\times 1)</math>.
495
496
Deve-se encontrar o vetor dos estimadores dos mínimos quadrado, <math display="inline">\hat{\mathit{\boldsymbol{\beta }}}</math>, que minimiza
497
498
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
499
|-
500
| 
501
{| style="text-align: center; margin:auto;width: 100%;"
502
|-
503
| <math>\mathit{\boldsymbol{L}}=\sum _{i=1}^{n}{\mathit{\boldsymbol{\epsilon }}}_{\mathit{\boldsymbol{i}}}^{\mathit{\boldsymbol{2}}}=</math><math>{\mathit{\boldsymbol{\epsilon }}}^{T}\mathit{\boldsymbol{\epsilon }}={\left( \mathit{\boldsymbol{Y}}-\mathit{\boldsymbol{X\beta }}\right) }^{T}\left( \mathit{\boldsymbol{Y}}-\right. </math><math>\left. \mathit{\boldsymbol{X\beta }}\right)</math> 
504
|}
505
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 25 )
506
|}
507
508
509
Desenvolvendo os cálculos chega-se a:
510
511
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
512
|-
513
| 
514
{| style="text-align: center; margin:auto;width: 100%;"
515
|-
516
| <math>\hat{\mathit{\boldsymbol{\beta }}}={\left( {\mathit{\boldsymbol{X}}}^{\mathit{\boldsymbol{T}}}\mathit{\boldsymbol{X}}\right) }^{-1}{\mathit{\boldsymbol{X}}}^{T}\mathit{\boldsymbol{Y}}</math>
517
|}
518
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 26 )
519
|}
520
521
522
Mais detalhes dos processos de cálculo podem ser lidos em Montgomery [7].
523
524
O modelo ajustado passa a ter a seguinte forma;
525
526
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
527
|-
528
| 
529
{| style="text-align: center; margin:auto;width: 100%;"
530
|-
531
| <math display="inline">{\hat{Y}}_{i}={\hat{\beta }}_{0}+\sum _{j=1}^{n}{\hat{\beta }}_{j}{x}_{ij}</math> 
532
|}
533
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 27 )
534
|}
535
536
537
Com <math display="inline">i=1,\, 2,\, \cdots ,n</math>.
538
539
Os testes de hipóteses utilizados são o estatístico de prova
540
541
“F” e os de coeficientes individuais “t” que podem ser compreendidos com detalhes em Montgomery [7].
542
543
As superfícies e seus os resultados estatísticos (Tabelas de 1 a 6) que serão usados nas análises elastoplásticas, obtidos em Vieira [1], são as seguintes:
544
545
{| class="formulaSCP" style="width: 72%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
546
|-
547
| 
548
{| style="text-align: center; margin:auto;width: 100%;"
549
|-
550
| <math display="inline">{f}_{1}=1,010{n}^{2}+0,968{m}_{y}^{2}+0,981{m}_{z}^{2}+</math><math>0,514n{m}_{y}+0,43n{m}_{z}-1=0</math> 
551
|}
552
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref533151259'></span>( 28 )
553
|}
554
555
556
557
'''Tabela 1'''. Prova de significância da superfície <math display="inline">{f}_{1}</math>
558
559
{| style="width: 62%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
560
|-
561
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
562
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
563
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
564
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
565
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
566
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
567
|-
568
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
569
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|23,860
570
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|3
571
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|7,953
572
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1183,674
573
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
574
|-
575
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
576
|  style="text-align: center;vertical-align: top;"|0,1400
577
|  style="text-align: center;vertical-align: top;"|21
578
|  style="text-align: center;vertical-align: top;"|0,007
579
|  style="text-align: center;vertical-align: top;"|
580
|  style="text-align: center;vertical-align: top;"|
581
|-
582
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
583
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24,000
584
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24
585
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
586
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
587
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
588
|-
589
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
590
|-
591
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
592
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Estimado'''
593
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
594
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
595
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
596
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
597
|-
598
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
599
|  style="text-align: center;vertical-align: top;"|1,1580
600
|  style="text-align: center;vertical-align: top;"|0,0377
601
|  style="text-align: center;vertical-align: top;"|30,740
602
|  style="text-align: center;vertical-align: top;"|0,000
603
|  style="text-align: center;vertical-align: top;"|
604
|-
605
|  style="text-align: right;vertical-align: top;"|<math>{m}_{y}^{2}</math>
606
|  style="text-align: center;vertical-align: top;"|1,1180
607
|  style="text-align: center;vertical-align: top;"|0,0387
608
|  style="text-align: center;vertical-align: top;"|28,900
609
|  style="text-align: center;vertical-align: top;"|0,000
610
|  style="text-align: center;vertical-align: top;"|
611
|-
612
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
613
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,1240
614
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0381
615
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|29,530
616
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
617
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
618
|}
619
620
621
{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
622
|-
623
| 
624
{| style="text-align: center; margin:auto;width: 100%;"
625
|-
626
| <math display="inline">{f}_{2}=1,158{n}^{2}+1,118{m}_{y}^{2}+1,124{m}_{z}^{2}-1=</math><math>0</math> 
627
|}
628
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref533151260'></span>( 29 )
629
|}
630
631
632
'''Tabela 2'''. Prova de significância da superfície <math display="inline">{f}_{2}</math>
633
634
{| style="width: 62%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
635
|-
636
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
637
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
638
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
639
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
640
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
641
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
642
|-
643
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
644
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|24,000
645
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5
646
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|4,800
647
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|55913,754
648
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
649
|-
650
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
651
|  style="text-align: center;vertical-align: top;"|0,000
652
|  style="text-align: center;vertical-align: top;"|19
653
|  style="text-align: center;vertical-align: top;"|0,000
654
|  style="text-align: center;vertical-align: top;"|
655
|  style="text-align: center;vertical-align: top;"|
656
|-
657
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
658
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24,000
659
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24
660
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
661
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
662
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
663
|-
664
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
665
|-
666
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
667
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
668
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
669
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
670
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
671
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
672
|-
673
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
674
|  style="text-align: center;vertical-align: top;"|1,0100
675
|  style="text-align: center;vertical-align: top;"|0,0056
676
|  style="text-align: center;vertical-align: top;"|179,500
677
|  style="text-align: center;vertical-align: top;"|0,000
678
|  style="text-align: center;vertical-align: top;"|
679
|-
680
|  style="text-align: right;vertical-align: top;"|<math>{m}_{y}^{2}</math>
681
|  style="text-align: center;vertical-align: top;"|0,9680
682
|  style="text-align: center;vertical-align: top;"|0,0086
683
|  style="text-align: center;vertical-align: top;"|113,100
684
|  style="text-align: center;vertical-align: top;"|0,000
685
|  style="text-align: center;vertical-align: top;"|
686
|-
687
|  style="text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
688
|  style="text-align: center;vertical-align: top;"|0,9810
689
|  style="text-align: center;vertical-align: top;"|0,0085
690
|  style="text-align: center;vertical-align: top;"|115,500
691
|  style="text-align: center;vertical-align: top;"|0,000
692
|  style="text-align: center;vertical-align: top;"|
693
|-
694
|  style="text-align: right;vertical-align: top;"|<math>n{m}_{y}</math>
695
|  style="text-align: center;vertical-align: top;"|0,5140
696
|  style="text-align: center;vertical-align: top;"|0,0312
697
|  style="text-align: center;vertical-align: top;"|16,500
698
|  style="text-align: center;vertical-align: top;"|0,000
699
|  style="text-align: center;vertical-align: top;"|
700
|-
701
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>n{m}_{z}</math>
702
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
703
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0305
704
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|14,100
705
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
706
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
707
|}
708
709
710
{| class="formulaSCP" style="width: 85%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
711
|-
712
| 
713
{| style="text-align: center; margin:auto;width: 100%;"
714
|-
715
| <math display="inline">{f}_{3}=1,\, 014{n}^{2}+0,966{m}_{y}^{2}+0,982{m}_{z}^{2}+</math><math>0,506n{m}_{y}+0,404n{m}_{z}+0,038{m}_{y}{m}_{z}-1=0</math> 
716
|}
717
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref533151262'></span>( 30 )
718
|}
719
720
721
'''Tabela 3.''' Prova de significância da superfície <math display="inline">{f}_{3}</math>
722
723
{| style="width: 62%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
724
|-
725
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
726
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
727
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
728
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
729
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
730
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
731
|-
732
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
733
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|24,000
734
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|6
735
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|4,000
736
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|53308,230
737
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
738
|-
739
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
740
|  style="text-align: center;vertical-align: top;"|0,000
741
|  style="text-align: center;vertical-align: top;"|18
742
|  style="text-align: center;vertical-align: top;"|0,000
743
|  style="text-align: center;vertical-align: top;"|
744
|  style="text-align: center;vertical-align: top;"|
745
|-
746
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
747
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24,000
748
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24
749
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
750
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
751
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
752
|-
753
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Prova dos coeficientes individuais'''
754
|-
755
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
756
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
757
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
758
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
759
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
760
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
761
|-
762
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
763
|  style="text-align: center;vertical-align: top;"|1,0140
764
|  style="text-align: center;vertical-align: top;"|0,00552
765
|  style="text-align: center;vertical-align: top;"|183,500
766
|  style="text-align: center;vertical-align: top;"|0,000
767
|  style="text-align: center;vertical-align: top;"|
768
|-
769
|  style="text-align: right;vertical-align: top;"|<math>{m}_{y}^{2}</math>
770
|  style="text-align: center;vertical-align: top;"|0,9660
771
|  style="text-align: center;vertical-align: top;"|0,00806
772
|  style="text-align: center;vertical-align: top;"|119,800
773
|  style="text-align: center;vertical-align: top;"|0,000
774
|  style="text-align: center;vertical-align: top;"|
775
|-
776
|  style="text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
777
|  style="text-align: center;vertical-align: top;"|0,9820
778
|  style="text-align: center;vertical-align: top;"|0,00795
779
|  style="text-align: center;vertical-align: top;"|123,500
780
|  style="text-align: center;vertical-align: top;"|0,000
781
|  style="text-align: center;vertical-align: top;"|
782
|-
783
|  style="text-align: right;vertical-align: top;"|<math>n{m}_{y}</math>
784
|  style="text-align: center;vertical-align: top;"|0,5060
785
|  style="text-align: center;vertical-align: top;"|0,02951
786
|  style="text-align: center;vertical-align: top;"|17,100
787
|  style="text-align: center;vertical-align: top;"|0,000
788
|  style="text-align: center;vertical-align: top;"|
789
|-
790
|  style="text-align: right;vertical-align: top;"|<math>n{m}_{z}</math>
791
|  style="text-align: center;vertical-align: top;"|0,4040
792
|  style="text-align: center;vertical-align: top;"|0,03146
793
|  style="text-align: center;vertical-align: top;"|12,800
794
|  style="text-align: center;vertical-align: top;"|0,000
795
|  style="text-align: center;vertical-align: top;"|
796
|-
797
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{y}{m}_{z}</math>
798
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0380
799
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,01980
800
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,900
801
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
802
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
803
|}
804
805
806
{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
807
|-
808
| 
809
{| style="text-align: center; margin:auto;width: 100%;"
810
|-
811
| <math display="inline">{f}_{4}=1,012{n}^{2}+1,027{m}_{z}^{2}-1=0</math> 
812
|}
813
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref531207210'></span>( 31 )
814
|}
815
816
817
'''Tabela 4'''. Prova de significância da superfície <math display="inline">{f}_{4}</math>
818
819
{| style="width: 66%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
820
|-
821
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
822
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
823
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
824
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
825
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
826
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
827
|-
828
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
829
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|12,000
830
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|2
831
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|6,000
832
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|84325,969
833
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
834
|-
835
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
836
|  style="text-align: center;vertical-align: top;"|0,000
837
|  style="text-align: center;vertical-align: top;"|10
838
|  style="text-align: center;vertical-align: top;"|0,000
839
|  style="text-align: center;vertical-align: top;"|
840
|  style="text-align: center;vertical-align: top;"|
841
|-
842
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
843
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12,000
844
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12
845
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
846
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
847
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
848
|-
849
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
850
|-
851
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
852
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
853
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
854
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
855
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
856
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
857
|-
858
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
859
|  style="text-align: center;vertical-align: top;"|1,0120
860
|  style="text-align: center;vertical-align: top;"|0,0043
861
|  style="text-align: center;vertical-align: top;"|235, 300
862
|  style="text-align: center;vertical-align: top;"|0,000
863
|  style="text-align: center;vertical-align: top;"|
864
|-
865
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
866
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,0270
867
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0044
868
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|233, 800
869
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
870
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
871
|}
872
873
874
{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
875
|-
876
| 
877
{| style="text-align: center; margin:auto;width: 100%;"
878
|-
879
| <math display="inline">{f}_{5}=1,\, 242{n}^{2}+1,\, 087{m}_{z}^{2}-1=0</math> 
880
|}
881
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref531207213'></span>( 32 )
882
|}
883
884
885
'''Tabela 5'''. Prova de significância da superfície <math display="inline">{f}_{5}</math>
886
887
{| style="width: 61%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
888
|-
889
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
890
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
891
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
892
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
893
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
894
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
895
|-
896
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
897
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|11,960
898
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|2
899
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5,979
900
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1455,211
901
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
902
|-
903
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
904
|  style="text-align: center;vertical-align: top;"|0,040
905
|  style="text-align: center;vertical-align: top;"|10
906
|  style="text-align: center;vertical-align: top;"|0,004
907
|  style="text-align: center;vertical-align: top;"|
908
|  style="text-align: center;vertical-align: top;"|
909
|-
910
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
911
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12,000
912
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12
913
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
914
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
915
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
916
|-
917
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
918
|-
919
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
920
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
921
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
922
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
923
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
924
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
925
|-
926
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
927
|  style="text-align: center;vertical-align: top;"|1,2420
928
|  style="text-align: center;vertical-align: top;"|0,0309
929
|  style="text-align: center;vertical-align: top;"|40,220
930
|  style="text-align: center;vertical-align: top;"|0,000
931
|  style="text-align: center;vertical-align: top;"|
932
|-
933
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
934
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,0870
935
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0355
936
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|30,610
937
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
938
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
939
|}
940
941
942
{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
943
|-
944
| 
945
{| style="text-align: center; margin:auto;width: 100%;"
946
|-
947
| <math display="inline">{f}_{6}=1,089n+\, 0,929{m}_{z}^{2}-1=0</math> 
948
|}
949
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref531207215'></span>( 33 )
950
|}
951
952
953
'''Tabela 6'''. Prova de significância da superfície <math display="inline">{f}_{6}</math>
954
955
{| style="width: 61%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
956
|-
957
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
958
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
959
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
960
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
961
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
962
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
963
|-
964
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
965
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|11,990
966
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|2
967
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5,996
968
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|8547,240
969
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
970
|-
971
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
972
|  style="text-align: center;vertical-align: top;"|0,010
973
|  style="text-align: center;vertical-align: top;"|10
974
|  style="text-align: center;vertical-align: top;"|0,001
975
|  style="text-align: center;vertical-align: top;"|
976
|  style="text-align: center;vertical-align: top;"|
977
|-
978
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
979
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12,000
980
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12
981
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
982
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
983
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
984
|-
985
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
986
|-
987
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
988
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
989
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
990
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
991
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
992
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
993
|-
994
|  style="text-align: right;vertical-align: top;"|<math>n</math>
995
|  style="text-align: center;vertical-align: top;"|1,0890
996
|  style="text-align: center;vertical-align: top;"|0,0112
997
|  style="text-align: center;vertical-align: top;"|97,600
998
|  style="text-align: center;vertical-align: top;"|0,000
999
|  style="text-align: center;vertical-align: top;"|
1000
|-
1001
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
1002
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,9290
1003
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0150
1004
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|62,040
1005
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
1006
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
1007
|}
1008
1009
===2.4 Análise elastoplástica de estruturas de pórticos===
1010
1011
Uma superfície de interação define o estado último de uma seção transversal e depende dos seguintes fatores:
1012
1013
:1. Forma geométrica da seção transversal;
1014
1015
:2. Combinação dos esforços seccionais que atuam na seção transversal;
1016
1017
:3. Teoria de viga utilizada.
1018
1019
Encontram-se soluções analíticas fechadas para determinados tipos de seções (I, Retangular, etc) com casos especiais de combinações de esforços, tais como momentos fletores e esforço normal Horne [8], Lubliner [3] e Neal [9]. Neste trabalho, assume-se uma superfície descrita na equação <span id='cite-_Ref529721691'></span>[[#_Ref529721691|( 20 )]] em função dos esforços seccionais.
1020
1021
A análise elastoplástica segue os conceitos apresentados no trabalho de Silva [10] com as seguintes considerações;
1022
1023
:1) Os esforços seccionais contidos no interior da superfície de interação geram somente deformações elásticas;
1024
1025
:2) Os esforços seccionais que estejam na superfície de interação geram deformações plásticas;
1026
1027
:3) Os esforços seccionais fora da superfície de interação representam estados de tensões inadmissíveis porque não se leva em conta o caso do endurecimento.
1028
1029
Durante o processo de aplicação do carregamento em passos de carga os esforços seccionais em alguns nós dos elementos da estrutura poderão sair da superfície de interação. Para trazer estes esforços seccionais de volta a superfície utiliza-se o método de Backward Euler que necessita das derivadas primeira e segunda da superfície em relação aos esforços seccionais.
1030
1031
====2.4.1 Derivadas de primeira ordem====
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1033
Baseando-se na equação <span id='cite-_Ref529721691'></span>[[#_Ref529721691|( 20 )]] são obtidas as derivadas de primeira ordem da superfície de interação em relação aos esforços seccionais:
1034
{| class="formulaSCP" style="width: 100%; text-align: center;" 
1035
|-
1036
| <math>f={\beta }_{1}{{\overline{f}}_{x}}^{{\beta }_{13}}+{\beta }_{2}{{\overline{f}}_{y}}^{{\beta }_{14}}+</math><math>{\beta }_{3}{{\overline{f}}_{z}}^{{\beta }_{15}}+{\beta }_{4}{{\overline{m}}_{x}}^{{\beta }_{16}}+</math><math>{\beta }_{5}{{\overline{m}}_{y}}^{{\beta }_{17}}+{\beta }_{6}{{\overline{m}}_{z}}^{{\beta }_{18}}+</math><math>{\beta }_{7}{{{\overline{f}}_{x}}^{{\beta }_{19}}\, {\overline{m}}_{x}}^{{\beta }_{20}}+</math><math>{\beta }_{8}{{{\overline{f}}_{x}}^{{\beta }_{21}}{\overline{m}}_{y}}^{{\beta }_{22}}+</math><math>{\beta }_{9}{{{\overline{f}}_{x}}^{{\beta }_{23}}{\overline{m}}_{z}}^{{\beta }_{24}}+</math><math>{\beta }_{10}{\, {\overline{m}}_{x}}^{{\beta }_{25}}{{\overline{m}}_{y}}^{{\beta }_{26}}+</math><math>{\beta }_{11}{\, {\overline{m}}_{x}}^{{\beta }_{27}}{{\overline{m}}_{z}}^{{\beta }_{28}}+</math><math>{\beta }_{12}{{\overline{m}}_{y}}^{{\beta }_{29}}{{\overline{m}}_{z}}^{{\beta }_{30}}-1=</math><math>0</math>
1037
|}
1038
1039
1040
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1041
|-
1042
| 
1043
{| style="text-align: center; margin:auto;width: 100%;"
1044
|-
1045
| <math>\frac{\partial f}{{f}_{x}}=\frac{1}{{f}_{x}}\left( s{f}_{x}\left( {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{19}}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{20}\, }s{m}_{x}{\beta }_{7}{\beta }_{19}+\right. \right. </math><math>\left. \left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{\beta 21}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{\beta 22}s{m}_{y}{\beta }_{8}{\beta }_{21}\, +\right. \right. </math><math>\left. \left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{23}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{24}}s{m}_{z}{\beta }_{9}{\beta }_{23}+\right. \right. </math><math>\left. \left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{13}}{\beta }_{1}{\beta }_{13}\right) \right)</math> 
1046
|}
1047
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 34 )
1048
|}
1049
1050
1051
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1052
|-
1053
| 
1054
{| style="text-align: center; margin:auto;width: 100%;"
1055
|-
1056
| <math>\frac{\partial f}{{f}_{y}}=\frac{s{f}_{y}{\beta }_{2}{\left( \frac{{f}_{y}}{{f}_{yp}}\right) }^{{\beta }_{14}}{\beta }_{14}}{{f}_{y}}</math>
1057
|}
1058
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 35 )
1059
|}
1060
1061
1062
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1063
|-
1064
| 
1065
{| style="text-align: center; margin:auto;width: 100%;"
1066
|-
1067
| <math>\frac{\partial f}{{f}_{z}}=\frac{s{f}_{z}{\beta }_{3}{\left( \frac{{f}_{z}}{{f}_{zp}}\right) }^{{\beta }_{15}}{\beta }_{15}}{{f}_{z}}</math>
1068
|}
1069
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 36 )
1070
|}
1071
1072
1073
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1074
|-
1075
| 
1076
{| style="text-align: center; margin:auto;width: 100%;"
1077
|-
1078
| <math>\frac{\partial f}{{m}_{x}}=\frac{1}{{m}_{x}}\left( s{m}_{x}\left( {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{25}}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{26}}s{m}_{y}{\beta }_{10}{\beta }_{25}+\right. \right. </math><math>\left. \left. {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{27}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{28}}s{m}_{z}{\beta }_{11}{\beta }_{27}+\right. \right. </math><math>\left. \left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{19}}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{20}}s{f}_{x}{\beta }_{7}{\beta }_{20}+\right. \right. </math><math>\left. \left. {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{16}}{\beta }_{4}{\beta }_{16}\right) \, \right) \, \,</math> 
1079
|}
1080
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 37 )
1081
|}
1082
1083
1084
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1085
|-
1086
| 
1087
{| style="text-align: center; margin:auto;width: 100%;"
1088
|-
1089
| <math>\frac{\partial f}{{m}_{y}}=\frac{1}{{m}_{y}}\left( s{m}_{y}\left( {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{25}}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{26}}s{m}_{x}{\beta }_{10}{\beta }_{26}+\right. \right. </math><math>\left. \left. {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{29}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{30}}s{m}_{z}{\beta }_{12}{\beta }_{29}+\right. \right. </math><math>\left. \left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{21}}\, {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{22}}s{f}_{x}{\beta }_{8}{\beta }_{22}+\right. \right. </math><math>\left. \left. {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{\beta 17}{\beta }_{5}{\beta }_{17}\right) \right) \, \,</math> 
1090
|}
1091
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 38 )
1092
|}
1093
1094
1095
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1096
|-
1097
| 
1098
{| style="text-align: center; margin:auto;width: 100%;"
1099
|-
1100
| <math>\frac{\partial f}{{m}_{z}}=\frac{1}{{m}_{z}}\left( s{m}_{z}\left( {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{27}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{28}}s{m}_{x}{\beta }_{11}{\beta }_{28}+\right. \right. </math><math>\left. \left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{23}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{24}}s{f}_{x}{\beta }_{9}{\beta }_{24}+\right. \right. </math><math>\left. \left. {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{29}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{30}}s{m}_{y}{\beta }_{12}{\beta }_{30}+\right. \right. </math><math>\left. \left. {\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{18}}{\beta }_{6}{\beta }_{18}\right) \, \right) \,</math> 
1101
|}
1102
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 39 )
1103
|}
1104
1105
1106
Onde
1107
1108
<math display="inline">s{f}_{i}=\frac{{f}_{i}}{\left| {f}_{i}\right| }</math> = sinal do esforço seccional de forças;
1109
1110
<math display="inline">s{m}_{i}=\frac{{m}_{i}}{\left| {m}_{i}\right| }</math> = sinal do esforço seccional de momentos;
1111
1112
A superfície de interação é assumida como um potencial plástico. As componentes são apresentadas na equação <span id='cite-_Ref529991859'></span>[[#_Ref529991859|( 40 )]]  na forma matricial para cada nó do elemento do fluxo plástico no nós do elemento durante o processo de carga.
1113
1114
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1115
|-
1116
| 
1117
{| style="text-align: center; margin:auto;width: 100%;"
1118
|-
1119
| <math display="inline">{\left\{ \frac{\partial f}{\partial {F}_{j}}\right\} }_{1}=</math><math>\left\{ \begin{matrix}\frac{\partial f}{\partial {F}_{x1}}\\\frac{\partial f}{\partial {F}_{y1}}\\\frac{\partial f}{\partial {F}_{z1}}\\\frac{\partial f}{\partial {M}_{x1}}\\\frac{\partial f}{\partial {M}_{y1}}\\\frac{\partial f}{\partial {M}_{z1}}\\\mathit{\boldsymbol{0}}\end{matrix}\right\}</math> ; <math display="inline">{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{2}=</math><math>\left\{ \begin{matrix}\mathit{\boldsymbol{0}}\\\frac{\partial f}{\partial {F}_{x2}}\\\frac{\partial f}{\partial {F}_{y2}}\\\frac{\partial f}{\partial {F}_{z2}}\\\frac{\partial f}{\partial {M}_{x2}}\\\frac{\partial f}{\partial {M}_{y2}}\\\frac{\partial f}{\partial {M}_{z2}}\end{matrix}\right\}</math>  
1120
|}
1121
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529991859'></span>( 40 )
1122
|}
1123
1124
1125
Onde <math display="inline">\mathit{\boldsymbol{0}}</math> é o vetor nulo de dimensão <math display="inline">(6\times 1)</math>.
1126
1127
====2.4.2 Derivadas de segunda ordem====
1128
1129
As derivadas de segunda ordem expressam o gradiente do vetor de fluxo plástico, obtido pela diferenciação de cada componente do vetores da equação <span id='cite-_Ref529991859'></span>[[#_Ref529991859|( 40 )]]. Desenvolvendo-se as derivadas, chega-se a:
1130
1131
'''Para''' <math display="inline">\partial {F}_{x}{F}_{k}</math>
1132
1133
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1134
|-
1135
| 
1136
{| style="text-align: center; margin:auto;width: 100%;"
1137
|-
1138
| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {F}_{x}}=\frac{1}{{f}_{x}^{2}}s{f}_{x}\left( {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{19}}{\beta }_{19}^{2}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{20}}s{m}_{x}{\beta }_{7}+\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{23}}{\beta }_{23}^{2}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{24}}\quad s{m}_{z}{\beta }_{9}+\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{21}}{\beta }_{21}^{2}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{22}}s{m}_{y}{\beta }_{8}-\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{19}}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{20}}s{m}_{x}{\beta }_{7}{\beta }_{19}-\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{23}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{24}}s{m}_{z}{\beta }_{9}{\beta }_{23}-\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{21}}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{22}}s{m}_{y}{\beta }_{8}{\beta }_{21}+\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{13}}{\beta }_{13}^{2}{\beta }_{1}-\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{13}}{\beta }_{1}{\beta }_{13}\right) \quad</math> 
1139
|}
1140
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 41 )
1141
|}
1142
1143
1144
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1145
|-
1146
| 
1147
{| style="text-align: center; margin:auto;width: 100%;"
1148
|-
1149
| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {F}_{y}}=0\quad</math> 
1150
|}
1151
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 42 )
1152
|}
1153
1154
1155
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1156
|-
1157
| 
1158
{| style="text-align: center; margin:auto;width: 100%;"
1159
|-
1160
| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {F}_{z}}=0\quad</math> 
1161
|}
1162
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 43 )
1163
|}
1164
1165
1166
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1167
|-
1168
| 
1169
{| style="text-align: center; margin:auto;width: 100%;"
1170
|-
1171
| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {M}_{x}}=\frac{s{f}_{x}{\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{19}}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{20}}{\beta }_{20}s{m}_{x}{\beta }_{7}{\beta }_{19}}{{f}_{x}{m}_{x}}</math>
1172
1173
<math>\quad</math> 
1174
|}
1175
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 44 )
1176
|}
1177
1178
1179
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1180
|-
1181
| 
1182
{| style="text-align: center; margin:auto;width: 100%;"
1183
|-
1184
| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {M}_{y}}=\frac{s{f}_{x}{\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{21}}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{22}}{\beta }_{22}s{m}_{y}{\beta }_{8}{\beta }_{21}}{{f}_{x}{m}_{y}}</math>
1185
1186
<math>\quad</math> 
1187
|}
1188
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 45 )
1189
|}
1190
1191
1192
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1193
|-
1194
| 
1195
{| style="text-align: center; margin:auto;width: 100%;"
1196
|-
1197
| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {M}_{z}}=\frac{s{f}_{x}{\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{23}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{24}}{\beta }_{24}s{m}_{z}{\beta }_{9}{\beta }_{23}}{{f}_{x}{m}_{z}}</math>
1198
1199
<math>\quad</math> 
1200
|}
1201
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 46 )
1202
|}
1203
1204
1205
'''Para''' <math display="inline">\partial {F}_{y}{F}_{k}</math>
1206
1207
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1208
|-
1209
| 
1210
{| style="text-align: center; margin:auto;width: 100%;"
1211
|-
1212
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {F}_{x}}=0\quad</math> 
1213
|}
1214
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 47 )
1215
|}
1216
1217
1218
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1219
|-
1220
| 
1221
{| style="text-align: center; margin:auto;width: 100%;"
1222
|-
1223
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {F}_{y}}=\frac{s{f}_{y}{\beta }_{2}{\left( \frac{{f}_{y}}{{f}_{yp}}\right) }^{{\beta }_{14}}{\beta }_{14}({\beta }_{14}-1)}{{f}_{y}^{2}}\quad</math> 
1224
|}
1225
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 48 )
1226
|}
1227
1228
1229
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1230
|-
1231
| 
1232
{| style="text-align: center; margin:auto;width: 100%;"
1233
|-
1234
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {F}_{z}}=0\quad</math> 
1235
|}
1236
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 49 )
1237
|}
1238
1239
1240
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1241
|-
1242
| 
1243
{| style="text-align: center; margin:auto;width: 100%;"
1244
|-
1245
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {M}_{x}}=0\quad</math> 
1246
|}
1247
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 50 )
1248
|}
1249
1250
1251
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1252
|-
1253
| 
1254
{| style="text-align: center; margin:auto;width: 100%;"
1255
|-
1256
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {M}_{y}}=0\quad</math> 
1257
|}
1258
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 51 )
1259
|}
1260
1261
1262
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1263
|-
1264
| 
1265
{| style="text-align: center; margin:auto;width: 100%;"
1266
|-
1267
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {M}_{z}}=0\quad</math> 
1268
|}
1269
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 52 )
1270
|}
1271
1272
1273
'''Para''' <math display="inline">\partial {F}_{z}{F}_{k}</math>
1274
1275
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1276
|-
1277
| 
1278
{| style="text-align: center; margin:auto;width: 100%;"
1279
|-
1280
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {F}_{x}}=0\quad</math> 
1281
|}
1282
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 53 )
1283
|}
1284
1285
1286
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1287
|-
1288
| 
1289
{| style="text-align: center; margin:auto;width: 100%;"
1290
|-
1291
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {F}_{y}}=0\quad</math> 
1292
|}
1293
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 54 )
1294
|}
1295
1296
1297
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1298
|-
1299
| 
1300
{| style="text-align: center; margin:auto;width: 100%;"
1301
|-
1302
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {F}_{z}}=\frac{s{f}_{z}{\beta }_{3}{\left( \frac{{f}_{z}}{{f}_{zp}}\right) }^{{\beta }_{15}}{\beta }_{15}({\beta }_{15}-1)\, }{{f}_{z}^{2}}\, \,</math> 
1303
|}
1304
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 55 )
1305
|}
1306
1307
1308
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1309
|-
1310
| 
1311
{| style="text-align: center; margin:auto;width: 100%;"
1312
|-
1313
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {M}_{x}}=0\quad</math> 
1314
|}
1315
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 56 )
1316
|}
1317
1318
1319
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1320
|-
1321
| 
1322
{| style="text-align: center; margin:auto;width: 100%;"
1323
|-
1324
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {M}_{y}}=0\quad</math> 
1325
|}
1326
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 57 )
1327
|-
1328
| 
1329
{| style="text-align: center; margin:auto;width: 100%;"
1330
|-
1331
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {M}_{z}}=0\quad</math> 
1332
|}
1333
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 58 )
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|}
1335
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1337
'''Para''' <math display="inline">\partial {M}_{x}{F}_{k}</math>
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {F}_{x}}=\frac{s{f}_{x}{\left( \frac{{f}_{x}}{{f}_{xp}\, \, }\right) }^{{\beta }_{19}}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{\beta 20}\beta 20\, s{m}_{x}\beta 7\beta 19}{{f}_{x}{m}_{x}}\quad</math> 
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|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 59 )
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|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
1355
| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {F}_{y}}=0\quad</math> 
1356
|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 60 )
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1362
|-
1363
| 
1364
{| style="text-align: center; margin:auto;width: 100%;"
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1366
| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {F}_{z}}=0\, \,</math> 
1367
|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 61 )
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1373
|-
1374
| 
1375
{| style="text-align: center; margin:auto;width: 100%;"
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|-
1377
| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {M}_{x}}=\frac{1}{{m}_{x}^{2}}s{m}_{x}\left( {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{25}}{\beta }_{25}^{2}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{26}}s{m}_{y}{\beta }_{10}+\right. </math><math>\left. {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{27}}{\beta }_{27}^{2}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{28}}s{m}_{z}{\beta }_{11}+\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{19}}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{20}}{\beta }_{20}^{2}s{f}_{x}{\beta }_{7}-\right. </math><math>\left. {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{25}}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{26}}s{m}_{y}{\beta }_{10}{\beta }_{25}-\right. </math><math>\left. {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{27}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{28}}s{m}_{z}{\beta }_{11}{\beta }_{27}-\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{19}}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{20}}s{f}_{x}{\beta }_{7}{\beta }_{20}+\right. </math><math>\left. {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{16}}{\beta }_{16}^{2}{\beta }_{4}-\right. </math><math>\left. {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{16}}{\beta }_{4}{\beta }_{16}\right) \, \,</math> 
1378
|}
1379
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 62 )
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1384
|-
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| 
1386
{| style="text-align: center; margin:auto;width: 100%;"
1387
|-
1388
| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {M}_{y}}=\frac{s{m}_{x}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{25}}\, {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{26}}{\beta }_{26}s{m}_{y}{\beta }_{10}{\beta }_{25}}{{m}_{x}{m}_{y}}</math>
1389
|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 63 )
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|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
1397
{| style="text-align: center; margin:auto;width: 100%;"
1398
|-
1399
| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {M}_{z}}=\frac{s{m}_{x}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{27}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{28}}{\beta }_{28}s{m}_{z}{\beta }_{11}{\beta }_{27}}{{m}_{x}{m}_{z}}</math>
1400
|}
1401
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 64 )
1402
|}
1403
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1405
'''Para''' <math display="inline">\partial {M}_{y}{F}_{k}</math>
1406
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1408
|-
1409
| 
1410
{| style="text-align: center; margin:auto;width: 100%;"
1411
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1412
| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {F}_{x}}=\frac{s{f}_{x}{\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{21}}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{22}}{\beta }_{22}s{m}_{y}{\beta }_{8}{\beta }_{21}}{{f}_{x}{m}_{y}}\, \,</math> 
1413
|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 65 )
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1419
|-
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| 
1421
{| style="text-align: center; margin:auto;width: 100%;"
1422
|-
1423
| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {F}_{y}}=0\quad</math> 
1424
|}
1425
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 66 )
1426
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1430
|-
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| 
1432
{| style="text-align: center; margin:auto;width: 100%;"
1433
|-
1434
| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {F}_{z}}=0\, \,</math> 
1435
|}
1436
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 67 )
1437
|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1441
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| 
1443
{| style="text-align: center; margin:auto;width: 100%;"
1444
|-
1445
| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {M}_{x}}=\frac{s{m}_{x}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{25}}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{26}}{\beta }_{26}s{m}_{y}{\beta }_{10}{\beta }_{25}}{{m}_{x}{m}_{y}}\, \,</math> 
1446
|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 68 )
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|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1452
|-
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| 
1454
{| style="text-align: center; margin:auto;width: 100%;"
1455
|-
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {M}_{y}}=\frac{1}{{m}_{y}^{2}}s{m}_{y}\left( {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{25}}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{26}}{\beta }_{26}^{2}s{m}_{x}{\beta }_{10}+\right. </math><math>\left. {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{29}}{\beta }_{29}^{2}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{30}}s{m}_{z}{\beta }_{12}+\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{21}}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{22}}{\beta }_{22}^{2}s{f}_{x}{\beta }_{8}-\right. </math><math>\left. \left( \frac{{m}_{x}}{{m}_{xp}}\right) {\beta }_{25}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{26}}s{m}_{x}{\beta }_{10}{\beta }_{26}-\right. </math><math>\left. {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{29}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{30}}s{m}_{z}{\beta }_{12}{\beta }_{29}-\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{21}}\left( \frac{{m}_{y}}{{m}_{yp}}\right) {\beta }_{22}s{f}_{x}{\beta }_{8}{\beta }_{22}+\right. </math><math>\left. {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{17}}{\beta }_{17}^{2}{\beta }_{5}-\right. </math><math>\left. {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{17}}{\beta }_{5}{\beta }_{17}\right)</math> 
1457
|}
1458
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 69 )
1459
|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
1465
{| style="text-align: center; margin:auto;width: 100%;"
1466
|-
1467
| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {M}_{z}}=\frac{s{m}_{y}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{29}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{30}}{\beta }_{30}s{m}_{z}{\beta }_{12}{\beta }_{29}}{{m}_{y}{m}_{z}}</math>
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|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 70 )
1470
|}
1471
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1473
'''Para''' <math display="inline">\partial {M}_{z}{F}_{k}</math>
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1476
|-
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| 
1478
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {F}_{x}}=\, \frac{s{f}_{x}{\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{23}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{24}}{\beta }_{24}s{m}_{z}{\beta }_{9}{\beta }_{23}}{{f}_{x}{m}_{z}}</math>
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|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 71 )
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|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
1489
{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {F}_{y}}=0\quad</math> 
1492
|}
1493
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 72 )
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|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
1500
{| style="text-align: center; margin:auto;width: 100%;"
1501
|-
1502
| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {F}_{z}}=0\, \,</math> 
1503
|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 73 )
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|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
1511
{| style="text-align: center; margin:auto;width: 100%;"
1512
|-
1513
| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {M}_{x}}=\frac{s{m}_{x}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{27}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{28}}{\beta }_{28}s{m}_{z}{\beta }_{11}{\beta }_{27}}{{m}_{x}{m}_{z}}\quad</math> 
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|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 74 )
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|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
1522
{| style="text-align: center; margin:auto;width: 100%;"
1523
|-
1524
| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {M}_{y}}=\frac{s{m}_{y}{\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{29}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{30}}{\beta }_{30}s{m}_{z}{\beta }_{12}{\beta }_{29}}{{m}_{y}{m}_{z}}</math>
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|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 75 )
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|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
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| 
1533
{| style="text-align: center; margin:auto;width: 100%;"
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|-
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {M}_{z}}=\frac{1}{{m}_{z}^{2}}s{m}_{z}\left( {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{27}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{28}}{\beta }_{28}^{2}s{m}_{x}{\beta }_{11}+\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{23}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{24}}{\beta }_{24}^{2}s{f}_{x}{\beta }_{9}+\right. </math><math>\left. {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{29}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{30}}{\beta }_{30}^{2}s{m}_{y}{\beta }_{12}-\right. </math><math>\left. {\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{{\beta }_{27}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{28}}s{m}_{x}{\beta }_{11}{\beta }_{28}-\right. </math><math>\left. {\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{23}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{24}}s{f}_{x}{\beta }_{9}{\beta }_{24}-\right. </math><math>\left. {\left( \frac{{m}_{y}}{{m}_{yp}}\right) }^{{\beta }_{29}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{30}}s{m}_{y}{\beta }_{12}{\beta }_{30}+\right. </math><math>\left. {\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{18}}{\beta }_{18}^{2}{\beta }_{6}-\right. </math><math>\left. {\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{18}}{\beta }_{6}\, {\beta }_{18}\right)</math> 
1536
|}
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|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 76 )
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{| class="formulaSCP" style="width: 78%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
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{| style="text-align: center; margin:auto;width: 100%;"
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|-
1546
| As 2ª derivadas na forma matricial podem ser expressas como
1547
1548
<math>{A}_{1}=\left[ \begin{matrix}\begin{matrix}\frac{{\partial }^{2}f}{\partial {F}_{x1}\partial {F}_{x1}}&\frac{{\partial }^{2}f}{\partial {F}_{x1}\partial {F}_{y1}}&\frac{{\partial }^{2}f}{\partial {F}_{x1}\partial {F}_{z1}}\\\frac{{\partial }^{2}f}{\partial {F}_{y1}\partial {F}_{x1}}&\frac{{\partial }^{2}f}{\partial {F}_{y1}\partial {F}_{y1}}&\frac{{\partial }^{2}f}{\partial {F}_{y1}\partial {F}_{z1}}\end{matrix}&\begin{matrix}\frac{{\partial }^{2}f}{\partial {F}_{x1}\partial {M}_{x1}}&\frac{{\partial }^{2}f}{\partial {F}_{x1}\partial {M}_{y1}}&\frac{{\partial }^{2}f}{\partial {F}_{x1}\partial {M}_{z1}}\\\frac{{\partial }^{2}f}{\partial {F}_{y1}\partial {M}_{x1}}&\frac{{\partial }^{2}f}{\partial {F}_{y1}\partial {M}_{y1}}&\frac{{\partial }^{2}f}{\partial {F}_{y1}\partial {M}_{z1}}\end{matrix}\\\begin{matrix}\frac{{\partial }^{2}f}{\partial {F}_{z1}\partial {F}_{x1}}&\frac{{\partial }^{2}f}{\partial {F}_{z1}\partial {F}_{y1}}&\frac{{\partial }^{2}f}{\partial {F}_{z1}\partial {F}_{z1}}\\\frac{{\partial }^{2}f}{\partial {M}_{x1}\partial {F}_{x1}}&\frac{{\partial }^{2}f}{\partial {M}_{x1}\partial {F}_{y1}}&\frac{{\partial }^{2}f}{\partial {M}_{x1}\partial {F}_{z1}}\end{matrix}&\begin{matrix}\frac{{\partial }^{2}f}{\partial {F}_{z1}\partial {M}_{x1}}&\frac{{\partial }^{2}f}{\partial {F}_{z1}\partial {M}_{y1}}&\frac{{\partial }^{2}f}{\partial {F}_{z1}\partial {M}_{z1}}\\\frac{{\partial }^{2}f}{\partial {M}_{x1}\partial {M}_{x1}}&\frac{{\partial }^{2}f}{\partial {M}_{x1}\partial {M}_{y1}}&\frac{{\partial }^{2}f}{\partial {M}_{x1}\partial {M}_{z1}}\end{matrix}\\\begin{matrix}\frac{{\partial }^{2}f}{\partial {M}_{y1}\partial {F}_{x1}}&\frac{{\partial }^{2}f}{\partial {M}_{y1}\partial {F}_{y1}}&\frac{{\partial }^{2}f}{\partial {M}_{y1}\partial {F}_{z1}}\\\frac{{\partial }^{2}f}{\partial {M}_{z1}\partial {F}_{x1}}&\frac{{\partial }^{2}f}{\partial {M}_{z1}\partial {F}_{y1}}&\frac{{\partial }^{2}f}{\partial {M}_{z1}\partial {F}_{z1}}\end{matrix}&\begin{matrix}\frac{{\partial }^{2}f}{\partial {M}_{y1}\partial {M}_{x1}}&\frac{{\partial }^{2}f}{\partial {M}_{y1}\partial {M}_{y1}}&\frac{{\partial }^{2}f}{\partial {M}_{y1}\partial {M}_{z1}}\\\frac{{\partial }^{2}f}{\partial {M}_{z1}\partial {M}_{x1}}&\frac{{\partial }^{2}f}{\partial {M}_{z1}\partial {M}_{y1}}&\frac{{\partial }^{2}f}{\partial {M}_{z1}\partial {M}_{z1}}\end{matrix}\end{matrix}\right]</math> 
1549
|}
1550
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 77 )
1551
|}
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{| class="formulaSCP" style="width: 45%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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|-
1556
| 
1557
{| style="text-align: center; margin:auto;width: 100%;"
1558
|-
1559
| <math>{\left[ \frac{{\partial }^{2}\mathit{\boldsymbol{f}}}{\partial {F}_{j}\partial {F}_{k}}\right] }_{1}=</math><math>\left[ \begin{matrix}{\mathit{\boldsymbol{A}}}_{\mathit{\boldsymbol{1}}}&\mathit{\boldsymbol{0}}\\\mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}\end{matrix}\right]</math> 
1560
|}
1561
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 78 )
1562
|}
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{| class="formulaSCP" style="width: 78%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1566
|-
1567
| 
1568
{| style="text-align: center; margin:auto;width: 100%;"
1569
|-
1570
| <math>{A}_{2}=\left[ \begin{matrix}\begin{matrix}\frac{{\partial }^{2}f}{\partial {F}_{x2}\partial {F}_{x2}}&\frac{{\partial }^{2}f}{\partial {F}_{x2}\partial {F}_{y2}}&\frac{{\partial }^{2}f}{\partial {F}_{x2}\partial {F}_{z2}}\\\frac{{\partial }^{2}f}{\partial {F}_{y2}\partial {F}_{x2}}&\frac{{\partial }^{2}f}{\partial {F}_{y2}\partial {F}_{y2}}&\frac{{\partial }^{2}f}{\partial {F}_{y2}\partial {F}_{z2}}\end{matrix}&\begin{matrix}\frac{{\partial }^{2}f}{\partial {F}_{x2}\partial {M}_{x2}}&\frac{{\partial }^{2}f}{\partial {F}_{x2}\partial {M}_{y2}}&\frac{{\partial }^{2}f}{\partial {F}_{x2}\partial {M}_{z2}}\\\frac{{\partial }^{2}f}{\partial {F}_{y2}\partial {M}_{x2}}&\frac{{\partial }^{2}f}{\partial {F}_{y2}\partial {M}_{y2}}&\frac{{\partial }^{2}f}{\partial {F}_{y2}\partial {M}_{z2}}\end{matrix}\\\begin{matrix}\frac{{\partial }^{2}f}{\partial {F}_{z2}\partial {F}_{x2}}&\frac{{\partial }^{2}f}{\partial {F}_{z2}\partial {F}_{y2}}&\frac{{\partial }^{2}f}{\partial {F}_{z2}\partial {F}_{z2}}\\\frac{{\partial }^{2}f}{\partial {M}_{x2}\partial {F}_{x2}}&\frac{{\partial }^{2}f}{\partial {M}_{x2}\partial {F}_{y2}}&\frac{{\partial }^{2}f}{\partial {M}_{x2}\partial {F}_{z2}}\end{matrix}&\begin{matrix}\frac{{\partial }^{2}f}{\partial {F}_{z2}\partial {M}_{x2}}&\frac{{\partial }^{2}f}{\partial {F}_{z2}\partial {M}_{y2}}&\frac{{\partial }^{2}f}{\partial {F}_{z2}\partial {M}_{z2}}\\\frac{{\partial }^{2}f}{\partial {M}_{x2}\partial {M}_{x2}}&\frac{{\partial }^{2}f}{\partial {M}_{x2}\partial {M}_{y2}}&\frac{{\partial }^{2}f}{\partial {M}_{x2}\partial {M}_{z2}}\end{matrix}\\\begin{matrix}\frac{{\partial }^{2}f}{\partial {M}_{y2}\partial {F}_{x2}}&\frac{{\partial }^{2}f}{\partial {M}_{y2}\partial {F}_{y2}}&\frac{{\partial }^{2}f}{\partial {M}_{y2}\partial {F}_{z2}}\\\frac{{\partial }^{2}f}{\partial {M}_{z2}\partial {F}_{x2}}&\frac{{\partial }^{2}f}{\partial {M}_{z2}\partial {F}_{y2}}&\frac{{\partial }^{2}f}{\partial {M}_{z2}\partial {F}_{z2}}\end{matrix}&\begin{matrix}\frac{{\partial }^{2}f}{\partial {M}_{y2}\partial {M}_{x2}}&\frac{{\partial }^{2}f}{\partial {M}_{y2}\partial {M}_{y2}}&\frac{{\partial }^{2}f}{\partial {M}_{y2}\partial {M}_{z2}}\\\frac{{\partial }^{2}f}{\partial {M}_{z2}\partial {M}_{x2}}&\frac{{\partial }^{2}f}{\partial {M}_{z2}\partial {M}_{y2}}&\frac{{\partial }^{2}f}{\partial {M}_{z2}\partial {M}_{z2}}\end{matrix}\end{matrix}\right]</math> 
1571
|}
1572
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 79 )
1573
|}
1574
1575
1576
{| class="formulaSCP" style="width: 45%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1577
|-
1578
| 
1579
{| style="text-align: center; margin:auto;width: 100%;"
1580
|-
1581
| <math>{\left[ \frac{{\partial }^{2}\mathit{\boldsymbol{f}}}{\partial {F}_{j}\partial {F}_{k}}\right] }_{2}=</math><math>\left[ \begin{matrix}\mathit{\boldsymbol{0}}&\mathit{\boldsymbol{0}}\\\mathit{\boldsymbol{0}}&{\mathit{\boldsymbol{A}}}_{\mathit{\boldsymbol{2}}}\end{matrix}\right]</math> 
1582
|}
1583
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 80 )
1584
|}
1585
1586
1587
Onde  <math display="inline">\mathit{\boldsymbol{0}}</math>''' '''é uma matriz de dimensão <math display="inline">(6\times 6)</math> com elementos nulos.
1588
1589
===2.5 Algoritmo de Retorno===
1590
1591
O algoritmo de retorno proposto por Silva [10] servirá para trazer de volta os esforços seccionais inadmissíveis, ou seja, os que saem da superfície de interação. O método de ''backward'' Euler será utilizado para trazer de volta a superfície estes esforços seccionais. Quando os esforços atingem a superfície se formam as rótulas plásticas.
1592
1593
Assume-se que exista uma combinação de esforços seccionais em um dos nós do elemento que esteja fora da superfície de interação. Usando o método de ''backward'' Euler para corrigir o vetor de forças nodais tem-se a seguinte forma:
1594
1595
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1596
|-
1597
| 
1598
{| style="text-align: center; margin:auto;width: 100%;"
1599
|-
1600
| <math>{\hat{\mathit{\boldsymbol{F}}}}_{i}={\mathit{\boldsymbol{F}}}_{i}^{trial}-{\lambda }_{1}{K}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}</math>
1601
|}
1602
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530073820'></span>( 81 )
1603
|}
1604
1605
1606
Onde
1607
1608
<math display="inline">{\hat{\mathit{\boldsymbol{F}}}}_{i}</math>= vetor de forças nodais corrigido;
1609
1610
<math display="inline">{\mathit{\boldsymbol{F}}}_{i}^{trial}</math> = vetor de força nodais estimado;
1611
1612
<math display="inline">{\lambda }_{1}</math>= multiplicador plástico do nó 1, de forma <math display="inline">{\lambda }_{1}\geq 0</math>;
1613
1614
<math display="inline">{\mathit{\boldsymbol{K}}}_{ij}</math>= matriz de rigidez do elemento;
1615
1616
<math display="inline">{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}</math>= vetor de fluxo plástico do nó 1.
1617
1618
O vetor de forças nodais estimado é expressado por:
1619
1620
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1621
|-
1622
| 
1623
{| style="text-align: center; margin:auto;width: 100%;"
1624
|-
1625
| <math>{\mathit{\boldsymbol{F}}}_{i}^{trial}={\overline{\mathit{\boldsymbol{F}}}}_{i}+{\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}</math>
1626
|}
1627
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530068060'></span>( 82 )
1628
|}
1629
1630
1631
Onde
1632
1633
<math display="inline">{\overline{\mathit{\boldsymbol{F}}}}_{i}</math> = vetor de forças nodais do último passo de carga convergido;
1634
1635
<math display="inline">d{\mathit{\boldsymbol{U}}}_{j}</math> = incrementos do campo de deslocamentos do nó.
1636
1637
O vetor <math display="inline">{F}_{i}^{trial}</math> é obtido da solução elástica dos incrementos de deslocamentos <math display="inline">d{U}_{j}</math> e da matriz de rigidez <math display="inline">{K}_{ij}</math> linear elástica do elemento de viga 3D. O vetor <math display="inline">{\overline{F}}_{i}</math> pode estar dentro, fora ou tocando a superfície de interação. Geralmente, os vetores de forças nodais, estimado ( <math display="inline">{F}_{i}^{trial}</math>) e o corrigido ( <math display="inline">{\hat{F}}_{i}</math>) não estão sobre a superfície de interação quando atingem a fase de escoamento. Usar-se-á um método iterativo para trazer os esforços seccionais a um estado de tensão que esteja na superfície de interação.
1638
1639
O algoritmo irá trabalhar com 2 (duas) possibilidades de formação de rótulas plásticas, ou seja, para 1 (um) nó ou os 2 (dois) nós.
1640
1641
====2.5.1 Algoritmo de retorno com 1 (um) vetor====
1642
1643
<span id='_Ref503810719'></span>O caso de formação de somente uma rótula plástica no elemento de viga emprega-se um vetor de fluxo plástico correspondente aos esforços seccionais que se encontra fora da superfície de interação, conforme <span id='cite-_Ref530062972'></span>[[#_Ref530062972|'''Figura 6''']].
1644
1645
{| style="width: 82%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
1646
|-
1647
|  style="text-align: center;vertical-align: top;width: 100%;"|[[Image:Draft_Vieira_908925676-image7.png|474px]] 
1648
|}
1649
1650
1651
<span id='_Ref530062972'></span>'''Figura 6'''. Retorno à superfície com um vetor.
1652
1653
O processo iterativo utiliza vetores de fluxo plástico atualizados para aproximar-se da superfície. Este procedimento é chamado de algoritmo de retorno.
1654
1655
Admite-se que os vetores de força nodais <math display="inline">{F}_{i}</math> (atual) e o corrigido <math display="inline">{\hat{F}}_{i}</math> não cumprem o critério de escoamento, ou seja, <math display="inline">f({F}_{i})>1</math> e <math display="inline">f({\hat{F}}_{i})>1.</math>
1656
1657
O vetor de forças residuais <math display="inline">{\mathit{\boldsymbol{r}}}_{i}</math> do processo iterativo será como
1658
1659
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1660
|-
1661
| 
1662
{| style="text-align: center; margin:auto;width: 100%;"
1663
|-
1664
| <math>{\mathit{\boldsymbol{r}}}_{i}={\mathit{\boldsymbol{F}}}_{i}-{\hat{\mathit{\boldsymbol{F}}}}_{i}=</math><math>{\mathit{\boldsymbol{F}}}_{i}-({\mathit{\boldsymbol{F}}}_{i}^{trial}-{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1})</math>
1665
|}
1666
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530063570'></span>( 83 )
1667
|}
1668
1669
1670
Desenvolvendo a equação <span id='cite-_Ref530063570'></span>[[#_Ref530063570|( 83 )]] numa série de Taylor até os termos de 1ª ordem e mantendo o vetor de forças nodais de partida <math display="inline">{\mathit{\boldsymbol{F}}}_{i}^{trial}</math> fixo, obtém-se um novo vetor de forças residuais <math display="inline">{\mathit{\boldsymbol{r}}}_{i}^{n+1}</math>, apresentado da seguinte forma:
1671
1672
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1673
|-
1674
| 
1675
{| style="text-align: center; margin:auto;width: 100%;"
1676
|-
1677
| <math>{\mathit{\boldsymbol{r}}}_{i}^{n+1}={\mathit{\boldsymbol{r}}}_{i}^{n}+d{\mathit{\boldsymbol{F}}}_{i}+</math><math>d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+</math><math>{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{1}d{\mathit{\boldsymbol{F}}}_{k}</math>
1678
|}
1679
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530064230'></span>( 84 )
1680
|}
1681
1682
1683
Onde
1684
1685
<math display="inline">n=0,1,2,\ldots</math>  = passo do processo iterativo.
1686
1687
<math display="inline">d{\mathit{\boldsymbol{F}}}_{i}</math> = variação do vetor de forças;
1688
1689
<math display="inline">d{\lambda }_{1}</math>= variação do multiplicador plástico;
1690
1691
<math display="inline">{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{1}d{\mathit{\boldsymbol{F}}}_{k}</math> = variação do vetor de fluxo (gradiente).
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1693
Aplicando a condição de que <math display="inline">{\mathit{\boldsymbol{r}}}_{i}^{n+1}=</math><math>\mathit{\boldsymbol{0}}</math>, a equação <span id='cite-_Ref530064230'></span>[[#_Ref530064230|( 84 )]] torna-se em
1694
1695
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1696
|-
1697
| 
1698
{| style="text-align: center; margin:auto;width: 100%;"
1699
|-
1700
| <math>\mathit{\boldsymbol{0}}={\mathit{\boldsymbol{r}}}_{i}^{n}+d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+</math><math>\left( {\delta }_{ik}+{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{1}\right) d{\mathit{\boldsymbol{F}}}_{k}</math>
1701
|}
1702
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530064567'></span>( 85 )
1703
|}
1704
1705
1706
Onde <math display="inline">{\delta }_{ik}</math> = Delta de Kronecker
1707
1708
Definindo-se o termo
1709
1710
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1711
|-
1712
| 
1713
{| style="text-align: center; margin:auto;width: 100%;"
1714
|-
1715
| <math>{\mathit{\boldsymbol{Q}}}_{ik}=\left( {\delta }_{ik}+{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{1}\right)</math> 
1716
|}
1717
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530075709'></span>( 86 )
1718
|}
1719
1720
1721
A equação <span id='cite-_Ref530064567'></span>[[#_Ref530064567|( 85 )]], torna-se:
1722
1723
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1724
|-
1725
| 
1726
{| style="text-align: center; margin:auto;width: 100%;"
1727
|-
1728
| <math>\mathit{\boldsymbol{0}}={\mathit{\boldsymbol{r}}}_{i}^{n}+d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+</math><math>{Q}_{ik}d{\mathit{\boldsymbol{F}}}_{k}</math>
1729
|}
1730
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 87 )
1731
|}
1732
1733
1734
Obtendo os termos da variação do vetor de força, chega-se a:
1735
1736
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1737
|-
1738
| 
1739
{| style="text-align: center; margin:auto;width: 100%;"
1740
|-
1741
| <math>{Q}_{ik}d{\mathit{\boldsymbol{F}}}_{k}=-\left( {\mathit{\boldsymbol{r}}}_{i}^{n}+\right. </math><math>\left. d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}\right)</math> 
1742
|}
1743
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 88 )
1744
|}
1745
1746
1747
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1748
|-
1749
| 
1750
{| style="text-align: center; margin:auto;width: 100%;"
1751
|-
1752
| <math>d{\mathit{\boldsymbol{F}}}_{k}=-{Q}_{ik}^{-1}\left( {\mathit{\boldsymbol{r}}}_{i}^{n}+\right. </math><math>\left. d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}\right)</math> 
1753
|}
1754
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530065198'></span>( 89 )
1755
|}
1756
1757
1758
Expandindo a superfície de interação, <math display="inline">\mathit{\boldsymbol{f}}</math> , numa série de Taylor até os termos de 1ª ordem entorno do vetor do vetor de forças nodais final ( <math display="inline">{F}_{k})</math>, obtém-se:
1759
1760
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1761
|-
1762
| 
1763
{| style="text-align: center; margin:auto;width: 100%;"
1764
|-
1765
| <math>{\mathit{\boldsymbol{f}}}_{1}^{n+1}={\mathit{\boldsymbol{f}}}_{1}^{n}+{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}d{\mathit{\boldsymbol{F}}}_{k}</math>
1766
|}
1767
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 90 )
1768
|}
1769
1770
1771
Tomando-se <math display="inline">{\mathit{\boldsymbol{f}}}_{1}^{n+1}=\mathit{\boldsymbol{0}}</math> e desenvolvendo a equação <span id='cite-_Ref530065198'></span>[[#_Ref530065198|( 89 )]] paro obter o multiplicador plástico, chega-se a:
1772
1773
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1774
|-
1775
| 
1776
{| style="text-align: center; margin:auto;width: 100%;"
1777
|-
1778
| <math>{\mathit{\boldsymbol{f}}}_{1}^{n}=-{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}d{\mathit{\boldsymbol{F}}}_{k}</math>
1779
|}
1780
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 91 )
1781
|}
1782
1783
1784
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1785
|-
1786
| 
1787
{| style="text-align: center; margin:auto;width: 100%;"
1788
|-
1789
| <math>{\mathit{\boldsymbol{f}}}_{1}^{n}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}\left[ {Q}_{ik}^{-1}\left( {\mathit{\boldsymbol{r}}}_{i}^{n}+\right. \right. </math><math>\left. \left. d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}\right) \right]</math> 
1790
|}
1791
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 92 )
1792
|}
1793
1794
1795
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1796
|-
1797
| 
1798
{| style="text-align: center; margin:auto;width: 100%;"
1799
|-
1800
| <math>{\mathit{\boldsymbol{f}}}_{1}^{n}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}{\mathit{\boldsymbol{r}}}_{i}^{n}+</math><math>{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}</math>
1801
|}
1802
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 93 )
1803
|}
1804
1805
1806
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1807
|-
1808
| 
1809
{| style="text-align: center; margin:auto;width: 100%;"
1810
|-
1811
| <math>{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}=</math><math>{\mathit{\boldsymbol{f}}}_{1}^{n}-{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}{\mathit{\boldsymbol{r}}}_{i}^{n}</math>
1812
|}
1813
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 94 )
1814
|}
1815
1816
1817
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1818
|-
1819
| 
1820
{| style="text-align: center; margin:auto;width: 100%;"
1821
|-
1822
| <math>d{\lambda }_{1}=\frac{{\mathit{\boldsymbol{f}}}_{1}^{n}-{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}{\mathit{\boldsymbol{r}}}_{i}^{n}}{{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}}</math>
1823
|}
1824
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 95 )
1825
|}
1826
1827
1828
O processo iterativo termina quando são alcançados os critérios de parada adotados:
1829
1830
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1831
|-
1832
| 
1833
{| style="text-align: center; margin:auto;width: 100%;"
1834
|-
1835
| <math>{r}^{norm}=\sqrt{\frac{\left\| {\mathit{\boldsymbol{r}}}_{i}\right\| }{\left\| {\mathit{\boldsymbol{F}}}_{i}^{trial}\right\| }}<Tol</math>
1836
1837
<math display="inline">{\mathit{\boldsymbol{f}}}^{norm}=\left| \mathit{\boldsymbol{f}}-\right. </math><math>\left. 1\right| <Tol</math> 
1838
|}
1839
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 96 )
1840
|}
1841
1842
1843
Onde
1844
1845
<math display="inline">\left\| {\mathit{\boldsymbol{r}}}_{i}\right\|</math>  = norma euclidiana do vetor de forças nodais;
1846
1847
<math display="inline">\left\| {\mathit{\boldsymbol{F}}}_{i}^{trial}\right\|</math>  = norma euclidiana do vetor de forças estimado;
1848
1849
<math display="inline">{\mathit{\boldsymbol{f}}}^{norm}</math>= vetor resíduo da superfície de interação;
1850
1851
<math display="inline">Tol</math> = tolerância adotada.
1852
1853
====2.5.2 Algoritmo de retorno com 2 (dois) vetores====
1854
1855
O caso da existência de duas rótulas plástica no elemento de viga usa 2 (dois) vetores de fluxo plástico, um para cada nó. O vetores seguem a premissa de que
1856
1857
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1858
|-
1859
| 
1860
{| style="text-align: center; margin:auto;width: 100%;"
1861
|-
1862
| <math display="inline">{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}({F}_{j})>1;\, {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{2}}}({F}_{j})>1</math> 
1863
|}
1864
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 97 )
1865
|}
1866
1867
1868
Durante o processo iterativo, usa-se dois vetores de fluxo para se aproximar da superfície de interação. Este procedimento é chamado algoritmo de retorno com 2 (dois) vetores. A interpretação geométrica é vista na <span id='cite-_Ref530324965'></span>[[#_Ref530324965|'''Figura 7''']].
1869
1870
{| style="width: 85%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
1871
|-
1872
|  style="text-align: center;vertical-align: top;"|''' [[Image:Draft_Vieira_908925676-image8.png|510px]] '''
1873
|}
1874
1875
1876
<span id='_Ref530324965'></span>'''Figura 7'''. Retorno à superfície com dois vetores.
1877
1878
O vetor nodal de partida é similar a equação   <span id='cite-_Ref530068060'></span>[[#_Ref530068060|( 82 )]]. O vetor de forças nodais para os dois nós corrigido é expressado como
1879
1880
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1881
|-
1882
| 
1883
{| style="text-align: center; margin:auto;width: 100%;"
1884
|-
1885
| <math>{\hat{\mathit{\boldsymbol{F}}}}_{i}={\mathit{\boldsymbol{F}}}_{i}^{trial}-{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}-</math><math>{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{2}</math>
1886
|}
1887
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 98 )
1888
|}
1889
1890
1891
Onde
1892
1893
<math display="inline">{\lambda }_{1}</math>e <math display="inline">{\lambda }_{2}</math> são os multiplicadores plásticos.
1894
1895
O vetor resíduo das forças tem a forma seguinte:
1896
1897
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1898
|-
1899
| 
1900
{| style="text-align: center; margin:auto;width: 100%;"
1901
|-
1902
| <math>{\mathit{\boldsymbol{r}}}_{i}={\mathit{\boldsymbol{F}}}_{i}-{\hat{\mathit{\boldsymbol{F}}}}_{i}=</math><math>{\mathit{\boldsymbol{F}}}_{i}-({\mathit{\boldsymbol{F}}}_{i}^{trial}-{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}-</math><math>{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{2})</math>
1903
|}
1904
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 99 )
1905
|}
1906
1907
1908
O vetor novo em função da série de Taylor com termos de 1ª ordem e <math display="inline">{\mathit{\boldsymbol{F}}}_{i}^{trial}</math> fixo é apresentado:
1909
1910
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1911
|-
1912
| 
1913
{| style="text-align: center; margin:auto;width: 100%;"
1914
|-
1915
| <math>{\mathit{\boldsymbol{r}}}_{i}^{n+1}={\mathit{\boldsymbol{r}}}_{i}^{n}+d{\mathit{\boldsymbol{F}}}_{i}+</math><math>d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+</math><math>{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{1}d{\mathit{\boldsymbol{F}}}_{k}+</math><math>d{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}+</math><math>{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{2}d{\mathit{\boldsymbol{F}}}_{k}</math>
1916
|}
1917
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 100 )
1918
|}
1919
1920
1921
Com a condição que <math display="inline">{\mathit{\boldsymbol{r}}}_{i}^{n+1}=</math><math>\mathit{\boldsymbol{0}}</math>, chega-se a:
1922
1923
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1924
|-
1925
| 
1926
{| style="text-align: center; margin:auto;width: 100%;"
1927
|-
1928
| <math>\mathit{\boldsymbol{0}}={\mathit{\boldsymbol{r}}}_{i}^{n}+d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+</math><math>d{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}+</math><math>\left( {\delta }_{ik}+{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{1}+\right. </math><math>\left. {\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{2}\right) d{\mathit{\boldsymbol{F}}}_{k}</math>
1929
|}
1930
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530068973'></span>( 101 )
1931
|}
1932
1933
1934
Adotando-se:
1935
1936
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1937
|-
1938
| 
1939
{| style="text-align: center; margin:auto;width: 100%;"
1940
|-
1941
| <math>{\mathit{\boldsymbol{Q}}}_{ik}=\left( {\delta }_{ik}+{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{1}+\right. </math><math>\left. {\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}\partial {\mathit{\boldsymbol{F}}}_{k}}\right\} }_{2}\right)</math> 
1942
|}
1943
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530069357'></span>( 102 )
1944
|}
1945
1946
1947
Isolando o termo <math display="inline">d{\mathit{\boldsymbol{F}}}_{k}</math> da equação <span id='cite-_Ref530068973'></span>[[#_Ref530068973|( 101 )]], obtém-se:
1948
1949
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1950
|-
1951
| 
1952
{| style="text-align: center; margin:auto;width: 100%;"
1953
|-
1954
| <math>d{\mathit{\boldsymbol{F}}}_{k}=-{Q}_{ik}^{-1}\left( {\mathit{\boldsymbol{r}}}_{i}^{n}+\right. </math><math>\left. d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+\right. </math><math>\left. d{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}\right)</math> 
1955
|}
1956
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 103 )
1957
|}
1958
1959
1960
Os termos iterativos da função de escoamento (superfícies) são apresentados como
1961
1962
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1963
|-
1964
| 
1965
{| style="text-align: center; margin:auto;width: 100%;"
1966
|-
1967
| <math>{\mathit{\boldsymbol{f}}}_{1}^{n+1}={\mathit{\boldsymbol{f}}}_{1}^{n}+{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}d{\mathit{\boldsymbol{F}}}_{k}</math>
1968
1969
<math>\, {\mathit{\boldsymbol{f}}}_{2}^{n+1}={\mathit{\boldsymbol{f}}}_{2}^{n}+{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}d{\mathit{\boldsymbol{F}}}_{k}</math>
1970
|}
1971
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 104 )
1972
|}
1973
1974
1975
Impondo o critério de que <math display="inline">{\mathit{\boldsymbol{f}}}_{1}^{n+1}=</math><math>\mathit{\boldsymbol{0}}</math> e <math display="inline">\, {\mathit{\boldsymbol{f}}}_{2}^{n+1}=</math><math>\mathit{\boldsymbol{0}}</math> e desenvolvendo a equação <span id='cite-_Ref530069357'></span>[[#_Ref530069357|( 102 )]], chega-se a:
1976
1977
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1978
|-
1979
| 
1980
{| style="text-align: center; margin:auto;width: 100%;"
1981
|-
1982
| <math>{\mathit{\boldsymbol{f}}}_{1}^{n}=-{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}d{\mathit{\boldsymbol{F}}}_{k}</math>
1983
1984
<math>{\mathit{\boldsymbol{f}}}_{2}^{n}=-{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}d{\mathit{\boldsymbol{F}}}_{k}</math>
1985
|}
1986
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 105 )
1987
|}
1988
1989
1990
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1991
|-
1992
| 
1993
{| style="text-align: center; margin:auto;width: 100%;"
1994
|-
1995
| <math>{\mathit{\boldsymbol{f}}}_{1}^{n}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}\left[ {Q}_{ik}^{-1}\left( {\mathit{\boldsymbol{r}}}_{i}^{n}+\right. \right. </math><math>\left. \left. d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+\right. \right. </math><math>\left. \left. d{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}\right) \right]</math>
1996
1997
<math>{\mathit{\boldsymbol{f}}}_{2}^{n}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}\left[ {Q}_{ik}^{-1}\left( {\mathit{\boldsymbol{r}}}_{i}^{n}+\right. \right. </math><math>\left. \left. d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+\right. \right. </math><math>\left. \left. d{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}\right) \right]</math> 
1998
|}
1999
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530069811'></span>( 106 )
2000
|}
2001
2002
2003
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2004
|-
2005
| 
2006
{| style="text-align: center; margin:auto;width: 100%;"
2007
|-
2008
| <math>{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+</math><math>{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}d{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}=</math><math>{\mathit{\boldsymbol{f}}}_{1}^{n}-{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}{\mathit{\boldsymbol{r}}}_{i}^{n}</math>
2009
|}
2010
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530070117'></span>( 107 )
2011
|}
2012
2013
2014
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2015
|-
2016
| 
2017
{| style="text-align: center; margin:auto;width: 100%;"
2018
|-
2019
| <math>{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}{Q}_{ik}^{-1}d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+</math><math>{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}{Q}_{ik}^{-1}d{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}=</math><math>{\mathit{\boldsymbol{f}}}_{2}^{n}-{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}{Q}_{ik}^{-1}{\mathit{\boldsymbol{r}}}_{i}^{n}</math>
2020
|}
2021
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530070122'></span>( 108 )
2022
|}
2023
2024
2025
Desenvolvendo o sistema de equações <span id='cite-_Ref530070117'></span>[[#_Ref530070117|( 107 )]] e <span id='cite-_Ref530070122'></span>[[#_Ref530070122|( 108 )]] no sistema matricial, obtêm-se os seguintes temos:
2026
2027
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2028
|-
2029
| 
2030
{| style="text-align: center; margin:auto;width: 100%;"
2031
|-
2032
| <math>\left[ \begin{matrix}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}&{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}\\{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}{Q}_{ik}^{-1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}&{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}{Q}_{ik}^{-1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}\end{matrix}\right] \left\{ \begin{matrix}d{\lambda }_{1}\\d{\lambda }_{2}\end{matrix}\right\} =</math><math>\left\{ \begin{matrix}{\mathit{\boldsymbol{f}}}_{1}^{n}-{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}{Q}_{ik}^{-1}{\mathit{\boldsymbol{r}}}_{i}^{n}\\{\mathit{\boldsymbol{f}}}_{2}^{n}-{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}{Q}_{ik}^{-1}{\mathit{\boldsymbol{r}}}_{i}^{n}\end{matrix}\right\}</math> 
2033
|}
2034
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530072897'></span>( 109 )
2035
|}
2036
2037
2038
Reapresentando a equação <span id='cite-_Ref530072897'></span>[[#_Ref530072897|( 109 )]] na forma sintética:
2039
2040
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2041
|-
2042
| 
2043
{| style="text-align: center; margin:auto;width: 100%;"
2044
|-
2045
| <math>\left[ \begin{matrix}{a}_{11}&{a}_{12}\\{a}_{21}&{a}_{22}\end{matrix}\right] \left\{ \begin{matrix}d{\lambda }_{1}\\d{\lambda }_{2}\end{matrix}\right\} =</math><math>\left\{ \begin{matrix}{b}_{1}\\{b}_{2}\end{matrix}\right\}</math> 
2046
|}
2047
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530072993'></span>( 110 )
2048
|}
2049
2050
2051
A solução do sistema da equação <span id='cite-_Ref530072993'></span>[[#_Ref530072993|( 110 )]] é a seguinte:
2052
2053
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2054
|-
2055
| 
2056
{| style="text-align: center; margin:auto;width: 100%;"
2057
|-
2058
| <math>\left\{ \begin{matrix}d{\lambda }_{1}\\d{\lambda }_{2}\end{matrix}\right\} =\left[ \begin{matrix}\frac{{a}_{22}{b}_{1}-{a}_{12}{b}_{2}}{{a}_{11}{a}_{22}-{a}_{12}{a}_{21}}\\\frac{{a}_{11}{b}_{2}-{a}_{21}{b}_{1}}{{a}_{11}{a}_{22}-{a}_{12}{a}_{21}}\end{matrix}\right]</math> 
2059
|}
2060
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 111 )
2061
|}
2062
2063
2064
O processo iterativo segue procedimentos similares ao caso com um 1 (um) vetor:
2065
2066
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2067
|-
2068
| 
2069
{| style="text-align: center; margin:auto;width: 100%;"
2070
|-
2071
| <math>{r}^{norm}=\sqrt{\frac{\left\| {\mathit{\boldsymbol{r}}}_{i}\right\| }{\left\| {\mathit{\boldsymbol{F}}}_{i}^{trial}\right\| }}<Tol</math>
2072
2073
<math>{\mathit{\boldsymbol{f}}}_{1}^{norm}=\left| {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}-\right. </math><math>\left. 1\right| <Tol</math>
2074
2075
<math display="inline">{\mathit{\boldsymbol{f}}}_{2}^{norm}=\left| {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{2}}}-\right. </math><math>\left. 1\right| <Tol</math> 
2076
|}
2077
|  style="text-align: right;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 112 )
2078
|}
2079
2080
2081
<span id='_Ref4583347'></span>
2082
2083
===2.6 Matriz de rigidez consistente===
2084
2085
O processo iterativo utiliza o método de Newton-Raphson para determinar a configuração de equilíbrio do sistema estrutural. A manutenção da convergência quadrática faz necessário a obtenção de uma matriz de rigidez consistente para os 2 (dois) vetores. Uma rótula plástica usará o algoritmo com um vetor e para 2 (duas) o algoritmo com dois vetores.
2086
2087
====2.6.1 Algoritmo de retorno com um vetor ====
2088
2089
Usando a equação <span id='cite-_Ref530073820'></span>[[#_Ref530073820|( 81 )]] e <span id='cite-_Ref530068060'></span>[[#_Ref530068060|( 82 )]] como ponto de partida:
2090
2091
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2092
|-
2093
| 
2094
{| style="text-align: center; margin:auto;width: 100%;"
2095
|-
2096
| <math>{\mathit{\boldsymbol{F}}}_{i}={\mathit{\boldsymbol{F}}}_{i}^{trial}-{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}</math>
2097
2098
<math>{\mathit{\boldsymbol{F}}}_{i}^{trial}={\overline{\mathit{\boldsymbol{F}}}}_{i}+{\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}</math>
2099
2100
com <math display="inline">{\lambda }_{1}>0</math>
2101
|}
2102
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530073960'></span>( 113 )
2103
|}
2104
2105
2106
Aplicando-se o diferencial total na equação <span id='cite-_Ref530073960'></span>[[#_Ref530073960|( 113 )]], chega-se a:
2107
2108
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2109
|-
2110
| 
2111
{| style="text-align: center; margin:auto;width: 100%;"
2112
|-
2113
| <math>d{\mathit{\boldsymbol{F}}}_{i}=d({\overline{F}}_{i}+{\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j})-</math><math>d\left( {\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}\right) d{F}_{k}</math>
2114
|}
2115
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 114 )
2116
|-
2117
| 
2118
{| style="text-align: center; margin:auto;width: 100%;"
2119
|-
2120
| <math>d{\mathit{\boldsymbol{F}}}_{i}=d{\overline{F}}_{i}+{\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}-</math><math>d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}-</math><math>{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{2}\mathit{\boldsymbol{f}}}{\partial {F}_{j}\partial {F}_{k}}\right\} }_{1}d{F}_{k}</math>
2121
|}
2122
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 115 )
2123
|}
2124
2125
2126
Adotando <math display="inline">\, d{\overline{F}}_{i}=\mathit{\boldsymbol{0}}</math>
2127
2128
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2129
|-
2130
| 
2131
{| style="text-align: center; margin:auto;width: 100%;"
2132
|-
2133
| <math>\mathit{\boldsymbol{0}}={\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}-</math><math>d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}-</math><math>d{F}_{i}-{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{2}\mathit{\boldsymbol{f}}}{\partial {F}_{j}\partial {F}_{k}}\right\} }_{1}d{F}_{k}</math>
2134
|}
2135
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 116 )
2136
|}
2137
2138
2139
Isolando o termo <math display="inline">d{F}_{k}</math>
2140
2141
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2142
|-
2143
| 
2144
{| style="text-align: center; margin:auto;width: 100%;"
2145
|-
2146
| <math>\mathit{\boldsymbol{0}}={\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}-</math><math>d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}-</math><math>\left( {\delta }_{ik}+{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{{\partial }^{2}\mathit{\boldsymbol{f}}}{\partial {F}_{j}\partial {F}_{k}}\right\} }_{1}\right) d{F}_{k}</math>
2147
|}
2148
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 117 )
2149
|}
2150
2151
2152
Usando <math display="inline">{Q}_{ik}</math> (equação <span id='cite-_Ref530075709'></span>[[#_Ref530075709|( 86 )]]):
2153
2154
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2155
|-
2156
| 
2157
{| style="text-align: center; margin:auto;width: 100%;"
2158
|-
2159
| <math>\mathit{\boldsymbol{0}}={\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}-</math><math>d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}-</math><math>{\mathit{\boldsymbol{Q}}}_{ik}d{F}_{k}</math>
2160
|}
2161
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 118 )
2162
|}
2163
2164
2165
Isolando o termo <math display="inline">d{F}_{k}</math>:
2166
2167
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2168
|-
2169
| 
2170
{| style="text-align: center; margin:auto;width: 100%;"
2171
|-
2172
| <math>d{F}_{k}={\mathit{\boldsymbol{Q}}}_{ik}^{-1}\left( {\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}-\right. </math><math>\left. d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}\right)</math> 
2173
|}
2174
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530076047'></span>( 119 )
2175
|}
2176
2177
2178
Adotando o termo:
2179
2180
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2181
|-
2182
| 
2183
{| style="text-align: center; margin:auto;width: 100%;"
2184
|-
2185
| <math>{\mathit{\boldsymbol{R}}}_{ij}={\mathit{\boldsymbol{Q}}}_{ik}^{-1}{\mathit{\boldsymbol{K}}}_{ij}</math>
2186
|}
2187
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530077061'></span>( 120 )
2188
|}
2189
2190
2191
A equação <span id='cite-_Ref530076047'></span>[[#_Ref530076047|( 119 )]] torna-se em:
2192
2193
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2194
|-
2195
| 
2196
{| style="text-align: center; margin:auto;width: 100%;"
2197
|-
2198
| <math>d{\mathit{\boldsymbol{F}}}_{i}={\mathit{\boldsymbol{R}}}_{ij}\left( d{\mathit{\boldsymbol{U}}}_{j}-\right. </math><math>\left. d{\lambda }_{1}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}\right)</math> 
2199
|}
2200
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530076227'></span>( 121 )
2201
|}
2202
2203
2204
O vetor de forças nodais final tem que cumprir a condição <math display="inline">{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}\left( {F}_{i}\right) =</math><math>\mathit{\boldsymbol{0}}</math>. Desta maneira, diferencia-se a equação <span id='cite-_Ref530076227'></span>[[#_Ref530076227|( 121 )]] e obtém-se:
2205
2206
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2207
|-
2208
| 
2209
{| style="text-align: center; margin:auto;width: 100%;"
2210
|-
2211
| <math>{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}d{\mathit{\boldsymbol{F}}}_{i}=</math><math>\mathit{\boldsymbol{0}}</math>
2212
|}
2213
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 122 )
2214
|}
2215
2216
2217
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2218
|-
2219
| 
2220
{| style="text-align: center; margin:auto;width: 100%;"
2221
|-
2222
| <math>{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}d{F}_{i}=</math><math>{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}{\mathit{\boldsymbol{R}}}_{ij}\left( d{\mathit{\boldsymbol{U}}}_{j}-\right. </math><math>\left. d{\lambda }_{1}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}\right)</math> 
2223
|}
2224
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 123 )
2225
|}
2226
2227
2228
Isolando o termo <math display="inline">d{\lambda }_{1}</math>:
2229
2230
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2231
|-
2232
| 
2233
{| style="text-align: center; margin:auto;width: 100%;"
2234
|-
2235
| <math>d{\lambda }_{1}=\frac{{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{1}{\mathit{\boldsymbol{R}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}}{{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{1}{\mathit{\boldsymbol{R}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}}</math>
2236
|}
2237
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530077047'></span>( 124 )
2238
|}
2239
2240
2241
A matriz de rigidez consistente é obtida, trabalhando com as equações <span id='cite-_Ref530077047'></span>[[#_Ref530077047|( 124 )]] e <span id='cite-_Ref530076227'></span>[[#_Ref530076227|( 121 )]]:
2242
2243
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2244
|-
2245
| 
2246
{| style="text-align: center; margin:auto;width: 100%;"
2247
|-
2248
| <math>d{\mathit{\boldsymbol{F}}}_{i}=\left( {\mathit{\boldsymbol{R}}}_{ij}-\frac{{\mathit{\boldsymbol{R}}}_{im}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{m}}\right\} }_{1}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{n}}\right\} }_{1}{\mathit{\boldsymbol{R}}}_{nj}}{{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{m}}\right\} }_{1}{\mathit{\boldsymbol{R}}}_{mn}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{n}}\right\} }_{1}}\right) d{\mathit{\boldsymbol{U}}}_{j}</math>
2249
|}
2250
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 125 )
2251
|}
2252
2253
2254
Com
2255
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2257
|-
2258
| 
2259
{| style="text-align: center; margin:auto;width: 100%;"
2260
|-
2261
| <math>{\mathit{\boldsymbol{K}}}_{ij}^{AL}=\left( {\mathit{\boldsymbol{R}}}_{ij}-\frac{{\mathit{\boldsymbol{R}}}_{im}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{m}}\right\} }_{1}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{n}}\right\} }_{1}{\mathit{\boldsymbol{R}}}_{nj}}{{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{m}}\right\} }_{1}{\mathit{\boldsymbol{R}}}_{mn}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{n}}\right\} }_{1}}\right)</math> 
2262
|}
2263
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 126 )
2264
|}
2265
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2267
====2.6.2 Algoritmo com dois vetores de retorno====
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2269
Os procedimentos similares são realizados para obter a matriz de rigidez consistente para dois vetores como por exemplo Silva [10] e Vieira [6].
2270
2271
A formulação para os dois vetores tem a seguinte forma:
2272
2273
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2274
|-
2275
| 
2276
{| style="text-align: center; margin:auto;width: 100%;"
2277
|-
2278
| <math>d{\mathit{\boldsymbol{F}}}_{i}={\mathit{\boldsymbol{R}}}_{ij}\left( \begin{matrix}d{\mathit{\boldsymbol{U}}}_{j}-d{\lambda }_{1}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}\\-d{\lambda }_{2}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{2}\end{matrix}\right)</math> 
2279
|}
2280
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530079964'></span>( 127 )
2281
|}
2282
2283
2284
As condições do vetor de forças nodais final são que
2285
2286
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2287
|-
2288
|  style="text-align: center;"|<math display="inline">{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}\left( {F}_{i}\right) =</math><math>\mathit{\boldsymbol{0}}</math>''' e ''' <math display="inline">{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{2}}}\left( {F}_{i}\right) =</math><math>\mathit{\boldsymbol{0}}</math>''' '''
2289
|  style="text-align: center;vertical-align: bottom;"|( 128 )
2290
|}
2291
2292
2293
Os termos dos multiplicadores plásticos são os seguintes:
2294
2295
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2296
|-
2297
| 
2298
{| style="text-align: center; margin:auto;width: 100%;"
2299
|-
2300
| <math>\left\{ \begin{matrix}d{\lambda }_{1}\\d{\lambda }_{2}\end{matrix}\right\} =\left[ \begin{matrix}\frac{{c}_{1}{b}_{22}-{c}_{2}{b}_{12}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}\\\frac{{c}_{2}{b}_{11}-{c}_{1}{b}_{21}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}\end{matrix}\right]</math> 
2301
|}
2302
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530079552'></span>( 129 )
2303
|}
2304
2305
2306
Com
2307
2308
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2309
|-
2310
| 
2311
{| style="text-align: center; margin:auto;width: 100%;"
2312
|-
2313
| <math>{c}_{1}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{1}{R}_{ij}d{U}_{j}</math>
2314
2315
<math>{c}_{2}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{2}{R}_{ij}d{U}_{j}</math>
2316
2317
<math>{b}_{11}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{1}{R}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}</math>
2318
2319
<math>{b}_{12}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{1}{R}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{2}</math>
2320
2321
<math>{b}_{21}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{2}{R}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}</math>
2322
2323
<math>{b}_{22}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{2}{R}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{2}</math>
2324
|}
2325
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 130 )
2326
|}
2327
2328
2329
Se ocorrer o caso dos multiplicadores plásticos for <math display="inline">d{\lambda }_{1}<0</math> ou <math display="inline">d{\lambda }_{2}<0</math> será atribuído o valor zero e desativa-se a rótula plástica correspondente ao caso negativo.
2330
2331
Desenvolvendo os termos das equações <span id='cite-_Ref530079964'></span>[[#_Ref530079964|( 127 )]] e  <span id='cite-_Ref530079552'></span>[[#_Ref530079552|( 129 )]]:
2332
2333
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2334
|-
2335
| 
2336
{| style="text-align: center; margin:auto;width: 100%;"
2337
|-
2338
| <math>{\mathit{\boldsymbol{K}}}_{ij}^{AL}={\mathit{\boldsymbol{R}}}_{ij}-\left( {d}_{1}{R}_{im}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{m}}\right\} }_{1}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{n}}\right\} }_{1}{\mathit{\boldsymbol{R}}}_{nj}-\right. </math><math>\left. {d}_{2}{R}_{im}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{m}}\right\} }_{1}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{n}}\right\} }_{2}{\mathit{\boldsymbol{R}}}_{nj}\right) -</math><math>\left( {d}_{3}{R}_{im}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{m}}\right\} }_{2}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{n}}\right\} }_{2}{\mathit{\boldsymbol{R}}}_{nj}-\right. </math><math>\left. {d}_{4}{R}_{im}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{m}}\right\} }_{2}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{n}}\right\} }_{1}{\mathit{\boldsymbol{R}}}_{nj}\right)</math> 
2339
|}
2340
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 131 )
2341
|}
2342
2343
2344
Com
2345
2346
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2347
|-
2348
| 
2349
{| style="text-align: center; margin:auto;width: 100%;"
2350
|-
2351
| <math>{d}_{1}=\frac{{b}_{22}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>
2352
2353
<math>{d}_{2}=\frac{{b}_{12}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>
2354
2355
<math>{d}_{3}=\frac{{b}_{11}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>
2356
2357
<math>{d}_{4}=\frac{{b}_{21}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>
2358
2359
2360
|}
2361
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 132 )
2362
|}
2363
2364
2365
<span id='_Ref477377056'></span>
2366
2367
===2.7 Caracterização dos casos===
2368
2369
Os casos abordam a formulação apresentada com o intuito de verificar a viabilidade das superfícies de interação obtidas pelo modelo de dano com a regressão linear múltipla. Também, pretende-se usar as informações estatísticas para comparar a qualidade das funções obtidas e suas análises elastoplásticas.
2370
2371
O '''caso 1''' é baseado nos dados do trabalho de Thai e Kim [11] que trata de um pórtico plano, conforme <span id='cite-_Ref530127321'></span>[[#_Ref530127321|Figura 8]] e tabelas de 7 a 10 que apresentam as características do caso como coordenadas, características físicas, propriedades do material e cargas aplicadas, respectivamente.
2372
2373
==Propriedades do Pórtico Plano (PP):==
2374
2375
{| style="width: 100%;border-collapse: collapse;" 
2376
|-
2377
|  style="text-align: center;width: 100%;"|[[Image:Draft_Vieira_908925676-image9.png|600px]]
2378
2379
2380
|}
2381
2382
2383
<span id='_Ref530127321'></span>'''Figura 8'''. Pórtico Plano de Thai e Kim<span style="text-align: center; font-size: 75%;">.</span>
2384
2385
<span id='_Ref457417233'></span>
2386
2387
<span id='_Ref494057582'></span>'''Tabela 7''' - Coordenadas do pórtico plano - Thai e Kim.
2388
2389
{| style="width: 58%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2390
|-
2391
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">''''''</span>
2392
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{X}}</math>
2393
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Y}}</math>
2394
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Z}}</math>
2395
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2396
|-
2397
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2398
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{cm}}</math><span style="text-align: center; font-size: 75%;">''')'''</span>
2399
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{cm}}</math><span style="text-align: center; font-size: 75%;">''')'''</span>
2400
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{cm}}</math><span style="text-align: center; font-size: 75%;">''')'''</span>
2401
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2402
|-
2403
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2404
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2405
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2406
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2407
|  style="text-align: center;vertical-align: top;"|
2408
|-
2409
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2410
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2411
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
2412
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2413
|  style="text-align: center;vertical-align: top;"|
2414
|-
2415
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2416
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
2417
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
2418
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2419
|  style="text-align: center;vertical-align: top;"|
2420
|-
2421
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2422
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
2423
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2424
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2425
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
2426
|}
2427
2428
2429
<span id='_Ref457417961'></span><span id='_Ref494057719'></span>
2430
2431
<span id='_Ref500759207'></span>'''Tabela 8''' – Características físicas do pórtico plano - Thai e Kim.
2432
2433
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2434
|-
2435
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Elem.}}</math>
2436
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, I}</math>
2437
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, F}</math>
2438
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{E}</math>
2439
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\sigma }}}_{\mathit{\boldsymbol{y}}}</math>
2440
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{\nu }</math>
2441
|-
2442
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2443
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2444
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2445
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{kN/c}}{\mathit{\boldsymbol{m}}}^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{)}}</math>
2446
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: left;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{kN/c}}{\mathit{\boldsymbol{m}}}^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{)}}</math>
2447
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2448
|-
2449
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2450
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2451
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2452
|  style="text-align: center;vertical-align: bottom;"|
2453
|  style="text-align: center;vertical-align: top;"|
2454
|  colspan='2'  style="text-align: center;vertical-align: bottom;"|
2455
|-
2456
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2457
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2458
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2459
|  style="text-align: center;vertical-align: bottom;"|1961,3
2460
|  style="text-align: center;vertical-align: top;"|9,8
2461
|  colspan='2'  style="text-align: center;vertical-align: bottom;"|0,170
2462
|-
2463
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2464
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2465
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2466
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2467
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
2468
|  colspan='2'  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2469
|}
2470
2471
2472
<span id='_Ref457418210'></span>
2473
2474
Os termos da <span id='cite-_Ref500759207'></span>[[#_Ref500759207|Tabela 8]] tem a seguinte descrição:
2475
2476
<math display="inline">E</math> = módulo de elasticidade;
2477
2478
<math display="inline">\nu</math>  = coeficiente de Poisson;
2479
2480
<math display="inline">{\sigma }_{y}</math>= tensão de escoamento.
2481
2482
<span id='_Ref494057745'></span>'''Tabela 9''' – Propriedades do material do pórtico plano - Thai e Kim.
2483
2484
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2485
|-
2486
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Elemento'''
2487
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Seção transversal'''
2488
|-
2489
|  rowspan='5' style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 100%;">1,2,3 e 4</span>
2490
|  style="text-align: center;"|<math>A=800,000\, c{m}^{2}</math>
2491
|-
2492
|  style="text-align: center;"|<math display="inline">b=20\, cm</math><span style="text-align: center; font-size: 75%;"> e </span> <math display="inline">h=</math><math>40\, cm</math>
2493
|-
2494
|  style="text-align: center;"|<math>{I}_{z}=106666,667\, c{m}^{4}</math>
2495
|-
2496
|  style="text-align: center;"|<math>{F}_{xp}=7840,000\, kN</math>
2497
|-
2498
|  style="border-bottom: 1pt solid black;text-align: center;"|<math>{M}_{zp}=78400,000\, kN\times cm</math>
2499
|}
2500
2501
2502
As cargas aplicadas são as seguintes:
2503
2504
'''Tabela 10''' – Cargas aplicadas do pórtico plano - Thai e Kim.
2505
2506
{| style="width: 58%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2507
|-
2508
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{N\acute{o}}}</math>
2509
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{Dir}</math>
2510
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{Valor\, (kN)}</math>
2511
|-
2512
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2513
|  style="text-align: center;vertical-align: bottom;"|<math>{F}_{X}</math>
2514
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1,000</span>
2515
|-
2516
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2517
|  style="text-align: center;vertical-align: bottom;"|<math>{F}_{y}</math>
2518
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">-1,000</span>
2519
|-
2520
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100;">3</span>
2521
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>{F}_{y}</math>
2522
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">-1,000</span>
2523
|}
2524
2525
2526
As funções utilizadas no caso são as <math display="inline">{f}_{4}</math>, <math display="inline">{f}_{5}</math> e <math display="inline">{f}_{6}</math> , equações <span id='cite-_Ref531207210'></span>[[#_Ref531207210|( 31 )]], <span id='cite-_Ref531207213'></span>[[#_Ref531207213|( 32 )]] e <span id='cite-_Ref531207215'></span>[[#_Ref531207215|( 33 )]], respectivamente.
2527
2528
O '''caso 2''' é baseado no pórtico espacial com dados dos trabalhos de Thai e Kim [11] e Argyris [12], conforme a <span id='cite-_Ref457421631'></span>[[#_Ref457421631|<span style="text-align: center; font-size: 100%;">Figura 9</span>]].
2529
2530
{| style="width: 100%;border-collapse: collapse;" 
2531
|-
2532
|  style="vertical-align: top;"|<span id='_Ref457421631'></span> [[Image:Draft_Vieira_908925676-image10.png|600px]] '''Figura 9''' – Pórtico espacial de 2 (dois) pavimentos.
2533
2534
2535
|}
2536
2537
2538
As propriedades do caso são apresentadas nas <span id='cite-_Ref457422104'></span>[[#_Ref457422104|Tabela 11]] a <span id='cite-_Ref532329759'></span>[[#_Ref532329759|Tabela 13]].
2539
2540
<span id='_Ref457422104'></span>'''Tabela 11''' - Coordenadas do pórtico espacial - Thai e Kim.
2541
2542
{| style="width: 58%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2543
|-
2544
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">''''''</span>
2545
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{X}}</math>
2546
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Y}}</math>
2547
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Z}}</math>
2548
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2549
|-
2550
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2551
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{cm}}</math><span style="text-align: center; font-size: 75%;">''')'''</span>
2552
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{cm}}</math><span style="text-align: center; font-size: 75%;">''')'''</span>
2553
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{cm}}</math><span style="text-align: center; font-size: 75%;">''')'''</span>
2554
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2555
|-
2556
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2557
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2558
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2559
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2560
|  style="text-align: center;vertical-align: top;"|
2561
|-
2562
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2563
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2564
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2565
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2566
|  style="text-align: center;vertical-align: top;"|
2567
|-
2568
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2569
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2570
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2571
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2572
|  style="text-align: center;vertical-align: top;"|
2573
|-
2574
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2575
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2576
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2577
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2578
|  style="text-align: center;vertical-align: top;"|
2579
|-
2580
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2581
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2582
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2583
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2584
|  style="text-align: center;vertical-align: top;"|
2585
|-
2586
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2587
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2588
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2589
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2590
|  style="text-align: center;vertical-align: top;"|
2591
|-
2592
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2593
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2594
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2595
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2596
|  style="text-align: center;vertical-align: top;"|
2597
|-
2598
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2599
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2600
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2601
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2602
|  style="text-align: center;vertical-align: top;"|
2603
|-
2604
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2605
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2606
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2607
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2608
|  style="text-align: center;vertical-align: top;"|
2609
|-
2610
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2611
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2612
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2613
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2614
|  style="text-align: center;vertical-align: top;"|
2615
|-
2616
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2617
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2618
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2619
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2620
|  style="text-align: center;vertical-align: top;"|
2621
|-
2622
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2623
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2624
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2625
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2626
|  style="text-align: center;vertical-align: top;"|
2627
|-
2628
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2629
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2630
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2631
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">150,000</span>
2632
|  style="text-align: center;vertical-align: top;"|
2633
|-
2634
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2635
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2636
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2637
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">150,000</span>
2638
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
2639
|}
2640
2641
2642
<span id='_Ref500757194'></span>
2643
2644
'''Tabela 12''' - Características físicas do pórtico espacial - Thai e Kim.
2645
2646
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2647
|-
2648
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span id='_Ref500758784'></span> <math>\mathit{\boldsymbol{Elem.}}</math>
2649
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, I}</math>
2650
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, F}</math>
2651
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{E}</math>
2652
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\sigma }}}_{\mathit{\boldsymbol{p}}}</math>
2653
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{\nu }</math>
2654
|-
2655
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2656
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2657
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2658
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{kN/c}}{\mathit{\boldsymbol{m}}}^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{)}}</math>
2659
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: left;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{kN/c}}{\mathit{\boldsymbol{m}}}^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{)}}</math>
2660
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2661
|-
2662
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2663
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2664
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2665
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|1961,3
2666
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|9,8
2667
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|0,170
2668
|-
2669
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2670
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2671
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2672
|-
2673
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2674
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2675
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2676
|-
2677
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2678
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2679
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2680
|-
2681
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2682
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2683
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2684
|-
2685
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2686
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2687
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2688
|-
2689
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2690
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2691
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2692
|-
2693
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2694
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2695
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2696
|-
2697
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2698
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2699
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2700
|-
2701
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2702
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2703
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2704
|-
2705
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2706
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2707
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2708
|-
2709
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2710
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2711
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2712
|-
2713
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2714
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2715
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2716
|-
2717
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2718
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2719
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2720
|-
2721
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">15</span>
2722
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2723
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2724
|-
2725
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">16</span>
2726
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2727
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2728
|-
2729
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">17</span>
2730
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2731
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2732
|-
2733
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">18</span>
2734
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2735
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2736
|}
2737
2738
2739
<span id='_Ref532329759'></span>'''Tabela 13''' - Propriedades do material do pórtico espacial - Thai e Kim.
2740
2741
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2742
|-
2743
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Elemento'''
2744
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Seção transversal'''
2745
|-
2746
|  rowspan='11' style="text-align: center;border-bottom: 1pt solid black;"|<span style="text-align: center; font-size: 100%;">1 a 18</span>
2747
|  style="text-align: center;"|<math>A=800,000\, c{m}^{2}</math>
2748
|-
2749
|  style="text-align: center;"|<math display="inline">b=20\, cm</math><span style="text-align: center; font-size: 75%;"> e </span> <math display="inline">h=</math><math>40\, cm</math>
2750
|-
2751
|  style="text-align: center;"|<math>{I}_{z}=1,067\times {10}^{5}\, c{m}^{4}</math>
2752
|-
2753
|  style="text-align: center;"|<math>{I}_{x}=1,067\times {10}^{5}\, c{m}^{4}</math>
2754
|-
2755
|  style="text-align: center;"|<math>{I}_{y}=2,667\times {10}^{4}\, c{m}^{4}</math>
2756
|-
2757
|  style="text-align: center;"|<math>{M}_{zp}=7,840\times {10}^{4}\, kN\times cm</math>
2758
|-
2759
|  style="text-align: center;"|<math>{M}_{xp}=6,533\times {10}^{4}\, kN\times cm</math>
2760
|-
2761
|  style="text-align: center;"|<math>{M}_{yp}=3,920\times {10}^{4}\, kN\times cm</math>
2762
|-
2763
|  style="text-align: center;"|<math>{F}_{zp}=4,526\times {10}^{3}\, kN</math>
2764
|-
2765
|  style="text-align: center;"|<math>{F}_{xp}=7,840\times {10}^{3}\, kN</math>
2766
|-
2767
|  style="border-bottom: 1pt solid black;text-align: center;"|<math>{F}_{yp}=4,526\times {10}^{3}\, kN</math>
2768
|}
2769
2770
2771
As cargas aplicadas são as seguintes:
2772
2773
'''Tabela 14''' – Cargas aplicadas do pórtico espacial - Thai e Kim.
2774
2775
{| style="width: 80%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2776
|-
2777
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{N\acute{o}}}</math>
2778
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{x}}}\mathit{\boldsymbol{\, (kN)}}</math>
2779
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''' '''</span> <math display="inline">{\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{y}}}\mathit{\boldsymbol{\, (kN)}}</math>
2780
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{z}}}\mathit{\boldsymbol{\, (kN)}}</math>
2781
|-
2782
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2783
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1,000</span>
2784
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2,000</span>
2785
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2786
|-
2787
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2788
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,250</span>
2789
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2,000</span>
2790
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2791
|-
2792
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2793
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,250</span>
2794
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2,000</span>
2795
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2796
|-
2797
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2798
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1,000</span>
2799
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2,000</span>
2800
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2801
|-
2802
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2803
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3,000</span>
2804
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-0,500</span>
2805
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2806
|-
2807
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2808
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,500</span>
2809
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-0,500</span>
2810
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2811
|-
2812
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2813
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,500</span>
2814
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-0,500</span>
2815
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2816
|-
2817
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2818
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3,000</span>
2819
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-0,500</span>
2820
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2821
|-
2822
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2823
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2824
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1,000</span>
2825
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2826
|-
2827
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2828
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2829
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1,000</span>
2830
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2831
|}
2832
2833
2834
As funções utilizadas no caso são as <math display="inline">{f}_{1}</math>, <math display="inline">{f}_{2}</math> e <math display="inline">{f}_{3}</math> , equações <span id='cite-_Ref533151259'></span>[[#_Ref533151259|( 28 )]], <span id='cite-_Ref533151260'></span>[[#_Ref533151260|( 29 )]] e <span id='cite-_Ref533151262'></span>[[#_Ref533151262|( 30 )]], respectivamente.
2835
2836
O '''caso 3''' é baseado no pórtico espacial com dados dos trabalhos de Argyris et al [12] e Park e Lee [13], conforme a  <span id='cite-_Ref533152504'></span>[[#_Ref533152504|Figura 10]].
2837
2838
{| style="width: 100%;border-collapse: collapse;" 
2839
|-
2840
|  style="text-align: center;vertical-align: top;width: 100%;"|[[Image:Draft_Vieira_908925676-image11.png|600px]]
2841
2842
2843
|}
2844
2845
2846
<span id='_Ref533152504'></span>'''Figura 10''' – Pórtico em domo - Argyris et al.
2847
2848
As propriedades do caso são apresentadas nas <span id='cite-_Ref534577723'></span>[[#_Ref534577723|Tabela 15]] a <span id='cite-_Ref534577726'></span>[[#_Ref534577726|Tabela 18]].
2849
2850
<span id='_Ref534577723'></span>'''Tabela 15''' - Coordenadas do <span style="text-align: center; font-size: 75%;">pórtico em domo - </span>Argyris et al.
2851
2852
{| style="width: 65%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2853
|-
2854
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">''''''</span>
2855
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{X}}</math>
2856
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Y}}</math>
2857
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Z}}</math>
2858
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2859
|-
2860
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2861
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{cm}}</math><span style="text-align: center; font-size: 75%;">''')'''</span>
2862
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{cm}}</math><span style="text-align: center; font-size: 75%;">''')'''</span>
2863
|  style="border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{cm}}</math><span style="text-align: center; font-size: 75%;">''')'''</span>
2864
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2865
|-
2866
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2867
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2868
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">610,000</span>
2869
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2870
|  style="text-align: center;vertical-align: top;"|
2871
|-
2872
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2873
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">628,500</span>
2874
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2875
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1088,500</span>
2876
|  style="text-align: center;vertical-align: top;"|
2877
|-
2878
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2879
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1257,000</span>
2880
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2881
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2882
|  style="text-align: center;vertical-align: top;"|
2883
|-
2884
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2885
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">628,500</span>
2886
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2887
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1088,500</span>
2888
|  style="text-align: center;vertical-align: top;"|
2889
|-
2890
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2891
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-628,500</span>
2892
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2893
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1088,500</span>
2894
|  style="text-align: center;vertical-align: top;"|
2895
|-
2896
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2897
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1257,000</span>
2898
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2899
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2900
|  style="text-align: center;vertical-align: top;"|
2901
|-
2902
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2903
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-628,500</span>
2904
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2905
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">1088,500</span>
2906
|  style="text-align: center;vertical-align: top;"|
2907
|-
2908
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2909
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">1219,000</span>
2910
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2911
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">2111,500</span>
2912
|  style="text-align: center;vertical-align: top;"|
2913
|-
2914
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2915
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">2438,000</span>
2916
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2917
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2918
|  style="text-align: center;vertical-align: top;"|
2919
|-
2920
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2921
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">1219,000</span>
2922
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2923
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2111,500</span>
2924
|  style="text-align: center;vertical-align: top;"|
2925
|-
2926
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2927
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1219,000</span>
2928
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2929
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2111,500</span>
2930
|  style="text-align: center;vertical-align: top;"|
2931
|-
2932
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2933
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2438,000</span>
2934
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2935
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2936
|  style="text-align: center;vertical-align: top;"|
2937
|-
2938
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2939
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1219,000</span>
2940
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2941
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">2111,500</span>
2942
|  style="text-align: center;vertical-align: top;"|
2943
|}
2944
2945
2946
'''Tabela 16''' - Características físicas do <span style="text-align: center; font-size: 75%;">pórtico em domo - </span>Argyris et al.
2947
2948
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2949
|-
2950
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Elem.}}</math>
2951
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, I}</math>
2952
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, F}</math>
2953
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{E}</math>
2954
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>{\mathit{\boldsymbol{\sigma }}}_{\mathit{\boldsymbol{p}}}</math>
2955
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{\nu }</math>
2956
|-
2957
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2958
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2959
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2960
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{kN/c}}{\mathit{\boldsymbol{m}}}^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{)}}</math>
2961
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: left;"|<span style="text-align: center; font-size: 75%;">'''('''</span> <math display="inline">\mathit{\boldsymbol{kN/c}}{\mathit{\boldsymbol{m}}}^{\mathit{\boldsymbol{2}}}\mathit{\boldsymbol{)}}</math>
2962
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2963
|-
2964
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2965
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2966
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2967
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|2068,0
2968
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|8,0
2969
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|0,1716
2970
|-
2971
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2972
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2973
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2974
|-
2975
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2976
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2977
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2978
|-
2979
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2980
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2981
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2982
|-
2983
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2984
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2985
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2986
|-
2987
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2988
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2989
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2990
|-
2991
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2992
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2993
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2994
|-
2995
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2996
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2997
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2998
|-
2999
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
3000
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
3001
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
3002
|-
3003
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
3004
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
3005
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
3006
|-
3007
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
3008
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
3009
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
3010
|-
3011
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
3012
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
3013
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
3014
|-
3015
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
3016
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
3017
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
3018
|-
3019
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
3020
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
3021
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
3022
|-
3023
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">15</span>
3024
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
3025
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
3026
|-
3027
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">16</span>
3028
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
3029
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
3030
|-
3031
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">17</span>
3032
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
3033
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
3034
|-
3035
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">18</span>
3036
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
3037
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
3038
|}
3039
3040
3041
'''Tabela 17''' - Propriedades do material do <span style="text-align: center; font-size: 75%;">pórtico em domo - </span>Argyris et al.
3042
3043
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3044
|-
3045
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Elemento'''
3046
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Seção transversal'''
3047
|-
3048
|  rowspan='11' style="text-align: center;border-bottom: 1pt solid black;"|<span style="text-align: center; font-size: 100%;">1 a 18</span>
3049
|  style="text-align: center;"|<math>A=9272,000\, c{m}^{2}</math>
3050
|-
3051
|  style="text-align: center;"|<math display="inline">b=76\, cm</math><span style="text-align: center; font-size: 75%;"> e </span> <math display="inline">h=</math><math>122\, cm</math>
3052
|-
3053
|  style="text-align: center;"|<math>{I}_{z}=1,150\times {10}^{7}\, c{m}^{4}</math>
3054
|-
3055
|  style="text-align: center;"|<math>{I}_{x}=1,596\times {10}^{7}\, c{m}^{4}</math>
3056
|-
3057
|  style="text-align: center;"|<math>{I}_{y}=4,463\times {10}^{6}\, c{m}^{4}</math>
3058
|-
3059
|  style="text-align: center;"|<math>{M}_{zp}=2,262\times {10}^{6}\, kN\times cm</math>
3060
|-
3061
|  style="text-align: center;"|<math>{M}_{xp}=6,533\times {10}^{9}\, kN\times cm</math>
3062
|-
3063
|  style="text-align: center;"|<math>{M}_{yp}=1,409\times {10}^{6}\, kN\times cm</math>
3064
|-
3065
|  style="text-align: center;"|<math>{F}_{zp}=4,282\times {10}^{4}\, kN</math>
3066
|-
3067
|  style="text-align: center;"|<math>{F}_{xp}=7,416\times {10}^{4}\, kN</math>
3068
|-
3069
|  style="border-bottom: 1pt solid black;text-align: center;"|<math>{F}_{yp}=4,282\times {10}^{4}\, kN</math>
3070
|}
3071
3072
3073
As cargas aplicadas são as seguintes:
3074
3075
<span id='_Ref534577726'></span>'''Tabela 18''' – Cargas aplicadas do <span style="text-align: center; font-size: 75%;">pórtico em domo - </span>Argyris et al.
3076
3077
{| style="width: 80%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3078
|-
3079
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{N\acute{o}}}</math>
3080
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{x}}}\mathit{\boldsymbol{\, (kN)}}</math>
3081
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''' '''</span> <math display="inline">{\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{y}}}\mathit{\boldsymbol{\, (kN)}}</math>
3082
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{z}}}\mathit{\boldsymbol{\, (kN)}}</math>
3083
|-
3084
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
3085
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
3086
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1,000</span>
3087
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
3088
|}
3089
3090
3091
As funções utilizadas no caso são as <math display="inline">{f}_{1}</math>, <math display="inline">{f}_{2}</math> e <math display="inline">{f}_{3}</math> , equações <span id='cite-_Ref533151259'></span>[[#_Ref533151259|( 28 )]], <span id='cite-_Ref533151260'></span>[[#_Ref533151260|( 29 )]] e <span id='cite-_Ref533151262'></span>[[#_Ref533151262|( 30 )]], respectivamente.
3092
3093
==3 RESULTADOS E DISCUSSÃO==
3094
3095
Os resultados e discussões dos estudos de caso 1 a 3 são apresentados.
3096
3097
===3.1 Caso 1===
3098
3099
Os resultados dos estudos, '''s. '''O número de elementos plastificados foram 3 (três) para todas as funções e a quantidade de rótulas 5 (cinco). Os caminhos da formação das rótulas são apresentados na <span id='cite-_Ref4574898'></span>[[#_Ref4574898|'''Tabela 19''']] e <span id='cite-_Ref457419276'></span>[[#_Ref457419276|'''Figura 11''']], com as 3 (três) funções <math display="inline">{f}_{4},\, {f}_{5}\, e\, {f}_{6}\, \,</math> mostram que os resultados estão mais próximos da solução do ABAQUS de 20 elementos de Thai e Kim [11] que teve o fator de carga entre 1 e 1,2. Do ponto de vista estatístico, a melhor solução seria a função <math display="inline">{f}_{4}</math>, depois <math display="inline">{f}_{6}</math> e por último <math display="inline">{f}_{5}</math>. No entanto, a carga limite de <math display="inline">{f}_{5}</math> é que se aproxima melhor aos resultados de Thai e Kim [11] com fatores de carga próximos a 0,8. Se levarmos em conta a solução de 20 elementos do ABACUS de Thai e Kim [11], a função <math display="inline">{f}_{4}</math> foi de fato a melhor corroborando com os resultados estatísticos. O número de elementos plastificados foram 3 (três) para todas as funções e a quantidade de rótulas 5 (cinco). Os caminhos da formação das rótulas são apresentados na <span id='cite-_Ref4574898'></span>[[#_Ref4574898|'''Tabela 19''']].
3100
3101
<span id='_Ref4574898'></span>'''Tabela 19''' – Rótulas plásticas – pórtico plano - Thai e Kim.
3102
3103
{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3104
|-
3105
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span id='_Hlk4522464'></span>
3106
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3107
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{4}</math>
3108
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3109
|-
3110
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: left; font-size: 75%;">'''Elemento'''</span>
3111
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3112
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3113
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3114
|-
3115
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3116
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3117
|  style="text-align: center;vertical-align: bottom;"|<math>0,108246\times {10}^{5}</math>
3118
|  rowspan='5' style="text-align: center;border-bottom: 1pt solid black;"|<math>309,146</math>
3119
|-
3120
|  style="text-align: center;vertical-align: bottom;"|
3121
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3122
|  style="text-align: center;vertical-align: bottom;"|<math>0,841207\times {10}^{4}</math>
3123
|-
3124
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: left; font-size: 75%;">2</span>
3125
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3126
|  style="text-align: center;vertical-align: bottom;"|<math>0,227389\times {10}^{2}</math>
3127
|-
3128
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: left; font-size: 75%;">3</span>
3129
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3130
|  style="text-align: center;vertical-align: bottom;"|<math>0,844737\times {10}^{4}</math>
3131
|-
3132
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
3133
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3134
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,108410\times {10}^{5}</math>
3135
|}
3136
3137
3138
{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3139
|-
3140
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3141
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3142
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{5}</math>
3143
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3144
|-
3145
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3146
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3147
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3148
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3149
|-
3150
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3151
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3152
|  style="text-align: center;vertical-align: bottom;"|<math>0,106273\times {10}^{5}</math>
3153
|  rowspan='5' style="border-bottom: 1pt solid black;text-align: center;"|<math>300,431</math>
3154
|-
3155
|  style="text-align: center;vertical-align: bottom;"|
3156
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3157
|  style="text-align: center;vertical-align: bottom;"|<math>0,208395\times {10}^{4}</math>
3158
|-
3159
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3160
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3161
|  style="text-align: center;vertical-align: bottom;"|<math>0,626480\times {10}^{4}</math>
3162
|-
3163
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3164
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3165
|  style="text-align: center;vertical-align: bottom;"|<math>0,836829\times {10}^{4}</math>
3166
|-
3167
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
3168
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3169
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,106452\times {10}^{5}</math>
3170
|-
3171
|-
3172
|}
3173
3174
3175
{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3176
|-
3177
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3178
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3179
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{6}</math>
3180
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3181
|-
3182
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3183
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3184
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3185
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3186
|-
3187
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3188
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3189
|  style="text-align: center;vertical-align: bottom;"|<math>0,114845\times {10}^{5}</math>
3190
|  rowspan='5' style="text-align: center;border-bottom: 1pt solid black;"|<math>318,103</math>
3191
|-
3192
|  style="text-align: center;vertical-align: bottom;"|
3193
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3194
|  style="text-align: center;vertical-align: bottom;"|<math>0,254762\times {10}^{4}</math>
3195
|-
3196
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3197
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3198
|  style="text-align: center;vertical-align: bottom;"|<math>0,604007\times {10}^{4}</math>
3199
|-
3200
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3201
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3202
|  style="text-align: center;vertical-align: bottom;"|<math>0,899557\times {10}^{4}</math>
3203
|-
3204
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
3205
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3206
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,117705\times {10}^{5}</math>
3207
|}
3208
3209
3210
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
3211
 [[Image:Draft_Vieira_908925676-image12-c.png|438px]] </div>
3212
3213
<span id='_Ref457419276'></span>'''Figura 11''' – Gráfico carga versus deslocamento horizontal ( <math display="inline">{H}_{x})-n\acute{o}\, 2</math> - pórtico plano - Thai e Kim.
3214
3215
{| style="width: 100%;border-collapse: collapse;" 
3216
|-
3217
|  style="text-align: center;vertical-align: top;"|
3218
|}
3219
3220
===3.2 Caso 2===
3221
3222
Os resultados dos estudos, <span id='cite-_Ref4571218'></span>[[#_Ref4571218|'''Tabela 20''']] e <span id='cite-_Ref500765589'></span>[[#_Ref500765589|'''Figura 12''']], com as 3 (três) funções <math display="inline">{f}_{2},\, {f}_{3}\, e\, {f}_{1}\, \,</math> mostram que os resultados das cargas limites estão mais elevados do que os de Thai e Kim [11] que teve o valor de 128,82 <math display="inline">kN</math>. Do ponto de vista estatístico, a melhor solução seria a função <math display="inline">{f}_{2}</math>, depois <math display="inline">{f}_{3}</math> e por último <math display="inline">{f}_{1}</math>. Neste caso, a carga limite de <math display="inline">{f}_{2}</math> é a que realmente se aproxima melhor aos resultados de Thai e Kim [11], confirmando os resultados estatísticos com uma diferença relativa de 4,08% (( <math display="inline">{f}_{2}-</math><math>Thai)/Thai\, \times 100)</math>. Esta diferença pode ter ocorrido porque os esforços plásticos limites (momentos, cortantes e axial) não existem nos dados de Thai e Kim [11], porém são necessários na teoria apresentada. O número de elementos plastificados foram 10 (dez) para todas as funções e a quantidade de rótulas 12 (doze). Os caminhos da formação das rótulas são apresentados na <span id='cite-_Ref4571218'></span>[[#_Ref4571218|'''Tabela 20''']].
3223
3224
<span id='_Ref4571218'></span><span id='_Ref500765287'></span>'''Tabela 20''' - Rótulas plásticas – pórtico espacial - Thai e Kim
3225
3226
{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3227
|-
3228
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3229
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3230
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{1}</math>
3231
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3232
|-
3233
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3234
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3235
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3236
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3237
|-
3238
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3239
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3240
|  style="text-align: center;vertical-align: bottom;"|<math>0,345728\times {10}^{4}</math>
3241
|  rowspan='12' style="text-align: center;border-bottom: 1pt solid black;"|<math>141,886</math>
3242
|-
3243
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3244
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3245
|  style="text-align: center;vertical-align: bottom;"|<math>0,353734\times {10}^{4}</math>
3246
|-
3247
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3248
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3249
|  style="text-align: center;vertical-align: bottom;"|<math>0,348667\times {10}^{4}</math>
3250
|-
3251
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3252
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3253
|  style="text-align: center;vertical-align: bottom;"|<math>0,350863\times {10}^{4}</math>
3254
|-
3255
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3256
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3257
|  style="text-align: center;vertical-align: bottom;"|<math>0,308873\times {10}^{4}</math>
3258
|-
3259
|  style="text-align: center;vertical-align: bottom;"|
3260
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3261
|  style="text-align: center;vertical-align: bottom;"|<math>0,307316\times {10}^{4}</math>
3262
|-
3263
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3264
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3265
|  style="text-align: center;vertical-align: bottom;"|<math>0,306402\times {10}^{4}</math>
3266
|-
3267
|  style="text-align: center;vertical-align: bottom;"|
3268
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">8</span>
3269
|  style="text-align: center;vertical-align: bottom;"|<math>0,309575\times {10}^{4}</math>
3270
|-
3271
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3272
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3273
|  style="text-align: center;vertical-align: bottom;"|<math>0,899973\times {10}^{3}</math>
3274
|-
3275
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3276
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3277
|  style="text-align: center;vertical-align: bottom;"|<math>0,933775\times {10}^{3}</math>
3278
|-
3279
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3280
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3281
|  style="text-align: center;vertical-align: bottom;"|<math>0,984856\times {10}^{3}</math>
3282
|-
3283
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3284
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3285
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,849066\times {10}^{3}</math>
3286
|}
3287
3288
3289
{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3290
|-
3291
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3292
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3293
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{2}</math>
3294
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3295
|-
3296
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3297
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3298
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3299
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3300
|-
3301
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3302
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3303
|  style="text-align: center;vertical-align: bottom;"|<math>0,321786\times {10}^{4}</math>
3304
|  rowspan='12' style="border-bottom: 1pt solid black;text-align: center;"|<math>134,077</math>
3305
|-
3306
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3307
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3308
|  style="text-align: center;vertical-align: bottom;"|<math>0,326692\times {10}^{4}</math>
3309
|-
3310
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3311
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3312
|  style="text-align: center;vertical-align: bottom;"|<math>0,326692\times {10}^{4}</math>
3313
|-
3314
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3315
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3316
|  style="text-align: center;vertical-align: bottom;"|<math>0,321786\times {10}^{4}</math>
3317
|-
3318
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3319
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3320
|  style="text-align: center;vertical-align: bottom;"|<math>0,299500\times {10}^{4}</math>
3321
|-
3322
|  style="text-align: center;vertical-align: bottom;"|
3323
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3324
|  style="text-align: center;vertical-align: bottom;"|<math>0,298730\times {10}^{4}</math>
3325
|-
3326
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3327
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3328
|  style="text-align: center;vertical-align: bottom;"|<math>0,298730\times {10}^{4}</math>
3329
|-
3330
|  style="text-align: center;vertical-align: bottom;"|
3331
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">8</span>
3332
|  style="text-align: center;vertical-align: bottom;"|<math>0,299500\times {10}^{4}</math>
3333
|-
3334
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3335
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3336
|  style="text-align: center;vertical-align: bottom;"|<math>0,106747\times {10}^{4}</math>
3337
|-
3338
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3339
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3340
|  style="text-align: center;vertical-align: bottom;"|<math>0,107793\times {10}^{4}</math>
3341
|-
3342
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3343
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3344
|  style="text-align: center;vertical-align: bottom;"|<math>0,107793\times {10}^{4}</math>
3345
|-
3346
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3347
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3348
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,106747\times {10}^{4}</math>
3349
|}
3350
3351
3352
{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3353
|-
3354
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3355
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3356
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{3}</math>
3357
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3358
|-
3359
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3360
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3361
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3362
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3363
|-
3364
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3365
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3366
|  style="text-align: center;vertical-align: bottom;"|<math>0,345110\times {10}^{4}</math>
3367
|  rowspan='5' style="text-align: center;"|<math>141,900</math>
3368
|-
3369
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3370
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3371
|  style="text-align: center;vertical-align: bottom;"|<math>0,353048\times {10}^{4}</math>
3372
|-
3373
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3374
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3375
|  style="text-align: center;vertical-align: bottom;"|<math>0,348312\times {10}^{4}</math>
3376
|-
3377
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3378
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3379
|  style="text-align: center;vertical-align: bottom;"|<math>0,349939\times {10}^{4}</math>
3380
|-
3381
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3382
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3383
|  style="text-align: center;vertical-align: bottom;"|<math>0,309236\times {10}^{4}</math>
3384
|-
3385
|  style="text-align: center;vertical-align: bottom;"|
3386
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3387
|  style="text-align: center;vertical-align: bottom;"|<math>0,307839\times {10}^{4}</math>
3388
|  style="text-align: right;vertical-align: bottom;"|
3389
|-
3390
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3391
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3392
|  style="text-align: center;vertical-align: bottom;"|<math>0,306698\times {10}^{4}</math>
3393
|  style="text-align: right;vertical-align: bottom;"|
3394
|-
3395
|  style="text-align: center;vertical-align: bottom;"|
3396
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">8</span>
3397
|  style="text-align: center;vertical-align: bottom;"|<math>0,309736\times {10}^{4}</math>
3398
|  style="text-align: right;vertical-align: bottom;"|
3399
|-
3400
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3401
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3402
|  style="text-align: center;vertical-align: bottom;"|<math>0,905438\times {10}^{3}</math>
3403
|  style="text-align: right;vertical-align: bottom;"|
3404
|-
3405
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3406
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3407
|  style="text-align: center;vertical-align: bottom;"|<math>0,938623\times {10}^{3}</math>
3408
|  style="text-align: right;vertical-align: bottom;"|
3409
|-
3410
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3411
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3412
|  style="text-align: center;vertical-align: bottom;"|<math>0,983798\times {10}^{3}</math>
3413
|  style="text-align: right;vertical-align: bottom;"|
3414
|-
3415
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3416
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3417
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,855577\times {10}^{3}</math>
3418
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: bottom;"|
3419
|}
3420
3421
3422
<div id="_Ref457419942" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
3423
 [[Image:Draft_Vieira_908925676-image13-c.png|456px]] </div>
3424
3425
<span id='_Ref500765589'></span>'''Figura 12''' - Gráfico carga versus deslocamento horizontal ( <math display="inline">{P}_{x})-n\acute{o}\, 12</math> - pórtico espacial - Thai e Kim.
3426
3427
===3.3 Caso 3===
3428
3429
Os resultados dos estudos, <span id='cite-_Ref4573319'></span>[[#_Ref4573319|'''Tabela 21''']] e <span id='cite-_Ref4573339'></span>[[#_Ref4573339|'''Figura 13''']], com as 3 (três) funções <math display="inline">{f}_{2},\, {f}_{3}\, e\, {f}_{1}\, \,</math> mostram que os resultados das cargas limites estão próximos de Argyris et al [12], visualmente, porque o valor exato não é apresentado. Do ponto de vista estatístico, a melhor solução seria a função <math display="inline">{f}_{2}</math>, depois <math display="inline">{f}_{3}</math> e por último <math display="inline">{f}_{1}</math>. Neste caso, a carga limite de <math display="inline">{f}_{3}</math> é a que realmente se aproxima melhor aos resultados de Argyris et al [12], visualmente. Porém, esta função não conseguiu avançar o caminho de deslocamento em relação aos demais. Se avaliarmos a trajetória de deslocamento, a função <math display="inline">{f}_{2}</math> será a melhor e corroborará com os resultados estatísticos. O número de elementos plastificados foram 6 (seis) para as funções <math display="inline">{f}_{1}</math> e <math display="inline">{f}_{2}</math> e a quantidade de rótulas 12 (doze). Já a função <math display="inline">{f}_{3}</math> foram 6 (seis) elementos e 6 (seis) rótulas. Os caminhos da formação das rótulas são apresentados na <span id='cite-_Ref4573319'></span>[[#_Ref4573319|'''Tabela 21''']].
3430
3431
<span id='_Ref4573319'></span>'''Tabela 21''' - Rótulas plásticas – pórtico espacial - pórtico espacial - Argyris et al.
3432
3433
{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3434
|-
3435
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3436
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3437
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{1}</math>
3438
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3439
|-
3440
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3441
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3442
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3443
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (MN)}}</math>
3444
|-
3445
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3446
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3447
|  style="text-align: center;vertical-align: bottom;"|<math>0,549351</math>
3448
|  rowspan='12' style="border-bottom: 1pt solid black;text-align: center;"|<math>51,0815</math>
3449
|-
3450
|  style="text-align: center;vertical-align: bottom;"|
3451
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3452
|  style="text-align: center;vertical-align: bottom;"|<math>0,891167</math>
3453
|-
3454
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3455
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3456
|  style="text-align: center;vertical-align: bottom;"|<math>0,552499</math>
3457
|-
3458
|  style="text-align: center;vertical-align: bottom;"|
3459
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3460
|  style="text-align: center;vertical-align: bottom;"|<math>0,861987</math>
3461
|-
3462
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3463
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3464
|  style="text-align: center;vertical-align: bottom;"|<math>0,564206</math>
3465
|-
3466
|  style="text-align: center;vertical-align: bottom;"|
3467
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3468
|  style="text-align: center;vertical-align: bottom;"|<math>0,883022</math>
3469
|-
3470
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3471
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3472
|  style="text-align: center;vertical-align: bottom;"|<math>0,569621</math>
3473
|-
3474
|  style="text-align: center;vertical-align: bottom;"|
3475
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3476
|  style="text-align: center;vertical-align: bottom;"|<math>0,885590</math>
3477
|-
3478
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3479
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3480
|  style="text-align: center;vertical-align: bottom;"|<math>0,556237</math>
3481
|-
3482
|  style="text-align: center;vertical-align: bottom;"|
3483
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3484
|  style="text-align: center;vertical-align: bottom;"|<math>0,862806</math>
3485
|-
3486
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3487
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3488
|  style="text-align: center;vertical-align: bottom;"|<math>0,541887</math>
3489
|-
3490
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
3491
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3492
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,934919</math>
3493
|-
3494
|-
3495
|}
3496
3497
3498
{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3499
|-
3500
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3501
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3502
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{2}</math>
3503
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3504
|-
3505
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3506
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3507
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3508
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (MN)}}</math>
3509
|-
3510
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3511
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3512
|  style="text-align: center;vertical-align: bottom;"|<math>2,63534</math>
3513
|  rowspan='12' style="border-bottom: 1pt solid black;text-align: center;"|<math>54,4497</math>
3514
|-
3515
|  style="text-align: center;vertical-align: bottom;"|
3516
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3517
|  style="text-align: center;vertical-align: bottom;"|<math>2,62747</math>
3518
|-
3519
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3520
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3521
|  style="text-align: center;vertical-align: bottom;"|<math>2,64188</math>
3522
|-
3523
|  style="text-align: center;vertical-align: bottom;"|
3524
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3525
|  style="text-align: center;vertical-align: bottom;"|<math>2,63063</math>
3526
|-
3527
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3528
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3529
|  style="text-align: center;vertical-align: bottom;"|<math>2,64561</math>
3530
|-
3531
|  style="text-align: center;vertical-align: bottom;"|
3532
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3533
|  style="text-align: center;vertical-align: bottom;"|<math>2,63270</math>
3534
|-
3535
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3536
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3537
|  style="text-align: center;vertical-align: bottom;"|<math>2,64226</math>
3538
|-
3539
|  style="text-align: center;vertical-align: bottom;"|
3540
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3541
|  style="text-align: center;vertical-align: bottom;"|<math>2,63100</math>
3542
|-
3543
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3544
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3545
|  style="text-align: center;vertical-align: bottom;"|<math>2,63496</math>
3546
|-
3547
|  style="text-align: center;vertical-align: bottom;"|
3548
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3549
|  style="text-align: center;vertical-align: bottom;"|<math>2,62711</math>
3550
|-
3551
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3552
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3553
|  style="text-align: center;vertical-align: bottom;"|<math>2,63079</math>
3554
|-
3555
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
3556
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3557
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>2,62451</math>
3558
|-
3559
|}
3560
3561
3562
{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3563
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3564
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3565
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{3}</math>
3566
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3567
|-
3568
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3569
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3570
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3571
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (MN)}}</math>
3572
|-
3573
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3574
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3575
|  style="text-align: center;vertical-align: bottom;"|<math>0,227607</math>
3576
|  rowspan='6' style="border-bottom: 1pt solid black;text-align: center;"|<math>48,0914</math>
3577
|-
3578
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3579
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3580
|  style="text-align: center;vertical-align: bottom;"|<math>0,227118</math>
3581
|-
3582
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3583
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3584
|  style="text-align: center;vertical-align: bottom;"|<math>0,226262</math>
3585
|-
3586
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3587
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3588
|  style="text-align: center;vertical-align: bottom;"|<math>0,224668</math>
3589
|-
3590
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3591
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3592
|  style="text-align: center;vertical-align: bottom;"|<math>0,224136</math>
3593
|-
3594
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3595
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3596
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,225236</math>
3597
|}
3598
3599
3600
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
3601
 [[Image:Draft_Vieira_908925676-image14-c.png|444px]] </div>
3602
3603
<span id='_Ref4573339'></span>'''Figura 13''' - Gráfico carga versus deslocamento horizontal ( <math display="inline">{P}_{y})-n\acute{o}\, 1</math> - pórtico espacial - Argyris et al.
3604
3605
==4 CONCLUSÕES==
3606
3607
'''* '''As superfícies de escoamento em resultantes de tensões aplicadas para pórticos planos e espaciais obtiveram resultados satisfatórios tendo em vista os exemplos apresentados;
3608
3609
'''* '''Os resultados estatísticos conseguiram detectar as melhores funções, a saber, <math display="inline">{f}_{2}</math> e <math display="inline">{f}_{4}</math> para as análises desenvolvidas;
3610
3611
'''* '''O método apresentado permitiu o uso do modelo de dano de viga de Timoshenko 3D com bons resultados para estruturas de aço;
3612
3613
'''* '''A regressão linear múltipla se apresenta como uma solução viável para obter funções por análises numéricas e/ou experimentais;
3614
3615
'''* '''O processo de formação de rótulas plásticas foi similares para os casos 1 e 2, porém o caso 3 apresentou distinções entre as funções apresentadas com <math display="inline">{f}_{1}</math> e <math display="inline">{f}_{2}</math> similares e diferente para <math display="inline">{f}_{3}</math>. Isto impactou nos resultados das cargas limites e trajetória de deslocamento do caso 3.
3616
3617
==AGRADECIMENTOS==
3618
3619
À UFOB, CIMNE/UPC, PECC/UnB e a CAPES.
3620
3621
== REFERÊNCIAS==
3622
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3624
{| style="width: 100%;" 
3625
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3626
|  style="text-align: left;vertical-align: top;"|[1]
3627
|  style="vertical-align: top;"|VIEIRA, P. C. S. '''Geração de Superfícies de Interação pelo Método da Regressão Linear Múltipla com o Modelo de Dano em Vigas de Timoshenko 3D'''. Pub. E.TD- 006A/04. Departamento de Engenharia Civil e Ambiental, Universidade de Brasília. Brasília. 2004.
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|-
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|  style="text-align: left;vertical-align: top;"|[2]   
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|  style="vertical-align: top;"|HANGANU, A. D. '''Metodologia de Evaluación del Deterioro en Estructuras de Hormigón Armado'''. Monografia CIMNE nº 39. CIMNE, UPC. Barcelona. 1997.
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|-
3632
|  style="text-align: left;vertical-align: top;"|[3]   
3633
|  style="vertical-align: top;"|LUBLINER, J. '''Plasticity Theory'''. Nova Iorque: Macmillan Publishing Company, 1990.
3634
|-
3635
|  style="text-align: left;vertical-align: top;"|[4] 
3636
|  style="vertical-align: top;"|MRÁZIK A., ÉSKALOUD M., TOCHÁÉCEK M. '''Plastic Design of Steel Structures'''. Nova Iorque: E. Horwood: Halsted Press, 1987.
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|  style="text-align: left;vertical-align: top;"|[5] 
3639
|  style="vertical-align: top;"|CRISFIELD, M. A. '''A Consistent Co-rotational Formulation for Non-linear, Three-dimensional'''. Comp. Methods Appl. Mech. Engrg., v. 81, p. 131-150, 1990.
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|  style="text-align: left;vertical-align: top;"|[6]   
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|  style="vertical-align: top;"|VIEIRA P. C. S., SILVA W. T. M. '''Análise Elastoplástica de Estruturas Aporticadas com Superfícies de Interação Obtidas por Regressão Linear Múltipla'''. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, v. 3, p. 175–187, 2013.
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|  style="text-align: left;vertical-align: top;"|[7]   
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|  style="vertical-align: top;"|MONTGOMERY D. C., RUNGER G. C. '''Estatística Aplicada e Probabilidade para Engenheiros'''. Rio de Janeiro: LTC, 2016.
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|  style="text-align: left;vertical-align: top;"|[8]   
3648
|  style="vertical-align: top;"|HORNE, M. R. '''Plastic theory of structures'''. 2ª. ed. Oxford: Pergamon Press, 1972.
3649
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|  style="text-align: left;vertical-align: top;"|[9]
3651
|  style="vertical-align: top;"|NEAL, B. G. '''The plastic methods of structural analysis'''. Inglaterra: Chapman and Hall, 1977.
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|  style="text-align: left;vertical-align: top;"|[10]  
3654
|  style="vertical-align: top;"|SILVA, W. T. M. '''Análise Elastoplástica de Pórticos Espaciais Utilizando o Conceito de Rótula Plástica e o Método de Backward Euler.'''. Métodos Computacionais em Engenharia. Lisboa: [s.n.]. 2004.
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|  style="text-align: left;vertical-align: top;"|[11]   
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|  style="vertical-align: top;"|THAI H. T., KIM S. E. '''Nonlinear inelastic analysis of space frames'''. Journal of Constructional Steel Research, v. 67, p. 585–592, 2011.
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3659
|  style="text-align: left;vertical-align: top;"|[12]   
3660
|  style="vertical-align: top;"|ARGYRIS, J. H. et al. '''Finite Element Analysis of Two and Three-Dimensional Elasto-Plastic Frames - The Natural Approach'''. Comp. Method. in Applied Mechanics and Engineering, v. 35, p. 221-248, 1982.
3661
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3662
|  style="text-align: left;vertical-align: top;"|[13]
3663
|  style="vertical-align: top;"|PARK M. S., LEE B. C. '''Geometrically Non-Linear and Elastoplastic Three-Dimensional Shear Flexible Beam Element of Von-Mises-Type Hardening Material'''. International Journal for Numerical Methods in Engineering, v. 39, p. 383-408, 1996.
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|}
3665

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Document information

Published on 01/10/19
Accepted on 25/09/19
Submitted on 06/04/19

Volume 35, Issue 4, 2019
DOI: 10.23967/j.rimni.2019.09.009
Licence: CC BY-NC-SA license

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