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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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ANÁLISE ELASTOPLÁSTICA DE PÓRTICOS METÁLICOS COM SUPERFÍCIES DE INTERAÇÃO EM RESULTANTES DE TENSÕES OBTIDAS POR REGRESSÃO MÚLTIPLA.</div>
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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ELASTOPLASTIC ANALYSIS OF STEEL FRAMES WITH INTERACTION SURFACES IN STRESS RESULTANTS OBTAINED FOR MULTIPLE LINEAR REGRESSION. </div>
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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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'''RESUMO'''</div>
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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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<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
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'''ABSTRACT'''</div>
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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
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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
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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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| <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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com
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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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|}
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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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|-
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|  style="text-align: center;vertical-align: top;width: 100%;"|[[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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|}
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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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|  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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|-
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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]] '''
383
|}
384
385
386
<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:
389
390
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
391
|-
392
| 
393
{| style="text-align: center; margin:auto;width: 100%;"
394
|-
395
| <math>f={\beta }_{1}{n}^{2}+{\beta }_{2}m-1=0</math>
396
|}
397
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 19 )
398
|}
399
400
401
Onde
402
403
<math display="inline">n\, e\, m</math> = esforços normal e fletor adimensionais, respectivamente;
404
405
<math display="inline">{\beta }_{1}\, e\, {\beta }_{2}</math> = coeficientes obtidos pela regressão linear múltipla.
406
407
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].
408
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O modelo desenvolvido para a formulação pretendida tem a seguinte forma:
410
411
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
412
|-
413
| 
414
{| style="text-align: center; margin:auto;width: 100%;"
415
|-
416
| <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>
417
418
419
|}
420
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529721691'></span>( 20 )
421
|}
422
423
424
Onde:
425
426
<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;
427
428
<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;
429
430
<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;
431
432
<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;
433
434
<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;
435
436
<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.
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438
Na regressão as observações da equação <span id='cite-_Ref529721691'></span>[[#_Ref529721691|( 20 )]] podem ser apresentadas como
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
441
|-
442
| 
443
{| style="text-align: center; margin:auto;width: 100%;"
444
|-
445
| <math>Y={\beta }_{0}+{\beta }_{1}{x}_{i1}+{\beta }_{2}{x}_{i2}+\cdots +{\beta }_{k}{x}_{ik}+</math><math>{\epsilon }_{i}=0</math>
446
|}
447
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 21 )
448
|}
449
450
451
Com <math display="inline">i=1,2,\, \cdots ,n</math>
452
453
Onde
454
455
<math display="inline">n</math> = número de observações (ensaios);
456
457
<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;
460
461
<math display="inline">{\epsilon }_{i}</math> = erros do modelo.
462
463
O enfoque matricial da formulação é mostrado como segue:
464
465
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
466
|-
467
| 
468
{| style="text-align: center; margin:auto;width: 100%;"
469
|-
470
| <math>\mathit{\boldsymbol{Y}}=\mathit{\boldsymbol{X\beta }}+\mathit{\boldsymbol{\epsilon }}=</math><math>0</math>
471
|}
472
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 22 )
473
|}
474
475
476
Com
477
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{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
479
|-
480
| 
481
{| style="text-align: center; margin:auto;width: 100%;"
482
|-
483
| <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> 
484
|}
485
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 23 )
486
|}
487
488
489
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
490
|-
491
| 
492
{| style="text-align: center; margin:auto;width: 100%;"
493
|-
494
| <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> 
495
|}
496
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 24 )
497
|}
498
499
500
Onde
501
502
<math display="inline">\mathit{\boldsymbol{Y}}</math> = é o vetor de observações de dimensão <math display="inline">(n\times 1)</math>;
503
504
<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;
505
506
<math display="inline">\mathit{\boldsymbol{\beta }}</math> = é o vetor dos coeficientes de regressão de dimensão <math display="inline">(p\times 1)</math>;
507
508
<math display="inline">\mathit{\boldsymbol{\epsilon }}</math> = é o vetor dos erros aleatórios de dimensão <math display="inline">(n\times 1)</math>.
509
510
Deve-se encontrar o vetor dos estimadores dos mínimos quadrado, <math display="inline">\hat{\mathit{\boldsymbol{\beta }}}</math>, que minimiza
511
512
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
513
|-
514
| 
515
{| style="text-align: center; margin:auto;width: 100%;"
516
|-
517
| <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> 
518
|}
519
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 25 )
520
|}
521
522
523
Desenvolvendo os cálculos chega-se a:
524
525
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
526
|-
527
| 
528
{| style="text-align: center; margin:auto;width: 100%;"
529
|-
530
| <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>
531
|}
532
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 26 )
533
|}
534
535
536
Mais detalhes dos processos de cálculo podem ser lidos em Montgomery [7].
537
538
O modelo ajustado passa a ter a seguinte forma;
539
540
{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
541
|-
542
| 
543
{| style="text-align: center; margin:auto;width: 100%;"
544
|-
545
| <math display="inline">{\hat{Y}}_{i}={\hat{\beta }}_{0}+\sum _{j=1}^{n}{\hat{\beta }}_{j}{x}_{ij}</math> 
546
|}
547
|  style="vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 27 )
548
|}
549
550
551
Com <math display="inline">i=1,\, 2,\, \cdots ,n</math>.
552
553
Os testes de hipóteses utilizados são o estatístico de prova
554
555
“F” e os de coeficientes individuais “t” que podem ser compreendidos com detalhes em Montgomery [7].
556
557
As superfícies e seus os resultados estatísticos que serão usados nas análises elastoplásticas, obtidos em Vieira [1], são as seguintes:
558
559
{| class="formulaSCP" style="width: 72%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
560
|-
561
| 
562
{| style="text-align: center; margin:auto;width: 100%;"
563
|-
564
| <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> 
565
|}
566
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref533151259'></span>( 28 )
567
|}
568
569
570
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
571
'''Tabela 1'''. Prova de significância da superfície <math display="inline">{f}_{1}</math></div>
572
573
{| style="width: 61%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
574
|-
575
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
576
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
577
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
578
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
579
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
580
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
581
|-
582
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
583
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|23,860
584
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|3
585
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|7,953
586
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1183,674
587
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
588
|-
589
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
590
|  style="text-align: center;vertical-align: top;"|0,1400
591
|  style="text-align: center;vertical-align: top;"|21
592
|  style="text-align: center;vertical-align: top;"|0,007
593
|  style="text-align: center;vertical-align: top;"|
594
|  style="text-align: center;vertical-align: top;"|
595
|-
596
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
597
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24,000
598
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24
599
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
600
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
601
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
602
|-
603
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
604
|-
605
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
606
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Estimado'''
607
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
608
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
609
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
610
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
611
|-
612
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
613
|  style="text-align: center;vertical-align: top;"|1,1580
614
|  style="text-align: center;vertical-align: top;"|0,0377
615
|  style="text-align: center;vertical-align: top;"|30,740
616
|  style="text-align: center;vertical-align: top;"|0,000
617
|  style="text-align: center;vertical-align: top;"|
618
|-
619
|  style="text-align: right;vertical-align: top;"|<math>{m}_{y}^{2}</math>
620
|  style="text-align: center;vertical-align: top;"|1,1180
621
|  style="text-align: center;vertical-align: top;"|0,0387
622
|  style="text-align: center;vertical-align: top;"|28,900
623
|  style="text-align: center;vertical-align: top;"|0,000
624
|  style="text-align: center;vertical-align: top;"|
625
|-
626
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
627
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,1240
628
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0381
629
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|29,530
630
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
631
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
632
|}
633
634
635
{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
636
|-
637
| 
638
{| style="text-align: center; margin:auto;width: 100%;"
639
|-
640
| <math display="inline">{f}_{2}=1,158{n}^{2}+1,118{m}_{y}^{2}+1,124{m}_{z}^{2}-1=</math><math>0</math> 
641
|}
642
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref533151260'></span>( 29 )
643
|}
644
645
646
'''Tabela 2'''. Prova de significância da superfície <math display="inline">{f}_{2}</math>
647
648
{| style="width: 62%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
649
|-
650
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
651
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
652
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
653
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
654
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
655
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
656
|-
657
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
658
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|24,000
659
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5
660
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|4,800
661
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|55913,754
662
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
663
|-
664
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
665
|  style="text-align: center;vertical-align: top;"|0,000
666
|  style="text-align: center;vertical-align: top;"|19
667
|  style="text-align: center;vertical-align: top;"|0,000
668
|  style="text-align: center;vertical-align: top;"|
669
|  style="text-align: center;vertical-align: top;"|
670
|-
671
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
672
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24,000
673
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24
674
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
675
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
676
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
677
|-
678
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
679
|-
680
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
681
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
682
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
683
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
684
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
685
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
686
|-
687
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
688
|  style="text-align: center;vertical-align: top;"|1,0100
689
|  style="text-align: center;vertical-align: top;"|0,0056
690
|  style="text-align: center;vertical-align: top;"|179,500
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>{m}_{y}^{2}</math>
695
|  style="text-align: center;vertical-align: top;"|0,9680
696
|  style="text-align: center;vertical-align: top;"|0,0086
697
|  style="text-align: center;vertical-align: top;"|113,100
698
|  style="text-align: center;vertical-align: top;"|0,000
699
|  style="text-align: center;vertical-align: top;"|
700
|-
701
|  style="text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
702
|  style="text-align: center;vertical-align: top;"|0,9810
703
|  style="text-align: center;vertical-align: top;"|0,0085
704
|  style="text-align: center;vertical-align: top;"|115,500
705
|  style="text-align: center;vertical-align: top;"|0,000
706
|  style="text-align: center;vertical-align: top;"|
707
|-
708
|  style="text-align: right;vertical-align: top;"|<math>n{m}_{y}</math>
709
|  style="text-align: center;vertical-align: top;"|0,5140
710
|  style="text-align: center;vertical-align: top;"|0,0312
711
|  style="text-align: center;vertical-align: top;"|16,500
712
|  style="text-align: center;vertical-align: top;"|0,000
713
|  style="text-align: center;vertical-align: top;"|
714
|-
715
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>n{m}_{z}</math>
716
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
717
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0305
718
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|14,100
719
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
720
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
721
|}
722
723
724
{| class="formulaSCP" style="width: 85%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
725
|-
726
| 
727
{| style="text-align: center; margin:auto;width: 100%;"
728
|-
729
| <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> 
730
|}
731
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref533151262'></span>( 30 )
732
|}
733
734
735
'''Tabela 3.''' Prova de significância da superfície <math display="inline">{f}_{3}</math>
736
737
{| style="width: 62%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
738
|-
739
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
740
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
741
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
742
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
743
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
744
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
745
|-
746
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
747
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|24,000
748
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|6
749
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|4,000
750
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|53308,230
751
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
752
|-
753
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
754
|  style="text-align: center;vertical-align: top;"|0,000
755
|  style="text-align: center;vertical-align: top;"|18
756
|  style="text-align: center;vertical-align: top;"|0,000
757
|  style="text-align: center;vertical-align: top;"|
758
|  style="text-align: center;vertical-align: top;"|
759
|-
760
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
761
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24,000
762
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24
763
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
764
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
765
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
766
|-
767
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Prova dos coeficientes individuais'''
768
|-
769
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
770
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
771
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
772
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
773
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
774
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
775
|-
776
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
777
|  style="text-align: center;vertical-align: top;"|1,0140
778
|  style="text-align: center;vertical-align: top;"|0,00552
779
|  style="text-align: center;vertical-align: top;"|183,500
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>{m}_{y}^{2}</math>
784
|  style="text-align: center;vertical-align: top;"|0,9660
785
|  style="text-align: center;vertical-align: top;"|0,00806
786
|  style="text-align: center;vertical-align: top;"|119,800
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>{m}_{z}^{2}</math>
791
|  style="text-align: center;vertical-align: top;"|0,9820
792
|  style="text-align: center;vertical-align: top;"|0,00795
793
|  style="text-align: center;vertical-align: top;"|123,500
794
|  style="text-align: center;vertical-align: top;"|0,000
795
|  style="text-align: center;vertical-align: top;"|
796
|-
797
|  style="text-align: right;vertical-align: top;"|<math>n{m}_{y}</math>
798
|  style="text-align: center;vertical-align: top;"|0,5060
799
|  style="text-align: center;vertical-align: top;"|0,02951
800
|  style="text-align: center;vertical-align: top;"|17,100
801
|  style="text-align: center;vertical-align: top;"|0,000
802
|  style="text-align: center;vertical-align: top;"|
803
|-
804
|  style="text-align: right;vertical-align: top;"|<math>n{m}_{z}</math>
805
|  style="text-align: center;vertical-align: top;"|0,4040
806
|  style="text-align: center;vertical-align: top;"|0,03146
807
|  style="text-align: center;vertical-align: top;"|12,800
808
|  style="text-align: center;vertical-align: top;"|0,000
809
|  style="text-align: center;vertical-align: top;"|
810
|-
811
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{y}{m}_{z}</math>
812
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0380
813
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,01980
814
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,900
815
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
816
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
817
|}
818
819
820
{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
821
|-
822
| 
823
{| style="text-align: center; margin:auto;width: 100%;"
824
|-
825
| <math display="inline">{f}_{4}=1,012{n}^{2}+1,027{m}_{z}^{2}-1=0</math> 
826
|}
827
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref531207210'></span>( 31 )
828
|}
829
830
831
'''Tabela 4'''. Prova de significância da superfície <math display="inline">{f}_{4}</math>
832
833
{| style="width: 66%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
834
|-
835
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
836
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
837
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
838
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
839
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
840
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
841
|-
842
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
843
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|12,000
844
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|2
845
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|6,000
846
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|84325,969
847
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
848
|-
849
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
850
|  style="text-align: center;vertical-align: top;"|0,000
851
|  style="text-align: center;vertical-align: top;"|10
852
|  style="text-align: center;vertical-align: top;"|0,000
853
|  style="text-align: center;vertical-align: top;"|
854
|  style="text-align: center;vertical-align: top;"|
855
|-
856
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
857
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12,000
858
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12
859
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
860
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
861
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
862
|-
863
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
864
|-
865
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
866
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
867
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
868
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
869
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
870
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
871
|-
872
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
873
|  style="text-align: center;vertical-align: top;"|1,0120
874
|  style="text-align: center;vertical-align: top;"|0,0043
875
|  style="text-align: center;vertical-align: top;"|235, 300
876
|  style="text-align: center;vertical-align: top;"|0,000
877
|  style="text-align: center;vertical-align: top;"|
878
|-
879
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
880
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,0270
881
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0044
882
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|233, 800
883
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
884
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
885
|}
886
887
888
{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
889
|-
890
| 
891
{| style="text-align: center; margin:auto;width: 100%;"
892
|-
893
| <math display="inline">{f}_{5}=1,\, 242{n}^{2}+1,\, 087{m}_{z}^{2}-1=0</math> 
894
|}
895
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref531207213'></span>( 32 )
896
|}
897
898
899
'''Tabela 5'''. Prova de significância da superfície <math display="inline">{f}_{5}</math>
900
901
{| style="width: 61%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
902
|-
903
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
904
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
905
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
906
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
907
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
908
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
909
|-
910
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
911
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|11,960
912
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|2
913
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5,979
914
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1455,211
915
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
916
|-
917
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
918
|  style="text-align: center;vertical-align: top;"|0,040
919
|  style="text-align: center;vertical-align: top;"|10
920
|  style="text-align: center;vertical-align: top;"|0,004
921
|  style="text-align: center;vertical-align: top;"|
922
|  style="text-align: center;vertical-align: top;"|
923
|-
924
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
925
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12,000
926
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12
927
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
928
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
929
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
930
|-
931
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
932
|-
933
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
934
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
935
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
936
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
937
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
938
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
939
|-
940
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
941
|  style="text-align: center;vertical-align: top;"|1,2420
942
|  style="text-align: center;vertical-align: top;"|0,0309
943
|  style="text-align: center;vertical-align: top;"|40,220
944
|  style="text-align: center;vertical-align: top;"|0,000
945
|  style="text-align: center;vertical-align: top;"|
946
|-
947
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
948
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,0870
949
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0355
950
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|30,610
951
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
952
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
953
|}
954
955
956
{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
957
|-
958
| 
959
{| style="text-align: center; margin:auto;width: 100%;"
960
|-
961
| <math display="inline">{f}_{6}=1,089n+\, 0,929{m}_{z}^{2}-1=0</math> 
962
|}
963
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref531207215'></span>( 33 )
964
|}
965
966
967
'''Tabela 6'''. Prova de significância da superfície <math display="inline">{f}_{6}</math>
968
969
{| style="width: 61%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
970
|-
971
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
972
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
973
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
974
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
975
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
976
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
977
|-
978
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
979
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|11,990
980
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|2
981
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5,996
982
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|8547,240
983
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
984
|-
985
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
986
|  style="text-align: center;vertical-align: top;"|0,010
987
|  style="text-align: center;vertical-align: top;"|10
988
|  style="text-align: center;vertical-align: top;"|0,001
989
|  style="text-align: center;vertical-align: top;"|
990
|  style="text-align: center;vertical-align: top;"|
991
|-
992
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
993
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12,000
994
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12
995
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
996
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
997
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
998
|-
999
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
1000
|-
1001
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
1002
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
1003
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
1004
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
1005
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
1006
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
1007
|-
1008
|  style="text-align: right;vertical-align: top;"|<math>n</math>
1009
|  style="text-align: center;vertical-align: top;"|1,0890
1010
|  style="text-align: center;vertical-align: top;"|0,0112
1011
|  style="text-align: center;vertical-align: top;"|97,600
1012
|  style="text-align: center;vertical-align: top;"|0,000
1013
|  style="text-align: center;vertical-align: top;"|
1014
|-
1015
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
1016
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,9290
1017
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0150
1018
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|62,040
1019
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
1020
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
1021
|}
1022
1023
1024
===2.4 Análise elastoplástica de estruturas de pórticos===
1025
1026
Uma superfície de interação define o estado último de uma seção transversal e depende dos seguintes fatores:
1027
1028
:1. Forma geométrica da seção transversal;
1029
1030
:2. Combinação dos esforços seccionais que atuam na seção transversal;
1031
1032
:3. Teoria de viga utilizada.
1033
1034
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.
1035
1036
A análise elastoplástica segue os conceitos apresentados no trabalho de Silva [10] com as seguintes considerações;
1037
1038
:1) Os esforços seccionais contidos no interior da superfície de interação geram somente deformações elásticas;
1039
1040
:2) Os esforços seccionais que estejam na superfície de interação geram deformações plásticas;
1041
1042
:3) Os esforços seccionais fora da superfície de interação representam estados de tensões inadmissíveis porque não se leva em conta o caso do endurecimento.
1043
1044
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.
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====2.4.1 Derivadas de primeira ordem====
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1048
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:
1049
{| class="formulaSCP" style="width: 100%; text-align: center;" 
1050
|-
1051
| <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>
1052
|}
1053
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1055
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1056
|-
1057
| 
1058
{| style="text-align: center; margin:auto;width: 100%;"
1059
|-
1060
| <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> 
1061
|}
1062
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 34 )
1063
|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1067
|-
1068
| 
1069
{| style="text-align: center; margin:auto;width: 100%;"
1070
|-
1071
| <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>
1072
|}
1073
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 35 )
1074
|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1078
|-
1079
| 
1080
{| style="text-align: center; margin:auto;width: 100%;"
1081
|-
1082
| <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>
1083
|}
1084
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 36 )
1085
|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1089
|-
1090
| 
1091
{| style="text-align: center; margin:auto;width: 100%;"
1092
|-
1093
| <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> 
1094
|}
1095
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 37 )
1096
|}
1097
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1100
|-
1101
| 
1102
{| style="text-align: center; margin:auto;width: 100%;"
1103
|-
1104
| <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> 
1105
|}
1106
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 38 )
1107
|}
1108
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1111
|-
1112
| 
1113
{| style="text-align: center; margin:auto;width: 100%;"
1114
|-
1115
| <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> 
1116
|}
1117
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 39 )
1118
|}
1119
1120
1121
Onde
1122
1123
<math display="inline">s{f}_{i}=\frac{{f}_{i}}{\left| {f}_{i}\right| }</math> = sinal do esforço seccional de forças;
1124
1125
<math display="inline">s{m}_{i}=\frac{{m}_{i}}{\left| {m}_{i}\right| }</math> = sinal do esforço seccional de momentos;
1126
1127
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.
1128
1129
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1130
|-
1131
| 
1132
{| style="text-align: center; margin:auto;width: 100%;"
1133
|-
1134
| <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>  
1135
|}
1136
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529991859'></span>( 40 )
1137
|}
1138
1139
1140
Onde <math display="inline">\mathit{\boldsymbol{0}}</math> é o vetor nulo de dimensão <math display="inline">(6\times 1)</math>.
1141
1142
====2.4.2 Derivadas de segunda ordem====
1143
1144
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:
1145
1146
'''Para''' <math display="inline">\partial {F}_{x}{F}_{k}</math>
1147
1148
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1149
|-
1150
| 
1151
{| style="text-align: center; margin:auto;width: 100%;"
1152
|-
1153
| <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> 
1154
|}
1155
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 41 )
1156
|}
1157
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1159
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1160
|-
1161
| 
1162
{| style="text-align: center; margin:auto;width: 100%;"
1163
|-
1164
| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {F}_{y}}=0\quad</math> 
1165
|}
1166
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 42 )
1167
|}
1168
1169
1170
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1171
|-
1172
| 
1173
{| style="text-align: center; margin:auto;width: 100%;"
1174
|-
1175
| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {F}_{z}}=0\quad</math> 
1176
|}
1177
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 43 )
1178
|}
1179
1180
1181
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1182
|-
1183
| 
1184
{| style="text-align: center; margin:auto;width: 100%;"
1185
|-
1186
| <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>
1187
1188
<math>\quad</math> 
1189
|}
1190
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 44 )
1191
|}
1192
1193
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1195
|-
1196
| 
1197
{| style="text-align: center; margin:auto;width: 100%;"
1198
|-
1199
| <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>
1200
1201
<math>\quad</math> 
1202
|}
1203
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 45 )
1204
|}
1205
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}_{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>
1213
1214
<math>\quad</math> 
1215
|}
1216
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 46 )
1217
|}
1218
1219
1220
'''Para''' <math display="inline">\partial {F}_{y}{F}_{k}</math>
1221
1222
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1223
|-
1224
| 
1225
{| style="text-align: center; margin:auto;width: 100%;"
1226
|-
1227
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {F}_{x}}=0\quad</math> 
1228
|}
1229
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 47 )
1230
|}
1231
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1234
|-
1235
| 
1236
{| style="text-align: center; margin:auto;width: 100%;"
1237
|-
1238
| <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> 
1239
|}
1240
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 48 )
1241
|}
1242
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1245
|-
1246
| 
1247
{| style="text-align: center; margin:auto;width: 100%;"
1248
|-
1249
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {F}_{z}}=0\quad</math> 
1250
|}
1251
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 49 )
1252
|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1256
|-
1257
| 
1258
{| style="text-align: center; margin:auto;width: 100%;"
1259
|-
1260
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {M}_{x}}=0\quad</math> 
1261
|}
1262
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 50 )
1263
|}
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1267
|-
1268
| 
1269
{| style="text-align: center; margin:auto;width: 100%;"
1270
|-
1271
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {M}_{y}}=0\quad</math> 
1272
|}
1273
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 51 )
1274
|}
1275
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1278
|-
1279
| 
1280
{| style="text-align: center; margin:auto;width: 100%;"
1281
|-
1282
| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {M}_{z}}=0\quad</math> 
1283
|}
1284
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 52 )
1285
|}
1286
1287
1288
'''Para''' <math display="inline">\partial {F}_{z}{F}_{k}</math>
1289
1290
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1291
|-
1292
| 
1293
{| style="text-align: center; margin:auto;width: 100%;"
1294
|-
1295
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {F}_{x}}=0\quad</math> 
1296
|}
1297
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 53 )
1298
|}
1299
1300
1301
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1302
|-
1303
| 
1304
{| style="text-align: center; margin:auto;width: 100%;"
1305
|-
1306
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {F}_{y}}=0\quad</math> 
1307
|}
1308
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 54 )
1309
|}
1310
1311
1312
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1313
|-
1314
| 
1315
{| style="text-align: center; margin:auto;width: 100%;"
1316
|-
1317
| <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> 
1318
|}
1319
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 55 )
1320
|}
1321
1322
1323
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1324
|-
1325
| 
1326
{| style="text-align: center; margin:auto;width: 100%;"
1327
|-
1328
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {M}_{x}}=0\quad</math> 
1329
|}
1330
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 56 )
1331
|}
1332
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1334
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1335
|-
1336
| 
1337
{| style="text-align: center; margin:auto;width: 100%;"
1338
|-
1339
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {M}_{y}}=0\quad</math> 
1340
|}
1341
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 57 )
1342
|-
1343
| 
1344
{| style="text-align: center; margin:auto;width: 100%;"
1345
|-
1346
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {M}_{z}}=0\quad</math> 
1347
|}
1348
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 58 )
1349
|}
1350
1351
1352
'''Para''' <math display="inline">\partial {M}_{x}{F}_{k}</math>
1353
1354
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1355
|-
1356
| 
1357
{| style="text-align: center; margin:auto;width: 100%;"
1358
|-
1359
| <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> 
1360
|}
1361
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 59 )
1362
|}
1363
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1366
|-
1367
| 
1368
{| style="text-align: center; margin:auto;width: 100%;"
1369
|-
1370
| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {F}_{y}}=0\quad</math> 
1371
|}
1372
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 60 )
1373
|}
1374
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1376
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1377
|-
1378
| 
1379
{| style="text-align: center; margin:auto;width: 100%;"
1380
|-
1381
| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {F}_{z}}=0\, \,</math> 
1382
|}
1383
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 61 )
1384
|}
1385
1386
1387
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1388
|-
1389
| 
1390
{| style="text-align: center; margin:auto;width: 100%;"
1391
|-
1392
| <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> 
1393
|}
1394
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 62 )
1395
|}
1396
1397
1398
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1399
|-
1400
| 
1401
{| style="text-align: center; margin:auto;width: 100%;"
1402
|-
1403
| <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>
1404
|}
1405
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 63 )
1406
|}
1407
1408
1409
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1410
|-
1411
| 
1412
{| style="text-align: center; margin:auto;width: 100%;"
1413
|-
1414
| <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>
1415
|}
1416
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 64 )
1417
|}
1418
1419
1420
'''Para''' <math display="inline">\partial {M}_{y}{F}_{k}</math>
1421
1422
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1423
|-
1424
| 
1425
{| style="text-align: center; margin:auto;width: 100%;"
1426
|-
1427
| <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> 
1428
|}
1429
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 65 )
1430
|}
1431
1432
1433
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1434
|-
1435
| 
1436
{| style="text-align: center; margin:auto;width: 100%;"
1437
|-
1438
| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {F}_{y}}=0\quad</math> 
1439
|}
1440
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 66 )
1441
|}
1442
1443
1444
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1445
|-
1446
| 
1447
{| style="text-align: center; margin:auto;width: 100%;"
1448
|-
1449
| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {F}_{z}}=0\, \,</math> 
1450
|}
1451
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 67 )
1452
|}
1453
1454
1455
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1456
|-
1457
| 
1458
{| style="text-align: center; margin:auto;width: 100%;"
1459
|-
1460
| <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> 
1461
|}
1462
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 68 )
1463
|}
1464
1465
1466
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1467
|-
1468
| 
1469
{| style="text-align: center; margin:auto;width: 100%;"
1470
|-
1471
| <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> 
1472
|}
1473
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 69 )
1474
|}
1475
1476
1477
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1478
|-
1479
| 
1480
{| style="text-align: center; margin:auto;width: 100%;"
1481
|-
1482
| <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>
1483
|}
1484
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 70 )
1485
|}
1486
1487
1488
'''Para''' <math display="inline">\partial {M}_{z}{F}_{k}</math>
1489
1490
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1491
|-
1492
| 
1493
{| style="text-align: center; margin:auto;width: 100%;"
1494
|-
1495
| <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>
1496
|}
1497
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 71 )
1498
|}
1499
1500
1501
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1502
|-
1503
| 
1504
{| style="text-align: center; margin:auto;width: 100%;"
1505
|-
1506
| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {F}_{y}}=0\quad</math> 
1507
|}
1508
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 72 )
1509
|}
1510
1511
1512
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1513
|-
1514
| 
1515
{| style="text-align: center; margin:auto;width: 100%;"
1516
|-
1517
| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {F}_{z}}=0\, \,</math> 
1518
|}
1519
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 73 )
1520
|}
1521
1522
1523
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1524
|-
1525
| 
1526
{| style="text-align: center; margin:auto;width: 100%;"
1527
|-
1528
| <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> 
1529
|}
1530
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 74 )
1531
|}
1532
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1534
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1535
|-
1536
| 
1537
{| style="text-align: center; margin:auto;width: 100%;"
1538
|-
1539
| <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>
1540
|}
1541
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 75 )
1542
|}
1543
1544
1545
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1546
|-
1547
| 
1548
{| style="text-align: center; margin:auto;width: 100%;"
1549
|-
1550
| <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> 
1551
|}
1552
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 76 )
1553
|}
1554
1555
1556
{| class="formulaSCP" style="width: 78%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1557
|-
1558
| 
1559
{| style="text-align: center; margin:auto;width: 100%;"
1560
|-
1561
| As 2ª derivadas na forma matricial podem ser expressas como
1562
1563
<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> 
1564
|}
1565
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 77 )
1566
|}
1567
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1569
{| class="formulaSCP" style="width: 45%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1570
|-
1571
| 
1572
{| style="text-align: center; margin:auto;width: 100%;"
1573
|-
1574
| <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> 
1575
|}
1576
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 78 )
1577
|}
1578
1579
1580
{| class="formulaSCP" style="width: 78%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1581
|-
1582
| 
1583
{| style="text-align: center; margin:auto;width: 100%;"
1584
|-
1585
| <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> 
1586
|}
1587
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 79 )
1588
|}
1589
1590
1591
{| class="formulaSCP" style="width: 45%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1592
|-
1593
| 
1594
{| style="text-align: center; margin:auto;width: 100%;"
1595
|-
1596
| <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> 
1597
|}
1598
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 80 )
1599
|}
1600
1601
1602
Onde  <math display="inline">\mathit{\boldsymbol{0}}</math>''' '''é uma matriz de dimensão <math display="inline">(6\times 6)</math> com elementos nulos.
1603
1604
===2.5 Algoritmo de Retorno===
1605
1606
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.
1607
1608
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:
1609
1610
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1611
|-
1612
| 
1613
{| style="text-align: center; margin:auto;width: 100%;"
1614
|-
1615
| <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>
1616
|}
1617
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530073820'></span>( 81 )
1618
|}
1619
1620
1621
Onde
1622
1623
<math display="inline">{\hat{\mathit{\boldsymbol{F}}}}_{i}</math>= vetor de forças nodais corrigido;
1624
1625
<math display="inline">{\mathit{\boldsymbol{F}}}_{i}^{trial}</math> = vetor de força nodais estimado;
1626
1627
<math display="inline">{\lambda }_{1}</math>= multiplicador plástico do nó 1, de forma <math display="inline">{\lambda }_{1}\geq 0</math>;
1628
1629
<math display="inline">{\mathit{\boldsymbol{K}}}_{ij}</math>= matriz de rigidez do elemento;
1630
1631
<math display="inline">{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}</math>= vetor de fluxo plástico do nó 1.
1632
1633
O vetor de forças nodais estimado é expressado por:
1634
1635
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1636
|-
1637
| 
1638
{| style="text-align: center; margin:auto;width: 100%;"
1639
|-
1640
| <math>{\mathit{\boldsymbol{F}}}_{i}^{trial}={\overline{\mathit{\boldsymbol{F}}}}_{i}+{\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}</math>
1641
|}
1642
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530068060'></span>( 82 )
1643
|}
1644
1645
1646
Onde
1647
1648
<math display="inline">{\overline{\mathit{\boldsymbol{F}}}}_{i}</math> = vetor de forças nodais do último passo de carga convergido;
1649
1650
<math display="inline">d{\mathit{\boldsymbol{U}}}_{j}</math> = incrementos do campo de deslocamentos do nó.
1651
1652
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.
1653
1654
O algoritmo irá trabalhar com 2 (duas) possibilidades de formação de rótulas plásticas, ou seja, para 1 (um) nó ou os 2 (dois) nós.
1655
1656
====2.5.1 Algoritmo de retorno com 1 (um) vetor====
1657
1658
<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''']].
1659
1660
{| style="width: 82%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
1661
|-
1662
|  style="text-align: center;vertical-align: top;width: 100%;"|[[Image:Draft_Vieira_908925676-image7.png|474px]] 
1663
|}
1664
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1666
<span id='_Ref530062972'></span>'''Figura 6'''. Retorno à superfície com um vetor.
1667
1668
O processo iterativo utiliza vetores de fluxo plástico atualizados para aproximar-se da superfície. Este procedimento é chamado de algoritmo de retorno.
1669
1670
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>
1671
1672
O vetor de forças residuais <math display="inline">{\mathit{\boldsymbol{r}}}_{i}</math> do processo iterativo será como
1673
1674
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1675
|-
1676
| 
1677
{| style="text-align: center; margin:auto;width: 100%;"
1678
|-
1679
| <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>
1680
|}
1681
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530063570'></span>( 83 )
1682
|}
1683
1684
1685
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:
1686
1687
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1688
|-
1689
| 
1690
{| style="text-align: center; margin:auto;width: 100%;"
1691
|-
1692
| <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>
1693
|}
1694
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530064230'></span>( 84 )
1695
|}
1696
1697
1698
Onde
1699
1700
<math display="inline">n=0,1,2,\ldots</math>  = passo do processo iterativo.
1701
1702
<math display="inline">d{\mathit{\boldsymbol{F}}}_{i}</math> = variação do vetor de forças;
1703
1704
<math display="inline">d{\lambda }_{1}</math>= variação do multiplicador plástico;
1705
1706
<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).
1707
1708
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
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{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>
1716
|}
1717
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530064567'></span>( 85 )
1718
|}
1719
1720
1721
Onde <math display="inline">{\delta }_{ik}</math> = Delta de Kronecker
1722
1723
Definindo-se o termo
1724
1725
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1726
|-
1727
| 
1728
{| style="text-align: center; margin:auto;width: 100%;"
1729
|-
1730
| <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> 
1731
|}
1732
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530075709'></span>( 86 )
1733
|}
1734
1735
1736
A equação <span id='cite-_Ref530064567'></span>[[#_Ref530064567|( 85 )]], torna-se:
1737
1738
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1739
|-
1740
| 
1741
{| style="text-align: center; margin:auto;width: 100%;"
1742
|-
1743
| <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>
1744
|}
1745
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 87 )
1746
|}
1747
1748
1749
Obtendo os termos da variação do vetor de força, chega-se a:
1750
1751
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1752
|-
1753
| 
1754
{| style="text-align: center; margin:auto;width: 100%;"
1755
|-
1756
| <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> 
1757
|}
1758
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 88 )
1759
|}
1760
1761
1762
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1763
|-
1764
| 
1765
{| style="text-align: center; margin:auto;width: 100%;"
1766
|-
1767
| <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> 
1768
|}
1769
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530065198'></span>( 89 )
1770
|}
1771
1772
1773
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:
1774
1775
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1776
|-
1777
| 
1778
{| style="text-align: center; margin:auto;width: 100%;"
1779
|-
1780
| <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>
1781
|}
1782
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 90 )
1783
|}
1784
1785
1786
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:
1787
1788
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1789
|-
1790
| 
1791
{| style="text-align: center; margin:auto;width: 100%;"
1792
|-
1793
| <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>
1794
|}
1795
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 91 )
1796
|}
1797
1798
1799
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1800
|-
1801
| 
1802
{| style="text-align: center; margin:auto;width: 100%;"
1803
|-
1804
| <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> 
1805
|}
1806
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 92 )
1807
|}
1808
1809
1810
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1811
|-
1812
| 
1813
{| style="text-align: center; margin:auto;width: 100%;"
1814
|-
1815
| <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>
1816
|}
1817
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 93 )
1818
|}
1819
1820
1821
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1822
|-
1823
| 
1824
{| style="text-align: center; margin:auto;width: 100%;"
1825
|-
1826
| <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>
1827
|}
1828
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 94 )
1829
|}
1830
1831
1832
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1833
|-
1834
| 
1835
{| style="text-align: center; margin:auto;width: 100%;"
1836
|-
1837
| <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>
1838
|}
1839
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 95 )
1840
|}
1841
1842
1843
O processo iterativo termina quando são alcançados os critérios de parada adotados:
1844
1845
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1846
|-
1847
| 
1848
{| style="text-align: center; margin:auto;width: 100%;"
1849
|-
1850
| <math>{r}^{norm}=\sqrt{\frac{\left\| {\mathit{\boldsymbol{r}}}_{i}\right\| }{\left\| {\mathit{\boldsymbol{F}}}_{i}^{trial}\right\| }}<Tol</math>
1851
1852
<math display="inline">{\mathit{\boldsymbol{f}}}^{norm}=\left| \mathit{\boldsymbol{f}}-\right. </math><math>\left. 1\right| <Tol</math> 
1853
|}
1854
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 96 )
1855
|}
1856
1857
1858
Onde
1859
1860
<math display="inline">\left\| {\mathit{\boldsymbol{r}}}_{i}\right\|</math>  = norma euclidiana do vetor de forças nodais;
1861
1862
<math display="inline">\left\| {\mathit{\boldsymbol{F}}}_{i}^{trial}\right\|</math>  = norma euclidiana do vetor de forças estimado;
1863
1864
<math display="inline">{\mathit{\boldsymbol{f}}}^{norm}</math>= vetor resíduo da superfície de interação;
1865
1866
<math display="inline">Tol</math> = tolerância adotada.
1867
1868
====2.5.2 Algoritmo de retorno com 2 (dois) vetores====
1869
1870
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
1871
1872
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1873
|-
1874
| 
1875
{| style="text-align: center; margin:auto;width: 100%;"
1876
|-
1877
| <math display="inline">{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}({F}_{j})>1;\, {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{2}}}({F}_{j})>1</math> 
1878
|}
1879
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 97 )
1880
|}
1881
1882
1883
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''']].
1884
1885
{| style="width: 85%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
1886
|-
1887
|  style="text-align: center;vertical-align: top;"|''' [[Image:Draft_Vieira_908925676-image8.png|510px]] '''
1888
|}
1889
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1891
<span id='_Ref530324965'></span>'''Figura 7'''. Retorno à superfície com dois vetores.
1892
1893
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
1894
1895
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1896
|-
1897
| 
1898
{| style="text-align: center; margin:auto;width: 100%;"
1899
|-
1900
| <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>
1901
|}
1902
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 98 )
1903
|}
1904
1905
1906
Onde
1907
1908
<math display="inline">{\lambda }_{1}</math>e <math display="inline">{\lambda }_{2}</math> são os multiplicadores plásticos.
1909
1910
O vetor resíduo das forças tem a forma seguinte:
1911
1912
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1913
|-
1914
| 
1915
{| style="text-align: center; margin:auto;width: 100%;"
1916
|-
1917
| <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>
1918
|}
1919
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 99 )
1920
|}
1921
1922
1923
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:
1924
1925
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1926
|-
1927
| 
1928
{| style="text-align: center; margin:auto;width: 100%;"
1929
|-
1930
| <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>
1931
|}
1932
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 100 )
1933
|}
1934
1935
1936
Com a condição que <math display="inline">{\mathit{\boldsymbol{r}}}_{i}^{n+1}=</math><math>\mathit{\boldsymbol{0}}</math>, chega-se a:
1937
1938
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1939
|-
1940
| 
1941
{| style="text-align: center; margin:auto;width: 100%;"
1942
|-
1943
| <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>
1944
|}
1945
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530068973'></span>( 101 )
1946
|}
1947
1948
1949
Adotando-se:
1950
1951
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1952
|-
1953
| 
1954
{| style="text-align: center; margin:auto;width: 100%;"
1955
|-
1956
| <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> 
1957
|}
1958
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530069357'></span>( 102 )
1959
|}
1960
1961
1962
Isolando o termo <math display="inline">d{\mathit{\boldsymbol{F}}}_{k}</math> da equação <span id='cite-_Ref530068973'></span>[[#_Ref530068973|( 101 )]], obtém-se:
1963
1964
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1965
|-
1966
| 
1967
{| style="text-align: center; margin:auto;width: 100%;"
1968
|-
1969
| <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> 
1970
|}
1971
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 103 )
1972
|}
1973
1974
1975
Os termos iterativos da função de escoamento (superfícies) são apresentados como
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+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>
1983
1984
<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>
1985
|}
1986
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 104 )
1987
|}
1988
1989
1990
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:
1991
1992
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
1993
|-
1994
| 
1995
{| style="text-align: center; margin:auto;width: 100%;"
1996
|-
1997
| <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>
1998
1999
<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>
2000
|}
2001
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 105 )
2002
|}
2003
2004
2005
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2006
|-
2007
| 
2008
{| style="text-align: center; margin:auto;width: 100%;"
2009
|-
2010
| <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>
2011
2012
<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> 
2013
|}
2014
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530069811'></span>( 106 )
2015
|}
2016
2017
2018
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2019
|-
2020
| 
2021
{| style="text-align: center; margin:auto;width: 100%;"
2022
|-
2023
| <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>
2024
|}
2025
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530070117'></span>( 107 )
2026
|}
2027
2028
2029
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2030
|-
2031
| 
2032
{| style="text-align: center; margin:auto;width: 100%;"
2033
|-
2034
| <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>
2035
|}
2036
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530070122'></span>( 108 )
2037
|}
2038
2039
2040
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:
2041
2042
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2043
|-
2044
| 
2045
{| style="text-align: center; margin:auto;width: 100%;"
2046
|-
2047
| <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> 
2048
|}
2049
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530072897'></span>( 109 )
2050
|}
2051
2052
2053
Reapresentando a equação <span id='cite-_Ref530072897'></span>[[#_Ref530072897|( 109 )]] na forma sintética:
2054
2055
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2056
|-
2057
| 
2058
{| style="text-align: center; margin:auto;width: 100%;"
2059
|-
2060
| <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> 
2061
|}
2062
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530072993'></span>( 110 )
2063
|}
2064
2065
2066
A solução do sistema da equação <span id='cite-_Ref530072993'></span>[[#_Ref530072993|( 110 )]] é a seguinte:
2067
2068
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2069
|-
2070
| 
2071
{| style="text-align: center; margin:auto;width: 100%;"
2072
|-
2073
| <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> 
2074
|}
2075
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 111 )
2076
|}
2077
2078
2079
O processo iterativo segue procedimentos similares ao caso com um 1 (um) vetor:
2080
2081
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2082
|-
2083
| 
2084
{| style="text-align: center; margin:auto;width: 100%;"
2085
|-
2086
| <math>{r}^{norm}=\sqrt{\frac{\left\| {\mathit{\boldsymbol{r}}}_{i}\right\| }{\left\| {\mathit{\boldsymbol{F}}}_{i}^{trial}\right\| }}<Tol</math>
2087
2088
<math>{\mathit{\boldsymbol{f}}}_{1}^{norm}=\left| {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}-\right. </math><math>\left. 1\right| <Tol</math>
2089
2090
<math display="inline">{\mathit{\boldsymbol{f}}}_{2}^{norm}=\left| {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{2}}}-\right. </math><math>\left. 1\right| <Tol</math> 
2091
|}
2092
|  style="text-align: right;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 112 )
2093
|}
2094
2095
2096
<span id='_Ref4583347'></span>
2097
2098
===2.6 Matriz de rigidez consistente===
2099
2100
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.
2101
2102
====2.6.1 Algoritmo de retorno com um vetor ====
2103
2104
Usando a equação <span id='cite-_Ref530073820'></span>[[#_Ref530073820|( 81 )]] e <span id='cite-_Ref530068060'></span>[[#_Ref530068060|( 82 )]] como ponto de partida:
2105
2106
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2107
|-
2108
| 
2109
{| style="text-align: center; margin:auto;width: 100%;"
2110
|-
2111
| <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>
2112
2113
<math>{\mathit{\boldsymbol{F}}}_{i}^{trial}={\overline{\mathit{\boldsymbol{F}}}}_{i}+{\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}</math>
2114
2115
com <math display="inline">{\lambda }_{1}>0</math>
2116
|}
2117
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530073960'></span>( 113 )
2118
|}
2119
2120
2121
Aplicando-se o diferencial total na equação <span id='cite-_Ref530073960'></span>[[#_Ref530073960|( 113 )]], chega-se a:
2122
2123
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2124
|-
2125
| 
2126
{| style="text-align: center; margin:auto;width: 100%;"
2127
|-
2128
| <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>
2129
|}
2130
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 114 )
2131
|-
2132
| 
2133
{| style="text-align: center; margin:auto;width: 100%;"
2134
|-
2135
| <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>
2136
|}
2137
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 115 )
2138
|}
2139
2140
2141
Adotando <math display="inline">\, d{\overline{F}}_{i}=\mathit{\boldsymbol{0}}</math>
2142
2143
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2144
|-
2145
| 
2146
{| style="text-align: center; margin:auto;width: 100%;"
2147
|-
2148
| <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>
2149
|}
2150
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 116 )
2151
|}
2152
2153
2154
Isolando o termo <math display="inline">d{F}_{k}</math>
2155
2156
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2157
|-
2158
| 
2159
{| style="text-align: center; margin:auto;width: 100%;"
2160
|-
2161
| <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>
2162
|}
2163
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 117 )
2164
|}
2165
2166
2167
Usando <math display="inline">{Q}_{ik}</math> (equação <span id='cite-_Ref530075709'></span>[[#_Ref530075709|( 86 )]]):
2168
2169
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2170
|-
2171
| 
2172
{| style="text-align: center; margin:auto;width: 100%;"
2173
|-
2174
| <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>
2175
|}
2176
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 118 )
2177
|}
2178
2179
2180
Isolando o termo <math display="inline">d{F}_{k}</math>:
2181
2182
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2183
|-
2184
| 
2185
{| style="text-align: center; margin:auto;width: 100%;"
2186
|-
2187
| <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> 
2188
|}
2189
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530076047'></span>( 119 )
2190
|}
2191
2192
2193
Adotando o termo:
2194
2195
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2196
|-
2197
| 
2198
{| style="text-align: center; margin:auto;width: 100%;"
2199
|-
2200
| <math>{\mathit{\boldsymbol{R}}}_{ij}={\mathit{\boldsymbol{Q}}}_{ik}^{-1}{\mathit{\boldsymbol{K}}}_{ij}</math>
2201
|}
2202
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530077061'></span>( 120 )
2203
|}
2204
2205
2206
A equação <span id='cite-_Ref530076047'></span>[[#_Ref530076047|( 119 )]] torna-se em:
2207
2208
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2209
|-
2210
| 
2211
{| style="text-align: center; margin:auto;width: 100%;"
2212
|-
2213
| <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> 
2214
|}
2215
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530076227'></span>( 121 )
2216
|}
2217
2218
2219
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:
2220
2221
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2222
|-
2223
| 
2224
{| style="text-align: center; margin:auto;width: 100%;"
2225
|-
2226
| <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>
2227
|}
2228
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 122 )
2229
|}
2230
2231
2232
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2233
|-
2234
| 
2235
{| style="text-align: center; margin:auto;width: 100%;"
2236
|-
2237
| <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> 
2238
|}
2239
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 123 )
2240
|}
2241
2242
2243
Isolando o termo <math display="inline">d{\lambda }_{1}</math>:
2244
2245
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2246
|-
2247
| 
2248
{| style="text-align: center; margin:auto;width: 100%;"
2249
|-
2250
| <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>
2251
|}
2252
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530077047'></span>( 124 )
2253
|}
2254
2255
2256
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 )]]:
2257
2258
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2259
|-
2260
| 
2261
{| style="text-align: center; margin:auto;width: 100%;"
2262
|-
2263
| <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>
2264
|}
2265
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 125 )
2266
|}
2267
2268
2269
Com
2270
2271
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2272
|-
2273
| 
2274
{| style="text-align: center; margin:auto;width: 100%;"
2275
|-
2276
| <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> 
2277
|}
2278
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 126 )
2279
|}
2280
2281
2282
====2.6.2 Algoritmo com dois vetores de retorno====
2283
2284
Os procedimentos similares são realizados para obter a matriz de rigidez consistente para dois vetores como por exemplo Silva [10] e Vieira [6].
2285
2286
A formulação para os dois vetores tem a seguinte forma:
2287
2288
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2289
|-
2290
| 
2291
{| style="text-align: center; margin:auto;width: 100%;"
2292
|-
2293
| <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> 
2294
|}
2295
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530079964'></span>( 127 )
2296
|}
2297
2298
2299
As condições do vetor de forças nodais final são que
2300
2301
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2302
|-
2303
|  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>''' '''
2304
|  style="text-align: center;vertical-align: bottom;"|( 128 )
2305
|}
2306
2307
2308
Os termos dos multiplicadores plásticos são os seguintes:
2309
2310
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2311
|-
2312
| 
2313
{| style="text-align: center; margin:auto;width: 100%;"
2314
|-
2315
| <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> 
2316
|}
2317
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530079552'></span>( 129 )
2318
|}
2319
2320
2321
Com
2322
2323
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2324
|-
2325
| 
2326
{| style="text-align: center; margin:auto;width: 100%;"
2327
|-
2328
| <math>{c}_{1}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{1}{R}_{ij}d{U}_{j}</math>
2329
2330
<math>{c}_{2}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{2}{R}_{ij}d{U}_{j}</math>
2331
2332
<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>
2333
2334
<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>
2335
2336
<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>
2337
2338
<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>
2339
|}
2340
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 130 )
2341
|}
2342
2343
2344
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.
2345
2346
Desenvolvendo os termos das equações <span id='cite-_Ref530079964'></span>[[#_Ref530079964|( 127 )]] e  <span id='cite-_Ref530079552'></span>[[#_Ref530079552|( 129 )]]:
2347
2348
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2349
|-
2350
| 
2351
{| style="text-align: center; margin:auto;width: 100%;"
2352
|-
2353
| <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> 
2354
|}
2355
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 131 )
2356
|}
2357
2358
2359
Com
2360
2361
{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
2362
|-
2363
| 
2364
{| style="text-align: center; margin:auto;width: 100%;"
2365
|-
2366
| <math>{d}_{1}=\frac{{b}_{22}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>
2367
2368
<math>{d}_{2}=\frac{{b}_{12}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>
2369
2370
<math>{d}_{3}=\frac{{b}_{11}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>
2371
2372
<math>{d}_{4}=\frac{{b}_{21}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>
2373
2374
2375
|}
2376
|  style="text-align: center;vertical-align: bottom;width: 5px;text-align: right;white-space: nowrap;"|( 132 )
2377
|}
2378
2379
2380
<span id='_Ref477377056'></span>
2381
2382
===2.7 Caracterização dos casos===
2383
2384
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.
2385
2386
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]].
2387
2388
==Propriedades do Pórtico Plano (PP):==
2389
2390
{| style="width: 100%;border-collapse: collapse;" 
2391
|-
2392
|  style="text-align: center;width: 100%;"|[[Image:Draft_Vieira_908925676-image9.png|600px]]
2393
2394
2395
|}
2396
2397
2398
<span id='_Ref530127321'></span>'''Figura 8'''. Pórtico Plano de Thai e Kim<span style="text-align: center; font-size: 75%;">.</span>
2399
2400
<span id='_Ref457417233'></span>
2401
2402
<span id='_Ref494057582'></span>'''Tabela 7''' - Coordenadas do pórtico plano - Thai e Kim.
2403
2404
{| style="width: 58%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2405
|-
2406
|  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>
2407
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{X}}</math>
2408
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Y}}</math>
2409
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Z}}</math>
2410
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2411
|-
2412
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2413
|  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>
2414
|  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>
2415
|  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>
2416
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2417
|-
2418
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2419
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2420
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2421
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2422
|  style="text-align: center;vertical-align: top;"|
2423
|-
2424
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2425
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2426
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
2427
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2428
|  style="text-align: center;vertical-align: top;"|
2429
|-
2430
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2431
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
2432
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
2433
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2434
|  style="text-align: center;vertical-align: top;"|
2435
|-
2436
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2437
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
2438
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2439
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2440
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
2441
|}
2442
2443
2444
<span id='_Ref457417961'></span><span id='_Ref494057719'></span>
2445
2446
<span id='_Ref500759207'></span>'''Tabela 8''' – Características físicas do pórtico plano - Thai e Kim.
2447
2448
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2449
|-
2450
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Elem.}}</math>
2451
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, I}</math>
2452
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, F}</math>
2453
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{E}</math>
2454
|  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>
2455
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{\nu }</math>
2456
|-
2457
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2458
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2459
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2460
|  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>
2461
|  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>
2462
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2463
|-
2464
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2465
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2466
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2467
|  style="text-align: center;vertical-align: bottom;"|
2468
|  style="text-align: center;vertical-align: top;"|
2469
|  colspan='2'  style="text-align: center;vertical-align: bottom;"|
2470
|-
2471
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2472
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2473
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2474
|  style="text-align: center;vertical-align: bottom;"|1961,3
2475
|  style="text-align: center;vertical-align: top;"|9,8
2476
|  colspan='2'  style="text-align: center;vertical-align: bottom;"|0,170
2477
|-
2478
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2479
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2480
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2481
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2482
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
2483
|  colspan='2'  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2484
|}
2485
2486
2487
<span id='_Ref457418210'></span>
2488
2489
Os termos da <span id='cite-_Ref500759207'></span>[[#_Ref500759207|Tabela 8]] tem a seguinte descrição:
2490
2491
<math display="inline">E</math> = módulo de elasticidade;
2492
2493
<math display="inline">\nu</math>  = coeficiente de Poisson;
2494
2495
<math display="inline">{\sigma }_{y}</math>= tensão de escoamento.
2496
2497
<span id='_Ref494057745'></span>'''Tabela 9''' – Propriedades do material do pórtico plano - Thai e Kim.
2498
2499
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2500
|-
2501
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Elemento'''
2502
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Seção transversal'''
2503
|-
2504
|  rowspan='2' style="text-align: center;"|<span style="text-align: center; font-size: 100%;">1,2,3 e 4</span>
2505
|  style="text-align: center;"|<math>A=800,000\, c{m}^{2}</math>
2506
|-
2507
|  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>
2508
|-
2509
|  style="text-align: center;"|<math>{I}_{z}=106666,667\, c{m}^{4}</math>
2510
|-
2511
|  style="text-align: center;"|<math>{F}_{xp}=7840,000\, kN</math>
2512
|-
2513
|  style="border-bottom: 1pt solid black;text-align: center;"|<math>{M}_{zp}=78400,000\, kN\times cm</math>
2514
|}
2515
2516
2517
As cargas aplicadas são as seguintes:
2518
2519
'''Tabela 10''' – Cargas aplicadas do pórtico plano - Thai e Kim.
2520
2521
{| style="width: 58%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2522
|-
2523
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{N\acute{o}}}</math>
2524
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{Dir}</math>
2525
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{Valor\, (kN)}</math>
2526
|-
2527
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2528
|  style="text-align: center;vertical-align: bottom;"|<math>{F}_{X}</math>
2529
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1,000</span>
2530
|-
2531
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2532
|  style="text-align: center;vertical-align: bottom;"|<math>{F}_{y}</math>
2533
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">-1,000</span>
2534
|-
2535
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100;">3</span>
2536
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>{F}_{y}</math>
2537
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">-1,000</span>
2538
|}
2539
2540
2541
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.
2542
2543
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: 75%;">Figura 9</span>]].
2544
2545
{| style="width: 100%;border-collapse: collapse;" 
2546
|-
2547
|  style="vertical-align: top;"|<span id='_Ref457421631'></span> [[Image:Draft_Vieira_908925676-image10.png|600px]] '''Figura 9''' – Pórtico espacial de 2 (dois) pavimentos.
2548
2549
2550
|}
2551
2552
2553
As propriedades do caso são apresentadas nas <span id='cite-_Ref457422104'></span>[[#_Ref457422104|Tabela 11]] a <span id='cite-_Ref532329759'></span>[[#_Ref532329759|Tabela 13]].
2554
2555
<span id='_Ref457422104'></span>'''Tabela 11''' - Coordenadas do pórtico espacial - Thai e Kim.
2556
2557
{| style="width: 58%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2558
|-
2559
|  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>
2560
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{X}}</math>
2561
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Y}}</math>
2562
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Z}}</math>
2563
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2564
|-
2565
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2566
|  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>
2567
|  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>
2568
|  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>
2569
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2570
|-
2571
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2572
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2573
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2574
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2575
|  style="text-align: center;vertical-align: top;"|
2576
|-
2577
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2578
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2579
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2580
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2581
|  style="text-align: center;vertical-align: top;"|
2582
|-
2583
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2584
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2585
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2586
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2587
|  style="text-align: center;vertical-align: top;"|
2588
|-
2589
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2590
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2591
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2592
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2593
|  style="text-align: center;vertical-align: top;"|
2594
|-
2595
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2596
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2597
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2598
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2599
|  style="text-align: center;vertical-align: top;"|
2600
|-
2601
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2602
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2603
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2604
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2605
|  style="text-align: center;vertical-align: top;"|
2606
|-
2607
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2608
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2609
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2610
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2611
|  style="text-align: center;vertical-align: top;"|
2612
|-
2613
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2614
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2615
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2616
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2617
|  style="text-align: center;vertical-align: top;"|
2618
|-
2619
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2620
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2621
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2622
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2623
|  style="text-align: center;vertical-align: top;"|
2624
|-
2625
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2626
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2627
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2628
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2629
|  style="text-align: center;vertical-align: top;"|
2630
|-
2631
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2632
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2633
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2634
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2635
|  style="text-align: center;vertical-align: top;"|
2636
|-
2637
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2638
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2639
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2640
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">300,000</span>
2641
|  style="text-align: center;vertical-align: top;"|
2642
|-
2643
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2644
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2645
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2646
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">150,000</span>
2647
|  style="text-align: center;vertical-align: top;"|
2648
|-
2649
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2650
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">400,000</span>
2651
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">800,000</span>
2652
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">150,000</span>
2653
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
2654
|}
2655
2656
2657
<span id='_Ref500757194'></span>
2658
2659
'''Tabela 12''' - Características físicas do pórtico espacial - Thai e Kim.
2660
2661
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2662
|-
2663
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span id='_Ref500758784'></span> <math>\mathit{\boldsymbol{Elem.}}</math>
2664
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, I}</math>
2665
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, F}</math>
2666
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{E}</math>
2667
|  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>
2668
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{\nu }</math>
2669
|-
2670
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2671
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2672
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2673
|  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>
2674
|  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>
2675
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2676
|-
2677
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2678
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2679
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2680
|  rowspan='18' style="text-align: center;"|1961,3
2681
|  rowspan='18' style="text-align: center;"|9,8
2682
|  rowspan='18' style="text-align: center;"|0,170
2683
|-
2684
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2685
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2686
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2687
|-
2688
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2689
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2690
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2691
|-
2692
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2693
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2694
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2695
|-
2696
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2697
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2698
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2699
|-
2700
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2701
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2702
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2703
|-
2704
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2705
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2706
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2707
|-
2708
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2709
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2710
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2711
|-
2712
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2713
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2714
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2715
|-
2716
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2717
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2718
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2719
|-
2720
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2721
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2722
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2723
|-
2724
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2725
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2726
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2727
|-
2728
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2729
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2730
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2731
|-
2732
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2733
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2734
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2735
|-
2736
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">15</span>
2737
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2738
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2739
|-
2740
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">16</span>
2741
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2742
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2743
|-
2744
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">17</span>
2745
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2746
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2747
|-
2748
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">18</span>
2749
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2750
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2751
|}
2752
2753
2754
<span id='_Ref532329759'></span>'''Tabela 13''' - Propriedades do material do pórtico espacial - Thai e Kim.
2755
2756
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2757
|-
2758
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Elemento'''
2759
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Seção transversal'''
2760
|-
2761
|  rowspan='11' style="text-align: center;"|<span style="text-align: center; font-size: 100%;">1 a 18</span>
2762
|  style="text-align: center;"|<math>A=800,000\, c{m}^{2}</math>
2763
|-
2764
|  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>
2765
|-
2766
|  style="text-align: center;"|<math>{I}_{z}=1,067\times {10}^{5}\, c{m}^{4}</math>
2767
|-
2768
|  style="text-align: center;"|<math>{I}_{x}=1,067\times {10}^{5}\, c{m}^{4}</math>
2769
|-
2770
|  style="text-align: center;"|<math>{I}_{y}=2,667\times {10}^{4}\, c{m}^{4}</math>
2771
|-
2772
|  style="text-align: center;"|<math>{M}_{zp}=7,840\times {10}^{4}\, kN\times cm</math>
2773
|-
2774
|  style="text-align: center;"|<math>{M}_{xp}=6,533\times {10}^{4}\, kN\times cm</math>
2775
|-
2776
|  style="text-align: center;"|<math>{M}_{yp}=3,920\times {10}^{4}\, kN\times cm</math>
2777
|-
2778
|  style="text-align: center;"|<math>{F}_{zp}=4,526\times {10}^{3}\, kN</math>
2779
|-
2780
|  style="text-align: center;"|<math>{F}_{xp}=7,840\times {10}^{3}\, kN</math>
2781
|-
2782
|  style="border-bottom: 1pt solid black;text-align: center;"|<math>{F}_{yp}=4,526\times {10}^{3}\, kN</math>
2783
|}
2784
2785
2786
As cargas aplicadas são as seguintes:
2787
2788
'''Tabela 14''' – Cargas aplicadas do pórtico espacial - Thai e Kim.
2789
2790
{| style="width: 80%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2791
|-
2792
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{N\acute{o}}}</math>
2793
|  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>
2794
|  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>
2795
|  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>
2796
|-
2797
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</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%;">6</span>
2803
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,250</span>
2804
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2,000</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%;">7</span>
2808
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,250</span>
2809
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2,000</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%;">8</span>
2813
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1,000</span>
2814
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2,000</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%;">9</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%;">10</span>
2823
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,500</span>
2824
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-0,500</span>
2825
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2826
|-
2827
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2828
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,500</span>
2829
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-0,500</span>
2830
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2831
|-
2832
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2833
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3,000</span>
2834
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-0,500</span>
2835
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2836
|-
2837
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2838
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2839
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1,000</span>
2840
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2841
|-
2842
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
2843
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2844
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1,000</span>
2845
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2846
|}
2847
2848
2849
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.
2850
2851
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]].
2852
2853
{| style="width: 100%;border-collapse: collapse;" 
2854
|-
2855
|  style="text-align: center;vertical-align: top;width: 100%;"|[[Image:Draft_Vieira_908925676-image11.png|600px]]
2856
2857
2858
|}
2859
2860
2861
<span id='_Ref533152504'></span>'''Figura 10''' – Pórtico em domo - Argyris et al.
2862
2863
As propriedades do caso são apresentadas nas <span id='cite-_Ref534577723'></span>[[#_Ref534577723|Tabela 15]] a <span id='cite-_Ref534577726'></span>[[#_Ref534577726|Tabela 18]].
2864
2865
<span id='_Ref534577723'></span>'''Tabela 15''' - Coordenadas do <span style="text-align: center; font-size: 75%;">pórtico em domo - </span>Argyris et al.
2866
2867
{| style="width: 65%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2868
|-
2869
|  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>
2870
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{X}}</math>
2871
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Y}}</math>
2872
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>\mathit{\boldsymbol{Z}}</math>
2873
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2874
|-
2875
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
2876
|  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>
2877
|  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>
2878
|  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>
2879
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
2880
|-
2881
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2882
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2883
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">610,000</span>
2884
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2885
|  style="text-align: center;vertical-align: top;"|
2886
|-
2887
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2888
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">628,500</span>
2889
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2890
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1088,500</span>
2891
|  style="text-align: center;vertical-align: top;"|
2892
|-
2893
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2894
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1257,000</span>
2895
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2896
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2897
|  style="text-align: center;vertical-align: top;"|
2898
|-
2899
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2900
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">628,500</span>
2901
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2902
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1088,500</span>
2903
|  style="text-align: center;vertical-align: top;"|
2904
|-
2905
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2906
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-628,500</span>
2907
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2908
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1088,500</span>
2909
|  style="text-align: center;vertical-align: top;"|
2910
|-
2911
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
2912
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1257,000</span>
2913
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2914
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2915
|  style="text-align: center;vertical-align: top;"|
2916
|-
2917
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
2918
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-628,500</span>
2919
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">455,000</span>
2920
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">1088,500</span>
2921
|  style="text-align: center;vertical-align: top;"|
2922
|-
2923
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
2924
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">1219,000</span>
2925
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2926
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">2111,500</span>
2927
|  style="text-align: center;vertical-align: top;"|
2928
|-
2929
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
2930
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">2438,000</span>
2931
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2932
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2933
|  style="text-align: center;vertical-align: top;"|
2934
|-
2935
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
2936
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">1219,000</span>
2937
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2938
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2111,500</span>
2939
|  style="text-align: center;vertical-align: top;"|
2940
|-
2941
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
2942
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1219,000</span>
2943
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2944
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2111,500</span>
2945
|  style="text-align: center;vertical-align: top;"|
2946
|-
2947
|  style="text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
2948
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-2438,000</span>
2949
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2950
|  style="text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2951
|  style="text-align: center;vertical-align: top;"|
2952
|-
2953
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
2954
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">-1219,000</span>
2955
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">0,000</span>
2956
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<span style="text-align: center; font-size: 100%;">2111,500</span>
2957
|  style="text-align: center;vertical-align: top;"|
2958
|}
2959
2960
2961
'''Tabela 16''' - Características físicas do <span style="text-align: center; font-size: 75%;">pórtico em domo - </span>Argyris et al.
2962
2963
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
2964
|-
2965
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Elem.}}</math>
2966
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, I}</math>
2967
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{N\acute{o}\, F}</math>
2968
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{E}</math>
2969
|  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>
2970
|  colspan='2'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\boldsymbol{\nu }</math>
2971
|-
2972
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2973
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2974
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2975
|  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>
2976
|  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>
2977
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
2978
|-
2979
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</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%;">2</span>
2982
|  rowspan='18' style="text-align: center;"|2068,0
2983
|  rowspan='18' style="text-align: center;"|8,0
2984
|  rowspan='18' style="text-align: center;"|0,1716
2985
|-
2986
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
2987
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2988
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2989
|-
2990
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
2991
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2992
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2993
|-
2994
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
2995
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
2996
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2997
|-
2998
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
2999
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
3000
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
3001
|-
3002
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
3003
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">1</span>
3004
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
3005
|-
3006
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
3007
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
3008
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
3009
|-
3010
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
3011
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
3012
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
3013
|-
3014
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
3015
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
3016
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
3017
|-
3018
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
3019
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
3020
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
3021
|-
3022
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
3023
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
3024
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
3025
|-
3026
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
3027
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
3028
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
3029
|-
3030
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
3031
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">2</span>
3032
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">8</span>
3033
|-
3034
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">14</span>
3035
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">3</span>
3036
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">9</span>
3037
|-
3038
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">15</span>
3039
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">4</span>
3040
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">10</span>
3041
|-
3042
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">16</span>
3043
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">5</span>
3044
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">11</span>
3045
|-
3046
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">17</span>
3047
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">6</span>
3048
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">12</span>
3049
|-
3050
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">18</span>
3051
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">7</span>
3052
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 100%;">13</span>
3053
|}
3054
3055
3056
'''Tabela 17''' - Propriedades do material do <span style="text-align: center; font-size: 75%;">pórtico em domo - </span>Argyris et al.
3057
3058
{| style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3059
|-
3060
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Elemento'''
3061
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|'''Seção transversal'''
3062
|-
3063
|  rowspan='11' style="text-align: center;"|<span style="text-align: center; font-size: 100%;">1 a 18</span>
3064
|  style="text-align: center;"|<math>A=9272,000\, c{m}^{2}</math>
3065
|-
3066
|  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>
3067
|-
3068
|  style="text-align: center;"|<math>{I}_{z}=1,150\times {10}^{7}\, c{m}^{4}</math>
3069
|-
3070
|  style="text-align: center;"|<math>{I}_{x}=1,596\times {10}^{7}\, c{m}^{4}</math>
3071
|-
3072
|  style="text-align: center;"|<math>{I}_{y}=4,463\times {10}^{6}\, c{m}^{4}</math>
3073
|-
3074
|  style="text-align: center;"|<math>{M}_{zp}=2,262\times {10}^{6}\, kN\times cm</math>
3075
|-
3076
|  style="text-align: center;"|<math>{M}_{xp}=6,533\times {10}^{9}\, kN\times cm</math>
3077
|-
3078
|  style="text-align: center;"|<math>{M}_{yp}=1,409\times {10}^{6}\, kN\times cm</math>
3079
|-
3080
|  style="text-align: center;"|<math>{F}_{zp}=4,282\times {10}^{4}\, kN</math>
3081
|-
3082
|  style="text-align: center;"|<math>{F}_{xp}=7,416\times {10}^{4}\, kN</math>
3083
|-
3084
|  style="border-bottom: 1pt solid black;text-align: center;"|<math>{F}_{yp}=4,282\times {10}^{4}\, kN</math>
3085
|}
3086
3087
3088
As cargas aplicadas são as seguintes:
3089
3090
<span id='_Ref534577726'></span>'''Tabela 18''' – Cargas aplicadas do <span style="text-align: center; font-size: 75%;">pórtico em domo - </span>Argyris et al.
3091
3092
{| style="width: 80%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3093
|-
3094
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{N\acute{o}}}</math>
3095
|  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>
3096
|  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>
3097
|  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>
3098
|-
3099
|  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>
3100
|  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>
3101
|  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>
3102
|  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>
3103
|}
3104
3105
3106
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.
3107
3108
==3 RESULTADOS E DISCUSSÃO==
3109
3110
Os resultados e discussões dos estudos de caso 1 a 3 são apresentados.
3111
3112
===3.1 Caso 1===
3113
3114
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''']].
3115
3116
<span id='_Ref4574898'></span>'''Tabela 19''' – Rótulas plásticas – pórtico plano - Thai e Kim.
3117
3118
{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3119
|-
3120
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span id='_Hlk4522464'></span>
3121
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3122
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{4}</math>
3123
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3124
|-
3125
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: left; font-size: 75%;">'''Elemento'''</span>
3126
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3127
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3128
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3129
|-
3130
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3131
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3132
|  style="text-align: right;vertical-align: bottom;"|<math>0,108246\times {10}^{5}</math>
3133
|  rowspan='5' style="text-align: center;"|<math>309,146</math>
3134
|-
3135
|  style="text-align: center;vertical-align: bottom;"|
3136
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3137
|  style="text-align: right;vertical-align: bottom;"|<math>0,841207\times {10}^{4}</math>
3138
|-
3139
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: left; font-size: 75%;">2</span>
3140
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3141
|  style="text-align: right;vertical-align: bottom;"|<math>0,227389\times {10}^{2}</math>
3142
|-
3143
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: left; font-size: 75%;">3</span>
3144
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3145
|  style="text-align: right;vertical-align: bottom;"|<math>0,844737\times {10}^{4}</math>
3146
|-
3147
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
3148
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3149
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,108410\times {10}^{5}</math>
3150
|-
3151
|-
3152
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3153
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3154
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{5}</math>
3155
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3156
|-
3157
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3158
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3159
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3160
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3161
|-
3162
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3163
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3164
|  style="text-align: center;vertical-align: bottom;"|<math>0,106273\times {10}^{5}</math>
3165
|  rowspan='5' style="text-align: center;"|<math>300,431</math>
3166
|-
3167
|  style="text-align: center;vertical-align: bottom;"|
3168
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3169
|  style="text-align: center;vertical-align: bottom;"|<math>0,208395\times {10}^{4}</math>
3170
|-
3171
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3172
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3173
|  style="text-align: center;vertical-align: bottom;"|<math>0,626480\times {10}^{4}</math>
3174
|-
3175
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3176
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3177
|  style="text-align: center;vertical-align: bottom;"|<math>0,836829\times {10}^{4}</math>
3178
|-
3179
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
3180
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3181
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,106452\times {10}^{5}</math>
3182
|-
3183
|-
3184
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3185
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3186
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{6}</math>
3187
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3188
|-
3189
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3190
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3191
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3192
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3193
|-
3194
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3195
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3196
|  style="text-align: center;vertical-align: bottom;"|<math>0,114845\times {10}^{5}</math>
3197
|  rowspan='5' style="text-align: center;"|<math>318,103</math>
3198
|-
3199
|  style="text-align: center;vertical-align: bottom;"|
3200
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3201
|  style="text-align: center;vertical-align: bottom;"|<math>0,254762\times {10}^{4}</math>
3202
|-
3203
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3204
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3205
|  style="text-align: center;vertical-align: bottom;"|<math>0,604007\times {10}^{4}</math>
3206
|-
3207
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3208
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3209
|  style="text-align: center;vertical-align: bottom;"|<math>0,899557\times {10}^{4}</math>
3210
|-
3211
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
3212
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3213
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,117705\times {10}^{5}</math>
3214
|}
3215
3216
3217
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
3218
 [[Image:Draft_Vieira_908925676-image12-c.png|438px]] </div>
3219
3220
<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.
3221
3222
{| style="width: 100%;border-collapse: collapse;" 
3223
|-
3224
|  style="text-align: center;vertical-align: top;"|
3225
|}
3226
3227
===3.2 Caso 2===
3228
3229
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''']].
3230
3231
<span id='_Ref4571218'></span><span id='_Ref500765287'></span>'''Tabela 20''' - Rótulas plásticas – pórtico espacial - Thai e Kim
3232
3233
{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3234
|-
3235
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3236
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3237
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{1}</math>
3238
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3239
|-
3240
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3241
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3242
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3243
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3244
|-
3245
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3246
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3247
|  style="text-align: center;vertical-align: bottom;"|<math>0,345728\times {10}^{4}</math>
3248
|  rowspan='12' style="text-align: center;"|<math>141,886</math>
3249
|-
3250
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3251
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3252
|  style="text-align: center;vertical-align: bottom;"|<math>0,353734\times {10}^{4}</math>
3253
|-
3254
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3255
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3256
|  style="text-align: center;vertical-align: bottom;"|<math>0,348667\times {10}^{4}</math>
3257
|-
3258
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3259
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3260
|  style="text-align: center;vertical-align: bottom;"|<math>0,350863\times {10}^{4}</math>
3261
|-
3262
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3263
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3264
|  style="text-align: center;vertical-align: bottom;"|<math>0,308873\times {10}^{4}</math>
3265
|-
3266
|  style="text-align: center;vertical-align: bottom;"|
3267
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3268
|  style="text-align: center;vertical-align: bottom;"|<math>0,307316\times {10}^{4}</math>
3269
|-
3270
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3271
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3272
|  style="text-align: center;vertical-align: bottom;"|<math>0,306402\times {10}^{4}</math>
3273
|-
3274
|  style="text-align: center;vertical-align: bottom;"|
3275
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">8</span>
3276
|  style="text-align: center;vertical-align: bottom;"|<math>0,309575\times {10}^{4}</math>
3277
|-
3278
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3279
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3280
|  style="text-align: center;vertical-align: bottom;"|<math>0,899973\times {10}^{3}</math>
3281
|-
3282
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3283
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3284
|  style="text-align: center;vertical-align: bottom;"|<math>0,933775\times {10}^{3}</math>
3285
|-
3286
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3287
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3288
|  style="text-align: center;vertical-align: bottom;"|<math>0,984856\times {10}^{3}</math>
3289
|-
3290
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3291
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3292
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,849066\times {10}^{3}</math>
3293
|}
3294
3295
{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3296
|-
3297
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3298
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3299
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{2}</math>
3300
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3301
|-
3302
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3303
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3304
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3305
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3306
|-
3307
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3308
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3309
|  style="text-align: center;vertical-align: bottom;"|<math>0,321786\times {10}^{4}</math>
3310
|  rowspan='12' style="text-align: center;"|<math>134,077</math>
3311
|-
3312
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3313
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3314
|  style="text-align: center;vertical-align: bottom;"|<math>0,326692\times {10}^{4}</math>
3315
|-
3316
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3317
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3318
|  style="text-align: center;vertical-align: bottom;"|<math>0,326692\times {10}^{4}</math>
3319
|-
3320
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3321
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3322
|  style="text-align: center;vertical-align: bottom;"|<math>0,321786\times {10}^{4}</math>
3323
|-
3324
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3325
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3326
|  style="text-align: center;vertical-align: bottom;"|<math>0,299500\times {10}^{4}</math>
3327
|-
3328
|  style="text-align: center;vertical-align: bottom;"|
3329
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3330
|  style="text-align: center;vertical-align: bottom;"|<math>0,298730\times {10}^{4}</math>
3331
|-
3332
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3333
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3334
|  style="text-align: center;vertical-align: bottom;"|<math>0,298730\times {10}^{4}</math>
3335
|-
3336
|  style="text-align: center;vertical-align: bottom;"|
3337
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">8</span>
3338
|  style="text-align: center;vertical-align: bottom;"|<math>0,299500\times {10}^{4}</math>
3339
|-
3340
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3341
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3342
|  style="text-align: center;vertical-align: bottom;"|<math>0,106747\times {10}^{4}</math>
3343
|-
3344
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3345
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3346
|  style="text-align: center;vertical-align: bottom;"|<math>0,107793\times {10}^{4}</math>
3347
|-
3348
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3349
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3350
|  style="text-align: center;vertical-align: bottom;"|<math>0,107793\times {10}^{4}</math>
3351
|-
3352
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3353
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3354
|  style="text-align: center;vertical-align: bottom;"|<math>0,106747\times {10}^{4}</math>
3355
|-
3356
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3357
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3358
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{3}</math>
3359
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3360
|-
3361
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3362
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3363
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3364
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
3365
|-
3366
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3367
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3368
|  style="text-align: center;vertical-align: bottom;"|<math>0,345110\times {10}^{4}</math>
3369
|  rowspan='5' style="text-align: center;"|<math>141,900</math>
3370
|-
3371
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3372
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3373
|  style="text-align: center;vertical-align: bottom;"|<math>0,353048\times {10}^{4}</math>
3374
|-
3375
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3376
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3377
|  style="text-align: center;vertical-align: bottom;"|<math>0,348312\times {10}^{4}</math>
3378
|-
3379
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3380
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3381
|  style="text-align: center;vertical-align: bottom;"|<math>0,349939\times {10}^{4}</math>
3382
|-
3383
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3384
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3385
|  style="text-align: center;vertical-align: bottom;"|<math>0,309236\times {10}^{4}</math>
3386
|-
3387
|  style="text-align: center;vertical-align: bottom;"|
3388
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3389
|  style="text-align: center;vertical-align: bottom;"|<math>0,307839\times {10}^{4}</math>
3390
|  style="text-align: right;vertical-align: bottom;"|
3391
|-
3392
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3393
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3394
|  style="text-align: center;vertical-align: bottom;"|<math>0,306698\times {10}^{4}</math>
3395
|  style="text-align: right;vertical-align: bottom;"|
3396
|-
3397
|  style="text-align: center;vertical-align: bottom;"|
3398
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">8</span>
3399
|  style="text-align: center;vertical-align: bottom;"|<math>0,309736\times {10}^{4}</math>
3400
|  style="text-align: right;vertical-align: bottom;"|
3401
|-
3402
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3403
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">9</span>
3404
|  style="text-align: center;vertical-align: bottom;"|<math>0,905438\times {10}^{3}</math>
3405
|  style="text-align: right;vertical-align: bottom;"|
3406
|-
3407
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3408
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">10</span>
3409
|  style="text-align: center;vertical-align: bottom;"|<math>0,938623\times {10}^{3}</math>
3410
|  style="text-align: right;vertical-align: bottom;"|
3411
|-
3412
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3413
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">11</span>
3414
|  style="text-align: center;vertical-align: bottom;"|<math>0,983798\times {10}^{3}</math>
3415
|  style="text-align: right;vertical-align: bottom;"|
3416
|-
3417
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3418
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">12</span>
3419
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,855577\times {10}^{3}</math>
3420
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: bottom;"|
3421
|}
3422
3423
3424
<div id="_Ref457419942" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
3425
 [[Image:Draft_Vieira_908925676-image13-c.png|456px]] </div>
3426
3427
<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.
3428
3429
===3.3 Caso 3===
3430
3431
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''']].
3432
3433
<span id='_Ref4573319'></span>'''Tabela 21''' - Rótulas plásticas – pórtico espacial - pórtico espacial - Argyris et al.
3434
3435
{| style="width: 36%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
3436
|-
3437
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3438
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3439
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{1}</math>
3440
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3441
|-
3442
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3443
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3444
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3445
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (MN)}}</math>
3446
|-
3447
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3448
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3449
|  style="text-align: center;vertical-align: bottom;"|<math>0,549351</math>
3450
|  rowspan='12' style="text-align: center;"|<math>51,0815</math>
3451
|-
3452
|  style="text-align: center;vertical-align: bottom;"|
3453
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3454
|  style="text-align: center;vertical-align: bottom;"|<math>0,891167</math>
3455
|-
3456
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3457
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3458
|  style="text-align: center;vertical-align: bottom;"|<math>0,552499</math>
3459
|-
3460
|  style="text-align: center;vertical-align: bottom;"|
3461
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3462
|  style="text-align: center;vertical-align: bottom;"|<math>0,861987</math>
3463
|-
3464
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</span>
3465
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3466
|  style="text-align: center;vertical-align: bottom;"|<math>0,564206</math>
3467
|-
3468
|  style="text-align: center;vertical-align: bottom;"|
3469
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3470
|  style="text-align: center;vertical-align: bottom;"|<math>0,883022</math>
3471
|-
3472
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</span>
3473
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3474
|  style="text-align: center;vertical-align: bottom;"|<math>0,569621</math>
3475
|-
3476
|  style="text-align: center;vertical-align: bottom;"|
3477
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3478
|  style="text-align: center;vertical-align: bottom;"|<math>0,885590</math>
3479
|-
3480
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</span>
3481
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3482
|  style="text-align: center;vertical-align: bottom;"|<math>0,556237</math>
3483
|-
3484
|  style="text-align: center;vertical-align: bottom;"|
3485
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3486
|  style="text-align: center;vertical-align: bottom;"|<math>0,862806</math>
3487
|-
3488
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3489
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3490
|  style="text-align: center;vertical-align: bottom;"|<math>0,541887</math>
3491
|-
3492
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|
3493
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3494
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,934919</math>
3495
|}
3496
3497
3498
{| style="width: 36%;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="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="text-align: center;vertical-align: bottom;"|
3556
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">7</span>
3557
|  style="text-align: center;vertical-align: bottom;"|<math>2,62451</math>
3558
|-
3559
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3560
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3561
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{3}</math>
3562
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
3563
|-
3564
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
3565
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">''''''</span>
3566
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
3567
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (MN)}}</math>
3568
|-
3569
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3570
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3571
|  style="text-align: center;vertical-align: bottom;"|<math>0,227607</math>
3572
|  rowspan='6' style="text-align: center;"|<math>48,0914</math>
3573
|-
3574
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">2</span>
3575
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3576
|  style="text-align: center;vertical-align: bottom;"|<math>0,227118</math>
3577
|-
3578
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">3</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>0,226262</math>
3581
|-
3582
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">4</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>0,224668</math>
3585
|-
3586
|  style="text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">5</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>0,224136</math>
3589
|-
3590
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">6</span>
3591
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<span style="text-align: center; font-size: 75%;">1</span>
3592
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: bottom;"|<math>0,225236</math>
3593
|}
3594
3595
3596
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
3597
 [[Image:Draft_Vieira_908925676-image14-c.png|444px]] </div>
3598
3599
<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.
3600
3601
==4 CONCLUSÕES==
3602
3603
* As superfícies de escoamento em resultantes de tensões aplicadas para pórticos planos e espaciais obtiveram resultados satisfatórios tendo em vista os exemplos apresentados;
3604
3605
* 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;
3606
3607
* O método apresentado permitiu o uso do modelo de dano de viga de Timoshenko 3D com bons resultados para estruturas de aço;
3608
3609
'''* '''A regressão linear múltipla se apresenta como uma solução viável para obter funções por análises numéricas e/ou experimentais;
3610
3611
'''* '''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.
3612
3613
==AGRADECIMENTOS==
3614
3615
À UFOB, CIMNE/UPC, PECC/UnB e a CAPES.
3616
3617
==REFERÊNCIAS==
3618
3619
==5 REFERÊNCIAS==
3620
3621
3622
{| style="width: 100%;" 
3623
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3624
|  style="text-align: right;vertical-align: top;"|1.
3625
|  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.
3626
|-
3627
|  style="text-align: right;vertical-align: top;"|2.
3628
|  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.
3629
|-
3630
|  style="text-align: right;vertical-align: top;"|3.
3631
|  style="vertical-align: top;"|LUBLINER, J. '''Plasticity Theory'''. Nova Iorque: Macmillan Publishing Company, 1990.
3632
|-
3633
|  style="text-align: right;vertical-align: top;"|4.
3634
|  style="vertical-align: top;"|MRÁZIK, A.; ÉSKALOUD, M.; TOCHÁÉCEK, M. '''Plastic Design of Steel Structures'''. Nova Iorque: E. Horwood: Halsted Press, 1987.
3635
|-
3636
|  style="text-align: right;vertical-align: top;"|5.
3637
|  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: right;vertical-align: top;"|6.
3640
|  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.
3641
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3642
|  style="text-align: right;vertical-align: top;"|7.
3643
|  style="vertical-align: top;"|MONTGOMERY, D. C.; RUNGER, G. C. '''Estatística Aplicada e Probabilidade para Engenheiros'''. Rio de Janeiro: LTC, 2016.
3644
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3645
|  style="text-align: right;vertical-align: top;"|8.
3646
|  style="vertical-align: top;"|HORNE, M. R. '''Plastic theory of structures'''. 2ª. ed. Oxford: Pergamon Press, 1972.
3647
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3648
|  style="text-align: right;vertical-align: top;"|9.
3649
|  style="vertical-align: top;"|NEAL, B. G. '''The plastic methods of structural analysis'''. Inglaterra: Chapman and Hall, 1977.
3650
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3651
|  style="text-align: right;vertical-align: top;"|10.
3652
|  style="vertical-align: top;"|SILVA, W. T. M. '''Análise Elastoplástica de Pórticos Espaciais Utilizando o Conceito de Ró-'''. Métodos Computacionais em Engenharia. Lisboa: [s.n.]. 2004.
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3654
|  style="text-align: right;vertical-align: top;"|11.
3655
|  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.
3656
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3657
|  style="text-align: right;vertical-align: top;"|12.
3658
|  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.
3659
|-
3660
|  style="text-align: right;vertical-align: top;"|13.
3661
|  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.
3662
|}
3663

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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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