Elastoplastic analysis of steel frames with interaction surfaces in stress resultants obtained for multiple linear regression

==Resumo==

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.

'''Palavras-chave:''' Superfícies de interação, vigas de Timoshenko 3D, regressão linear múltipla, resultantes de tensões, pórticos metálicos, análise elastoplástica

==Abstract==

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.

'''Keywords: '''Interaction curves, 3D Timoshenko beams, multiple linear regression, stress resultants, steel frame, elastoplastic analysis

<span id='_Ref457256930'></span>

==1. Introdução==

A análise elastoplástica com pórticos espaciais ou planos necessita da função de escoamento que controla o término da fase elástica e o estado plástico da estrutura. Usar superfícies de interação em resultantes de tensões é de mais fácil entendimento para os projetistas porque geralmente os esforços seccionais são apresentados nestas resultantes, a saber, momentos, cortantes e axial. O modelo de dano em vigas de Timoshenko em vigas 3D permite obter os esforços em resultantes de tensões com a versatilidade de poder usá-lo para estruturas de concreto armado ou aço, de acordo com os parâmetros adotados. No trabalho de Vieira [1] faz-se a aplicação da tese doutoral de Hanganu [2] para o caso de estruturas de aço com a definição do limite de dano quando o valor da função de endurecimento (<math display="inline">k\left( d\right) )</math> for igual a máxima resistência ao cortante octaédrica (<math display="inline">{\tau }_{oct}^{m\acute{a}x}(d)</math>). As verificações realizadas demonstraram que os limites plásticos foram atendidos. Este trabalho fará a aplicação das superfícies de interação em pórticos planos e espaciais, de modo que seja verificado se a regressão linear múltipla consegue demostrar se a função é de boa utilidade ou não na análise elastoplástica.

==2. Métodos==

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]].

<span id='_Ref4583336'></span>

===2.1 Funções de Escoamento===

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:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>f={n}^{2}+{m}_{i}-1=0\, com\, i=x,y</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(1)
|}

onde:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math display="inline">n=\frac{N}{{N}_{xp}}</math>; <math display="inline">{m}_{i}=</math><math>\frac{{M}_{i}}{{M}_{ip}}</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(2)
|}

com

<math display="inline">N</math> = esforço normal atuante;

<math display="inline">{N}_{xp}</math> = esforço axial de plastificação;

<math display="inline">{M}_{i}</math> = esforço de momento atuante;

<math display="inline">{M}_{ip}</math> = esforço de momento de plastificação;

<math display="inline">x\, e\, y</math> = direções dos esforços no sistema de referência.

Outras funções apresentam as interações de esforços seccionais como segue:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>f={n}^{2}+\frac{s}{\sqrt{3}}{m}_{i}n+{m}_{i}^{2}-1=0\,</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(3)
|}

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>f={n}^{2}+3s{m}_{i}n+\frac{9}{4}{m}_{i}^{2}-1=0\,</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(4)
|}

com

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>s=\frac{{M}_{i}}{\left| {M}_{ip}\right| }\,</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(5)
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>f={m}_{z}+\frac{3}{4}{m}_{y}^{2}-1=0</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(6)
|}

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>f={m}_{i}^{2}+{f}_{i}^{2}-1=0</math>
|}
|  style="width: 5px;text-align: right;white-space: nowrap;"|(7)
|}

com
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <math>{f}_{i}=\frac{{F}_{i}}{{F}_{ip}}</math>
|}


<math display="inline">{F}_{i}</math> = força cortante atuante;

<math display="inline">{F}_{ip}</math> = força cortante de plastificação.

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].

===2.2 Modelo de dano ===

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]]:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>d=\frac{{S}_{n}-{\overline{S}}_{n}}{{S}_{n}}=1-\frac{{\overline{S}}_{n}}{{S}_{n}}</math>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529712878'></span>(8)
|}

onde

<math display="inline">{S}_{n}</math> = área total da seção;

<math display="inline">{\overline{S}}_{n}</math>= área resistente efetiva;

<math display="inline">{S}_{n}-{\overline{S}}_{n}\,</math>  = área ocupada pelas aberturas.


{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"|  [[Image:Review_967755629029_4053_test-image1.png|120px]] 
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figura 1'''. Superfície com dano
|}


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]]:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>\sigma S=\, \overline{\sigma }\boldsymbol{\, }\overline{S}</math>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529712885'></span>(9)
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|  style="text-align: center;padding:10px;"| [[Image:Review_967755629029_1811_test-image2.png|158px]] 
|  style="text-align: center;vertical-align: top;padding:10px;"| [[Image:Review_967755629029_8526_test-image3.png|158px]] 
|-
|  style="text-align: center;font-size: 75%;"|(a) Região real com dano
|  style="text-align: center;font-size: 75%;"|(b) Região equivalente sem dano
|- style="text-align: center; font-size: 75%;"
| colspan="2" style="padding:10px;"| '''Figura 2'''. Tensão de Cauchy <math display="inline">\sigma</math>  e tensão efetiva <math display="inline">\overline{\sigma }</math>
|}


Fazendo as relações entre as equações <span id='cite-_Ref529712878'></span>[[#_Ref529712878|(8)]] e <span id='cite-_Ref529712885'></span>[[#_Ref529712885|(9)]] obtém-se:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(10)
|}

onde:

<math display="inline">E</math> = módulo de elasticidade do material;

<math display="inline">\epsilon</math>  = deformação do material.

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:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529714747'></span>(11)
|}

com

<math display="inline">{\Psi }_{0}</math> = energia livre elástica de Helmholtz do material sem danos;

<math display="inline">\Psi</math>  = energia livre de Helmholtz para um modelo com dano isotérmico;

<math display="inline">{m}_{0}</math>= densidade na configuração material.

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

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>\left( \frac{1}{{m}_{o}}{\mathit{\boldsymbol{\sigma }}}^{T}-\frac{\partial \boldsymbol{\Psi }}{\partial \mathit{\boldsymbol{\epsilon }}}\right) =</math><math>0</math>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529778988'></span>(12)
|}

Desenvolvendo a equação <span id='cite-_Ref529778988'></span>[[#_Ref529778988|(12)]] chega-se a:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(13)
|}

onde <math display="inline">{\mathit{\boldsymbol{C}}}^{s}= \left( 1- d\right) {\mathit{\boldsymbol{C}}}^{0}</math> é a matriz constitutiva secante do material com dano.

Por consequência o termo restante da equação <span id='cite-_Ref529714747'></span>[[#_Ref529714747|(11)]] torna-se em

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>\overset{\cdot}{{\Xi }_{m}}={-\frac{\partial \boldsymbol{\Psi }}{\partial d}\overset{\cdot}{d}\boldsymbol{=\Psi }}_{0}\overset{\cdot}{d}\geq 0</math>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529779568'></span>(14)
|}


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>.

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.

O termo <math display="inline">n</math> é

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>n=\frac{{f}_{c}}{{f}_{t}}</math>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(15)
|}

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[Image:Review_967755629029_5411_test-image4.png|402px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figura 3'''. Função limite de dano no plano principal <math display="inline">{\sigma }_{1}-</math><math>{\sigma }_{2}</math>
|}


A equação que a representa é a seguinte:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math display="inline">\overline{F}=\, G\left( \overline{\sigma }\right) -G({f}_{c})\leq 0</math> 
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(16)
|}

onde <math display="inline">G\left( \chi \right)</math>  = função escalar, inversível, positiva e derivada positiva, a determinar.

A função de evolução do limite de dano, Hanganu [2], é mostrada na <span id='cite-_Ref529718486'></span>[[#_Ref529718486|Figura 4]].

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[Image:Review_967755629029_3845_test-image5.png|250px]] 
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figura 4'''. Representação da função <math display="inline">G\left( \overline{\sigma }\right)</math>
|}


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 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>

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>\kappa \, (d)={\tau }_{oct}^{m\acute{a}x}(d)</math>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(17)
|}

Este critério é representado na equação  <span id='cite-_Ref529821017'></span>[[#_Ref529821017|(18)]]:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529821017'></span>(18)
|}

===2.3 Superfícies de interação por regressão linear múltipla===

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].

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[Image:Review_967755629029_4024_test-image6.png|220px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figura 5'''. Pontos gerados para criar a função de escoamento (caso uniaxial)
|}


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:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>f={\beta }_{1}{n}^{2}+{\beta }_{2}m-1=0</math>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(19)
|}

onde

<math display="inline">n\, e\, m</math> = esforços normal e fletor adimensionais, respectivamente;

<math display="inline">{\beta }_{1}\, e\, {\beta }_{2}</math> = coeficientes obtidos pela regressão linear múltipla.

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].

O modelo desenvolvido para a formulação pretendida tem a seguinte forma:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529721691'></span>(20)
|}

onde:

<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;

<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;

<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;

<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;

<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;

<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.

Na regressão as observações da equação <span id='cite-_Ref529721691'></span>[[#_Ref529721691|(20)]] podem ser apresentadas como

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>Y={\beta }_{0}+{\beta }_{1}{x}_{i1}+{\beta }_{2}{x}_{i2}+\cdots +{\beta }_{k}{x}_{ik}+</math><math>{\epsilon }_{i}=0 \qquad i=1,2,\, \cdots ,n</math>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(21)
|}

onde

<math display="inline">n</math> = número de observações (ensaios);

<math display="inline">{\beta }_{k}</math> = coeficientes de regressão da resposta <math display="inline">Y</math>;

<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;

<math display="inline">{\epsilon }_{i}</math> = erros do modelo.

O enfoque matricial da formulação é mostrado como segue:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>\mathit{\boldsymbol{Y}}=\mathit{\boldsymbol{X\beta }}+\mathit{\boldsymbol{\epsilon }}=</math><math>0</math>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(22)
|}

com

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(23)
|}

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(24)
|}

onde

<math display="inline">\mathit{\boldsymbol{Y}}</math> = é o vetor de observações de dimensão <math display="inline">(n\times 1)</math>;

<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;

<math display="inline">\mathit{\boldsymbol{\beta }}</math> = é o vetor dos coeficientes de regressão de dimensão <math display="inline">(p\times 1)</math>;

<math display="inline">\mathit{\boldsymbol{\epsilon }}</math> = é o vetor dos erros aleatórios de dimensão <math display="inline">(n\times 1)</math>.

Deve-se encontrar o vetor dos estimadores dos mínimos quadrado, <math display="inline">\hat{\mathit{\boldsymbol{\beta }}}</math>, que minimiza

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(25)
|}

Desenvolvendo os cálculos chega-se a:

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(26)
|}

Mais detalhes dos processos de cálculo podem ser lidos em Montgomery [7].

O modelo ajustado passa a ter a seguinte forma;

{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math display="inline">{\hat{Y}}_{i}={\hat{\beta }}_{0}+\sum _{j=1}^{n}{\hat{\beta }}_{j}{x}_{ij}</math> 
|}
|  style="vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(27)
|}

Com <math display="inline">i=1,\, 2,\, \cdots ,n</math>.

Os testes de hipóteses utilizados são o estatístico de prova “F” e os de coeficientes individuais “t” que podem ser compreendidos com detalhes em Montgomery [7].

As superfícies e seus os resultados estatísticos (Tabelas de 1 a 6) que serão usados nas análises elastoplásticas, obtidos em Vieira [1], são as seguintes:

{| class="formulaSCP" style="width: 72%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref533151259'></span>(28)
|}

{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math display="inline">{f}_{2}=1,158{n}^{2}+1,118{m}_{y}^{2}+1,124{m}_{z}^{2}-1=</math><math>0</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref533151260'></span>(29)
|}

{| class="formulaSCP" style="width: 85%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref533151262'></span>(30)
|}

{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math display="inline">{f}_{4}=1,012{n}^{2}+1,027{m}_{z}^{2}-1=0</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref531207210'></span>(31)
|}

{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math display="inline">{f}_{5}=1,\, 242{n}^{2}+1,\, 087{m}_{z}^{2}-1=0</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref531207213'></span>(32)
|}

{| class="formulaSCP" style="width: 52%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math display="inline">{f}_{6}=1,089n+\, 0,929{m}_{z}^{2}-1=0</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref531207215'></span>(33)
|}


<div class="center" style="font-size: 75%;">'''Tabela 1'''. Prova de significância da superfície <math display="inline">{f}_{1}</math>
</div>

{| style="width: 62%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|23,860
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|3
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|7,953
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1183,674
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
|  style="text-align: center;vertical-align: top;"|0,1400
|  style="text-align: center;vertical-align: top;"|21
|  style="text-align: center;vertical-align: top;"|0,007
|  style="text-align: center;vertical-align: top;"|
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Estimado'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
|  style="text-align: center;vertical-align: top;"|1,1580
|  style="text-align: center;vertical-align: top;"|0,0377
|  style="text-align: center;vertical-align: top;"|30,740
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{m}_{y}^{2}</math>
|  style="text-align: center;vertical-align: top;"|1,1180
|  style="text-align: center;vertical-align: top;"|0,0387
|  style="text-align: center;vertical-align: top;"|28,900
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,1240
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0381
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|29,530
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|}


<div class="center" style="font-size: 75%;">'''Tabela 2'''. Prova de significância da superfície <math display="inline">{f}_{2}</math>
</div>

{| style="width: 62%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|24,000
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|4,800
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|55913,754
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|19
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
|  style="text-align: center;vertical-align: top;"|1,0100
|  style="text-align: center;vertical-align: top;"|0,0056
|  style="text-align: center;vertical-align: top;"|179,500
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{m}_{y}^{2}</math>
|  style="text-align: center;vertical-align: top;"|0,9680
|  style="text-align: center;vertical-align: top;"|0,0086
|  style="text-align: center;vertical-align: top;"|113,100
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
|  style="text-align: center;vertical-align: top;"|0,9810
|  style="text-align: center;vertical-align: top;"|0,0085
|  style="text-align: center;vertical-align: top;"|115,500
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>n{m}_{y}</math>
|  style="text-align: center;vertical-align: top;"|0,5140
|  style="text-align: center;vertical-align: top;"|0,0312
|  style="text-align: center;vertical-align: top;"|16,500
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>n{m}_{z}</math>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0305
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|14,100
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|}


<div class="center" style="font-size: 75%;">'''Tabela 3.''' Prova de significância da superfície <math display="inline">{f}_{3}</math>
</div>

{| style="width: 62%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|24,000
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|6
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|4,000
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|53308,230
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|18
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|24
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Prova dos coeficientes individuais'''
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
|  style="text-align: center;vertical-align: top;"|1,0140
|  style="text-align: center;vertical-align: top;"|0,00552
|  style="text-align: center;vertical-align: top;"|183,500
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{m}_{y}^{2}</math>
|  style="text-align: center;vertical-align: top;"|0,9660
|  style="text-align: center;vertical-align: top;"|0,00806
|  style="text-align: center;vertical-align: top;"|119,800
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
|  style="text-align: center;vertical-align: top;"|0,9820
|  style="text-align: center;vertical-align: top;"|0,00795
|  style="text-align: center;vertical-align: top;"|123,500
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>n{m}_{y}</math>
|  style="text-align: center;vertical-align: top;"|0,5060
|  style="text-align: center;vertical-align: top;"|0,02951
|  style="text-align: center;vertical-align: top;"|17,100
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>n{m}_{z}</math>
|  style="text-align: center;vertical-align: top;"|0,4040
|  style="text-align: center;vertical-align: top;"|0,03146
|  style="text-align: center;vertical-align: top;"|12,800
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{y}{m}_{z}</math>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0380
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,01980
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,900
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|}


<div class="center" style="font-size: 75%;">'''Tabela 4'''. Prova de significância da superfície <math display="inline">{f}_{4}</math>
</div>

{| style="width: 66%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|12,000
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|2
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|6,000
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|84325,969
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|10
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
|  style="text-align: center;vertical-align: top;"|1,0120
|  style="text-align: center;vertical-align: top;"|0,0043
|  style="text-align: center;vertical-align: top;"|235, 300
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,0270
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0044
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|233, 800
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|}


<div class="center" style="font-size: 75%;">'''Tabela 5'''. Prova de significância da superfície <math display="inline">{f}_{5}</math>
</div>

{| style="width: 61%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|11,960
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|2
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5,979
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|1455,211
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
|  style="text-align: center;vertical-align: top;"|0,040
|  style="text-align: center;vertical-align: top;"|10
|  style="text-align: center;vertical-align: top;"|0,004
|  style="text-align: center;vertical-align: top;"|
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>{n}^{2}</math>
|  style="text-align: center;vertical-align: top;"|1,2420
|  style="text-align: center;vertical-align: top;"|0,0309
|  style="text-align: center;vertical-align: top;"|40,220
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|1,0870
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0355
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|30,610
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|}


<div class="center" style="font-size: 75%;">'''Tabela 6'''. Prova de significância da superfície <math display="inline">{f}_{6}</math>
</div>

{| style="width: 61%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Fonte de variação'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Soma dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Graus de liberdade'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Média dos quadrados'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|<math>F</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>P>F</math>
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Regressão'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|11,990
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|2
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|5,996
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|8547,240
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: right;vertical-align: top;"|'''Erro (resíduo)'''
|  style="text-align: center;vertical-align: top;"|0,010
|  style="text-align: center;vertical-align: top;"|10
|  style="text-align: center;vertical-align: top;"|0,001
|  style="text-align: center;vertical-align: top;"|
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|'''Total'''
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|12
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  colspan='6'  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|'''Prova dos coeficientes individuais'''
|-
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Variáveis'''
|  style="border-top: 1pt solid black;text-align: right;vertical-align: top;"|'''Estimado'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|'''Erro'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>t</math>
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>P>\left| t\right|</math> 
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: top;"|<math>n</math>
|  style="text-align: center;vertical-align: top;"|1,0890
|  style="text-align: center;vertical-align: top;"|0,0112
|  style="text-align: center;vertical-align: top;"|97,600
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|<math>{m}_{z}^{2}</math>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,9290
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,0150
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|62,040
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|}

===2.4 Análise elastoplástica de estruturas de pórticos===

Uma superfície de interação define o estado último de uma seção transversal e depende dos seguintes fatores:

:1. Forma geométrica da seção transversal;

:2. Combinação dos esforços seccionais que atuam na seção transversal;

:3. Teoria de viga utilizada.

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.

A análise elastoplástica segue os conceitos apresentados no trabalho de Silva [10] com as seguintes considerações;

:1) Os esforços seccionais contidos no interior da superfície de interação geram somente deformações elásticas;

:2) Os esforços seccionais que estejam na superfície de interação geram deformações plásticas;

:3) Os esforços seccionais fora da superfície de interação representam estados de tensões inadmissíveis porque não se leva em conta o caso do endurecimento.

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.

====2.4.1 Derivadas de primeira ordem====

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:
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| <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>
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(34)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(35)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(36)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(37)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(38)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(39)
|}

onde

<math display="inline">s{f}_{i}=\frac{{f}_{i}}{\left| {f}_{i}\right| }</math> = sinal do esforço seccional de forças;

<math display="inline">s{m}_{i}=\frac{{m}_{i}}{\left| {m}_{i}\right| }</math> = sinal do esforço seccional de momentos;

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.

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <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>  
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref529991859'></span>(40)
|}

onde <math display="inline">\mathit{\boldsymbol{0}}</math> é o vetor nulo de dimensão <math display="inline">(6\times 1)</math>.

====2.4.2 Derivadas de segunda ordem====

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:


''Para'' <math display="inline">\partial {F}_{x}{F}_{k}</math>


{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(41)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {F}_{y}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(42)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{x}\partial {F}_{z}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(43)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(44)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(45)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(46)
|}


''Para'' <math display="inline">\partial {F}_{y}{F}_{k}</math>

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {F}_{x}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(47)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(48)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {F}_{z}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(49)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {M}_{x}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(50)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {M}_{y}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(51)
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{y}\partial {M}_{z}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(52)
|}


''Para'' <math display="inline">\partial {F}_{z}{F}_{k}</math>

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {F}_{x}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(53)
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {F}_{y}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(54)
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(55)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {M}_{x}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(56)
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{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {M}_{y}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(57)
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| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>\frac{{\partial }^{2}f}{\partial {F}_{z}\partial {M}_{z}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(58)
|}


''Para'' <math display="inline">\partial {M}_{x}{F}_{k}</math>

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {F}_{x}}=\frac{s{f}_{x}{\left( \frac{{f}_{x}}{{f}_{xp}\, \, }\right) }^{{\beta }_{19}}{\left( \frac{{m}_{x}}{{m}_{xp}}\right) }^{\beta 20}\beta 20\, s{m}_{x}\beta 7\beta 19}{{f}_{x}{m}_{x}}\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(59)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {F}_{y}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(60)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{x}\partial {F}_{z}}=0\, \,</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(61)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(62)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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{| style="text-align: center; margin:auto;width: 100%;"
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| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(63)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(64)
|}


''Para'' <math display="inline">\partial {M}_{y}{F}_{k}</math>

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(65)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {F}_{y}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(66)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{y}\partial {F}_{z}}=0\, \,</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(67)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(68)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(69)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(70)
|}


''Para'' <math display="inline">\partial {M}_{z}{F}_{k}</math>

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {F}_{x}}=\, \frac{s{f}_{x}{\left( \frac{{f}_{x}}{{f}_{xp}}\right) }^{{\beta }_{23}}{\left( \frac{{m}_{z}}{{m}_{zp}}\right) }^{{\beta }_{24}}{\beta }_{24}s{m}_{z}{\beta }_{9}{\beta }_{23}}{{f}_{x}{m}_{z}}</math>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(71)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
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| 
{| style="text-align: center; margin:auto;width: 100%;"
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| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {F}_{y}}=0\quad</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(72)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>\frac{{\partial }^{2}f}{\partial {M}_{z}\partial {F}_{z}}=0\, \,</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(73)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(74)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(75)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(76)
|}


As 2ª derivadas na forma matricial podem ser expressas como

{| class="formulaSCP" style="width: 78%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(77)
|}

{| class="formulaSCP" style="width: 45%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(78)
|}

{| class="formulaSCP" style="width: 78%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(79)
|}

{| class="formulaSCP" style="width: 45%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(80)
|}

onde  <math display="inline">\mathit{\boldsymbol{0}}</math> é uma matriz de dimensão <math display="inline">(6\times 6)</math> com elementos nulos.

===2.5 Algoritmo de Retorno===

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.

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:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530073820'></span>(81)
|}

onde

<math display="inline">{\hat{\mathit{\boldsymbol{F}}}}_{i}</math>= vetor de forças nodais corrigido;

<math display="inline">{\mathit{\boldsymbol{F}}}_{i}^{trial}</math> = vetor de força nodais estimado;

<math display="inline">{\lambda }_{1}</math>= multiplicador plástico do nó 1, de forma <math display="inline">{\lambda }_{1}\geq 0</math>;

<math display="inline">{\mathit{\boldsymbol{K}}}_{ij}</math>= matriz de rigidez do elemento;

<math display="inline">{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}</math>= vetor de fluxo plástico do nó 1.

O vetor de forças nodais estimado é expressado por:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{\mathit{\boldsymbol{F}}}_{i}^{trial}={\overline{\mathit{\boldsymbol{F}}}}_{i}+{\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}</math>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530068060'></span>(82)
|}

onde

<math display="inline">{\overline{\mathit{\boldsymbol{F}}}}_{i}</math> = vetor de forças nodais do último passo de carga convergido;

<math display="inline">d{\mathit{\boldsymbol{U}}}_{j}</math> = incrementos do campo de deslocamentos do nó.

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.

O algoritmo irá trabalhar com 2 (duas) possibilidades de formação de rótulas plásticas, ou seja, para 1 (um) nó ou os 2 (dois) nós.

====2.5.1 Algoritmo de retorno com 1 (um) vetor====

<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]].

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[Image:Review_967755629029_6569_test-image7.png|450px]] 
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figura 6'''. Retorno à superfície com um vetor
|}


O processo iterativo utiliza vetores de fluxo plástico atualizados para aproximar-se da superfície. Este procedimento é chamado de algoritmo de retorno.

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>

O vetor de forças residuais <math display="inline">{\mathit{\boldsymbol{r}}}_{i}</math> do processo iterativo será como

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{\mathit{\boldsymbol{r}}}_{i}={\mathit{\boldsymbol{F}}}_{i}-{\hat{\mathit{\boldsymbol{F}}}}_{i}=</math><math>{\mathit{\boldsymbol{F}}}_{i}-\left({\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}\right)</math>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530063570'></span>(83)
|}


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:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530064230'></span>(84)
|}

onde

<math display="inline">n=0,1,2,\ldots</math>  = passo do processo iterativo.

<math display="inline">d{\mathit{\boldsymbol{F}}}_{i}</math> = variação do vetor de forças;

<math display="inline">d{\lambda }_{1}</math>= variação do multiplicador plástico;

<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).

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

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530064567'></span>(85)
|}

onde <math display="inline">{\delta }_{ik}</math> = Delta de Kronecker.

Definindo-se o termo

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530075709'></span>(86)
|}


A equação <span id='cite-_Ref530064567'></span>[[#_Ref530064567|(85)]], torna-se:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(87)
|}


Obtendo os termos da variação do vetor de força, chega-se a:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{Q}_{ik}d{\mathit{\boldsymbol{F}}}_{k}=-\left( {\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}\right)</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(88)
|}


{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>d{\mathit{\boldsymbol{F}}}_{k}=-{Q}_{ik}^{-1}\left( {\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}\right)</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530065198'></span>(89)
|}


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:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(90)
|}


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:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(91)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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}+ 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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(92)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(93)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(94)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(95)
|}

O processo iterativo termina quando são alcançados os critérios de parada adotados:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{r}^{norm}=\sqrt{\frac{\left\| {\mathit{\boldsymbol{r}}}_{i}\right\| }{\left\| {\mathit{\boldsymbol{F}}}_{i}^{trial}\right\| }}<Tol</math>

<math display="inline">{\mathit{\boldsymbol{f}}}^{norm}=\left| \mathit{\boldsymbol{f}}-\right. </math><math>\left. 1\right| <Tol</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(96)
|}

onde

<math display="inline">\left\| {\mathit{\boldsymbol{r}}}_{i}\right\|</math>  = norma euclidiana do vetor de forças nodais;

<math display="inline">\left\| {\mathit{\boldsymbol{F}}}_{i}^{trial}\right\|</math>  = norma euclidiana do vetor de forças estimado;

<math display="inline">{\mathit{\boldsymbol{f}}}^{norm}</math>= vetor resíduo da superfície de interação;

<math display="inline">Tol</math> = tolerância adotada.

====2.5.2 Algoritmo de retorno com 2 (dois) vetores====

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

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math display="inline">{\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}({F}_{j})>1;\, {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{2}}}({F}_{j})>1</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(97)
|}

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]].

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"|[[Image:Review_967755629029_9634_test-image8.png|450px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figura 7'''. Retorno à superfície com dois vetores
|}


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

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(98)
|}

onde <math display="inline">{\lambda }_{1}</math> e <math display="inline">{\lambda }_{2}</math> são os multiplicadores plásticos.

O vetor resíduo das forças tem a forma seguinte:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{\mathit{\boldsymbol{r}}}_{i}={\mathit{\boldsymbol{F}}}_{i}-{\hat{\mathit{\boldsymbol{F}}}}_{i}={\mathit{\boldsymbol{F}}}_{i}-\left(({\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}-{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{2}\right)</math>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(99)
|}

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:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(100)
|}

Com a condição que <math display="inline">{\mathit{\boldsymbol{r}}}_{i}^{n+1}=</math><math>\mathit{\boldsymbol{0}}</math>, chega-se a:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530068973'></span>(101)
|}

Adotando-se:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530069357'></span>(102)
|}

Isolando o termo <math display="inline">d{\mathit{\boldsymbol{F}}}_{k}</math> da equação <span id='cite-_Ref530068973'></span>[[#_Ref530068973|(101)]], obtém-se:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>d{\mathit{\boldsymbol{F}}}_{k}=-{Q}_{ik}^{-1}\left( {\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}+ d{\lambda }_{2}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{2}\right)</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(103)
|}

Os termos iterativos da função de escoamento (superfícies) são apresentados como

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>

<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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(104)
|}


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:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>

<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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(105)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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}+ d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+ 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>

<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}+ d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{\mathit{\boldsymbol{j}}}}\right\} }_{1}+ 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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530069811'></span>(106)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530070117'></span>(107)
|}

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530070122'></span>(108)
|}


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:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530072897'></span>(109)
|}


Reapresentando a equação <span id='cite-_Ref530072897'></span>[[#_Ref530072897|(109)]] na forma sintética:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530072993'></span>(110)
|}


A solução do sistema da equação <span id='cite-_Ref530072993'></span>[[#_Ref530072993|(110)]] é a seguinte:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(111)
|}

O processo iterativo segue procedimentos similares ao caso com um 1 (um) vetor:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{r}^{norm}=\sqrt{\frac{\left\| {\mathit{\boldsymbol{r}}}_{i}\right\| }{\left\| {\mathit{\boldsymbol{F}}}_{i}^{trial}\right\| }}<Tol</math>

<math>{\mathit{\boldsymbol{f}}}_{1}^{norm}=\left| {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{1}}}-\right. </math><math>\left. 1\right| <Tol</math>

<math display="inline">{\mathit{\boldsymbol{f}}}_{2}^{norm}=\left| {\mathit{\boldsymbol{f}}}_{\mathit{\boldsymbol{2}}}-\right. </math><math>\left. 1\right| <Tol</math> 
|}
|  style="text-align: right;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(112)
|}


<span id='_Ref4583347'></span>

===2.6 Matriz de rigidez consistente===

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.

====2.6.1 Algoritmo de retorno com um vetor ====

Usando a equação <span id='cite-_Ref530073820'></span>[[#_Ref530073820|(81)]] e <span id='cite-_Ref530068060'></span>[[#_Ref530068060|(82)]] como ponto de partida:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>

<math>{\mathit{\boldsymbol{F}}}_{i}^{trial}={\overline{\mathit{\boldsymbol{F}}}}_{i}+{\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}</math>

com <math display="inline">{\lambda }_{1}>0</math>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530073960'></span>(113)
|}


Aplicando-se o diferencial total na equação <span id='cite-_Ref530073960'></span>[[#_Ref530073960|(113)]], chega-se a:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(114)
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(115)
|}

Adotando <math display="inline">\, d{\overline{F}}_{i}=\mathit{\boldsymbol{0}}</math>

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(116)
|}

Isolando o termo <math display="inline">d{F}_{k}</math>

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(117)
|}

Usando <math display="inline">{Q}_{ik}</math> (equação <span id='cite-_Ref530075709'></span>[[#_Ref530075709|(86)]]):

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(118)
|}

Isolando o termo <math display="inline">d{F}_{k}</math>:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>d{F}_{k}={\mathit{\boldsymbol{Q}}}_{ik}^{-1}\left( {\mathit{\boldsymbol{K}}}_{ij}d{\mathit{\boldsymbol{U}}}_{j}- d{\lambda }_{1}{\mathit{\boldsymbol{K}}}_{ij}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}\right)</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530076047'></span>(119)
|}

Adotando o termo:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{\mathit{\boldsymbol{R}}}_{ij}={\mathit{\boldsymbol{Q}}}_{ik}^{-1}{\mathit{\boldsymbol{K}}}_{ij}</math>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530077061'></span>(120)
|}

A equação <span id='cite-_Ref530076047'></span>[[#_Ref530076047|(119)]] torna-se em:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>d{\mathit{\boldsymbol{F}}}_{i}={\mathit{\boldsymbol{R}}}_{ij}\left( d{\mathit{\boldsymbol{U}}}_{j}- d{\lambda }_{1}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{j}}\right\} }_{1}\right)</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530076227'></span>(121)
|}

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:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(122)
|}


{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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}- d{\lambda }_{1}{\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {F}_{j}}\right\} }_{1}\right)</math> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(123)
|}

Isolando o termo <math display="inline">d{\lambda }_{1}</math>:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530077047'></span>(124)
|}

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)]]:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(125)
|}

com

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(126)
|}

====2.6.2 Algoritmo com dois vetores de retorno====

Os procedimentos similares são realizados para obter a matriz de rigidez consistente para dois vetores como por exemplo Silva [10] e Vieira [6].

A formulação para os dois vetores tem a seguinte forma:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530079964'></span>(127)
|}


As condições do vetor de forças nodais final são que

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
|<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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(128)
|}

Os termos dos multiplicadores plásticos são os seguintes:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|<span id='_Ref530079552'></span>(129)
|}

com

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{c}_{1}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{1}{R}_{ij}d{U}_{j}</math>

<math>{c}_{2}={\left\{ \frac{\partial \mathit{\boldsymbol{f}}}{\partial {\mathit{\boldsymbol{F}}}_{i}}\right\} }_{2}{R}_{ij}d{U}_{j}</math>

<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>

<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>

<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>

<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>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(130)
|}


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.

Desenvolvendo os termos das equações <span id='cite-_Ref530079964'></span>[[#_Ref530079964|(127)]] e  <span id='cite-_Ref530079552'></span>[[#_Ref530079552|(129)]]:

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <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> 
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(131)
|}

com

{| class="formulaSCP" style="width: 100%;margin: 1em auto 0.1em auto;border-collapse: collapse;width: 100%;text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;width: 100%;"
|-
| <math>{d}_{1}=\frac{{b}_{22}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>

<math>{d}_{2}=\frac{{b}_{12}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>

<math>{d}_{3}=\frac{{b}_{11}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>

<math>{d}_{4}=\frac{{b}_{21}}{{b}_{11}{b}_{22}-{b}_{12}{b}_{21}}</math>
|}
|  style="text-align: center;vertical-align: center;width: 5px;text-align: right;white-space: nowrap;"|(132)
|}


<span id='_Ref477377056'></span>

===2.7 Caracterização dos casos===

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.

O '''caso 1''' é baseado nos dados do trabalho de Thai e Kim [11] que trata de um pórtico plano, conforme <span id='cite-_Ref530127321'></span>[[#_Ref530127321|Figura 8]] e Tabelas de 7 a 10 que apresentam as características do caso como coordenadas, características físicas, propriedades do material e cargas aplicadas, respectivamente.

'''''Propriedades do Pórtico Plano (PP)'''''

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[Image:Review_967755629029_6129_test-image9.png|250px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figura 8'''. Pórtico Plano de Thai e Kim
|}

<span id='_Ref457417233'></span>


<div class="center" style="font-size: 75%;">'''Tabela 7'''. Coordenadas do pórtico plano - Thai e Kim
</div>

{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''Nó'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''X'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''Y'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''Z'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
|-
|  style="text-align: center;vertical-align: center;border-bottom: 1pt solid black;"|
|  style="text-align: center;border-bottom: 1pt solid black;"|('''cm''')
|  style="text-align: center;border-bottom: 1pt solid black;"|('''cm''')
|  style="text-align: center;border-bottom: 1pt solid black;"|('''cm''')
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">1</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">0,000</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">0,000</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">0,000</span>
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">2</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">0,000</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">0,000</span>
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">3</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">0,000</span>
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">4</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">1000,000</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">0,000</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 100%;">0,000</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|}



<div class="center" style="font-size: 75%;">'''Tabela 8'''. Características físicas do pórtico plano - Thai e Kim
</div>
<span id='_Ref457417961'></span><span id='_Ref494057719'></span>
<span id='_Ref500759207'></span>
{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;text-align: center;"|''''Elem.'''
|  style="border-top: 1pt solid black;text-align: center;"|'''No I'''
|  style="border-top: 1pt solid black;text-align: center;"|'''No F'''
|  style="border-top: 1pt solid black;text-align: center;"|'''E'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>\boldsymbol{\sigma }_{{y}}</math>
|  colspan='2'  style="border-top: 1pt solid black;text-align: center;"|<math>\boldsymbol{\nu }</math>
|-
|  style="text-align: center;border-bottom: 1pt solid black;"|
|  style="text-align: center;border-bottom: 1pt solid black;"|
|  style="text-align: center;border-bottom: 1pt solid black;"|
|  style="border-bottom: 1pt solid black;text-align: center;"|('''kN/cm'''<math>^{\bf 2}</math>)
|  colspan='1'  style="border-bottom: 1pt solid black;text-align: center;"|('''kN/cm'''<math>^{\bf 2}</math>)
|  colspan='1'  style="border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: top;"|
|  colspan='1'  style="text-align: center;vertical-align: center;"|
|-
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|1961,3
|  style="text-align: center;vertical-align: top;"|9,8
|  colspan='1'  style="text-align: center;vertical-align: center;"|0,170
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|3
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|3
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|4
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|  colspan='1'  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|
|}


<span id='_Ref457418210'></span>

Os termos da <span id='cite-_Ref500759207'></span>[[#_Ref500759207|Tabela 8]] tem a seguinte descrição:

<math display="inline">E</math> = módulo de elasticidade;

<math display="inline">\nu</math>  = coeficiente de Poisson;

<math display="inline">{\sigma }_{y}</math>= tensão de escoamento.

<span id='_Ref494057745'></span>

<div class="center" style="font-size: 75%;">'''Tabela 9'''. Propriedades do material do pórtico plano - Thai e Kim
</div>

{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|'''Elemento'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|'''Seção transversal'''
|-
|  rowspan='5' style="border-bottom: 1pt solid black;text-align: center;"|1,2,3 e 4
|  style="text-align: center;"|<math>A=800,000\, c{m}^{2}</math>
|-
|  style="text-align: center;"|<math display="inline">b=20\, cm</math> e <math display="inline">h=</math><math>40\, cm</math>
|-
|  style="text-align: center;"|<math>{I}_{z}=106666,667\, c{m}^{4}</math>
|-
|  style="text-align: center;"|<math>{F}_{xp}=7840,000\, kN</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;"|<math>{M}_{zp}=78400,000\, kN\times cm</math>
|}


As cargas aplicadas são as seguintes:

<div class="center" style="font-size: 75%;">'''Tabela 10'''. Cargas aplicadas do pórtico plano - Thai e Kim
</div>

{| style="width: 35%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Nó'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Dir'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Valor (kN)'''
|-
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|<math>{F}_{X}</math>
|  style="text-align: center;vertical-align: center;"|1,000
|-
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|<math>{F}_{y}</math>
|  style="text-align: center;vertical-align: center;"|-1,000
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|3
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>{F}_{y}</math>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|-1,000
|}


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.

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|Figura 9]].

<span id='_Ref457421631'></span>

{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[Image:Review_967755629029_7705_test-image10.png|300px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figura 9''' – Pórtico espacial de 2 (dois) pavimentos
|}


As propriedades do caso são apresentadas nas <span id='cite-_Ref457422104'></span>[[#_Ref457422104|Tabela 11]] a <span id='cite-_Ref532329759'></span>[[#_Ref532329759|Tabela 13]].

<span id='_Ref457422104'></span>
<div class="center" style="font-size: 75%;">'''Tabela 11'''. Coordenadas do pórtico espacial - Thai e Kim
</div>

{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''Nó'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''X'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''Y'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''Z'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|
|-
|  style="text-align: center;vertical-align: center;border-bottom: 1pt solid black;"|
|  style="text-align: center;border-bottom: 1pt solid black;"|('''cm''')
|  style="text-align: center;border-bottom: 1pt solid black;"|('''cm''')
|  style="text-align: center;border-bottom: 1pt solid black;"|('''cm''')
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|300,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|4
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|300,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|5
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|6
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|7
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|300,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|8
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|300,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|9
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|800,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|10
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|800,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|11
|  style="text-align: center;vertical-align: center;"|400,000
|  style="text-align: center;vertical-align: center;"|800,000
|  style="text-align: center;vertical-align: center;"|300,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|12
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|800,000
|  style="text-align: center;vertical-align: center;"|300,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|13
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|800,000
|  style="text-align: center;vertical-align: center;"|150,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: center;"|14
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|400,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|800,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|150,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|
|}


<span id='_Ref500757194'></span>
<div class="center" style="font-size: 75%;">'''Tabela 12'''. Características físicas do pórtico espacial - Thai e Kim
</div>

{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;text-align: center;"|''''Elem.'''
|  style="border-top: 1pt solid black;text-align: center;"|'''No I'''
|  style="border-top: 1pt solid black;text-align: center;"|'''No F'''
|  style="border-top: 1pt solid black;text-align: center;"|'''E'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>\boldsymbol{\sigma }_{{y}}</math>
|  colspan='1'  style="border-top: 1pt solid black;text-align: center;"|<math>\boldsymbol{\nu }</math>
|-
|  style="text-align: center;border-bottom: 1pt solid black;"|
|  style="text-align: center;border-bottom: 1pt solid black;"|
|  style="text-align: center;border-bottom: 1pt solid black;"|
|  style="border-bottom: 1pt solid black;text-align: center;"|('''kN/cm'''<math>^{\bf 2}</math>)
|  colspan='1'  style="border-bottom: 1pt solid black;text-align: center;"|('''kN/cm'''<math>^{\bf 2}</math>)
|  colspan='1'  style="border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|5
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|1961,3
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|9,8
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|0,170
|-
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|6
|-
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|7
|-
|  style="text-align: center;vertical-align: center;"|4
|  style="text-align: center;vertical-align: center;"|4
|  style="text-align: center;vertical-align: center;"|8
|-
|  style="text-align: center;vertical-align: center;"|5
|  style="text-align: center;vertical-align: center;"|5
|  style="text-align: center;vertical-align: center;"|6
|-
|  style="text-align: center;vertical-align: center;"|6
|  style="text-align: center;vertical-align: center;"|6
|  style="text-align: center;vertical-align: center;"|7
|-
|  style="text-align: center;vertical-align: center;"|7
|  style="text-align: center;vertical-align: center;"|7
|  style="text-align: center;vertical-align: center;"|8
|-
|  style="text-align: center;vertical-align: center;"|8
|  style="text-align: center;vertical-align: center;"|8
|  style="text-align: center;vertical-align: center;"|5
|-
|  style="text-align: center;vertical-align: center;"|9
|  style="text-align: center;vertical-align: center;"|5
|  style="text-align: center;vertical-align: center;"|9
|-
|  style="text-align: center;vertical-align: center;"|10
|  style="text-align: center;vertical-align: center;"|6
|  style="text-align: center;vertical-align: center;"|10
|-
|  style="text-align: center;vertical-align: center;"|11
|  style="text-align: center;vertical-align: center;"|7
|  style="text-align: center;vertical-align: center;"|11
|-
|  style="text-align: center;vertical-align: center;"|12
|  style="text-align: center;vertical-align: center;"|8
|  style="text-align: center;vertical-align: center;"|12
|-
|  style="text-align: center;vertical-align: center;"|13
|  style="text-align: center;vertical-align: center;"|9
|  style="text-align: center;vertical-align: center;"|10
|-
|  style="text-align: center;vertical-align: center;"|14
|  style="text-align: center;vertical-align: center;"|10
|  style="text-align: center;vertical-align: center;"|14
|-
|  style="text-align: center;vertical-align: center;"|15
|  style="text-align: center;vertical-align: center;"|14
|  style="text-align: center;vertical-align: center;"|11
|-
|  style="text-align: center;vertical-align: center;"|16
|  style="text-align: center;vertical-align: center;"|11
|  style="text-align: center;vertical-align: center;"|12
|-
|  style="text-align: center;vertical-align: center;"|17
|  style="text-align: center;vertical-align: center;"|9
|  style="text-align: center;vertical-align: center;"|13
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|18
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|12
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|13
|}


<div class="center" style="font-size: 75%;"><span id='_Ref532329759'></span>'''Tabela 13'''. Propriedades do material do pórtico espacial - Thai e Kim
</div>

{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|'''Elemento'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|'''Seção transversal'''
|-
|  rowspan='11' style="text-align: center;border-bottom: 1pt solid black;"|1 a 18
|  style="text-align: center;"|<math>A=800,000\, c{m}^{2}</math>
|-
|  style="text-align: center;"|<math display="inline">b=20\, cm</math> e  <math display="inline">h=</math><math>40\, cm</math>
|-
|  style="text-align: center;"|<math>{I}_{z}=1,067\times {10}^{5}\, c{m}^{4}</math>
|-
|  style="text-align: center;"|<math>{I}_{x}=1,067\times {10}^{5}\, c{m}^{4}</math>
|-
|  style="text-align: center;"|<math>{I}_{y}=2,667\times {10}^{4}\, c{m}^{4}</math>
|-
|  style="text-align: center;"|<math>{M}_{zp}=7,840\times {10}^{4}\, kN\times cm</math>
|-
|  style="text-align: center;"|<math>{M}_{xp}=6,533\times {10}^{4}\, kN\times cm</math>
|-
|  style="text-align: center;"|<math>{M}_{yp}=3,920\times {10}^{4}\, kN\times cm</math>
|-
|  style="text-align: center;"|<math>{F}_{zp}=4,526\times {10}^{3}\, kN</math>
|-
|  style="text-align: center;"|<math>{F}_{xp}=7,840\times {10}^{3}\, kN</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;"|<math>{F}_{yp}=4,526\times {10}^{3}\, kN</math>
|}


As cargas aplicadas são as seguintes:

<div class="center" style="font-size: 75%;">'''Tabela 14'''. Cargas aplicadas do pórtico espacial - Thai e Kim
</div>

{| style="width: 50%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Nó'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{{\boldsymbol{F}}}_{{\boldsymbol{x}}}</math> '''(kN)'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math display="inline">{{\boldsymbol{F}}}_{{\boldsymbol{y}}}</math> '''(kN)'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{{\boldsymbol{F}}}_{{\boldsymbol{z}}}</math> '''(kN)'''
|-
|  style="text-align: center;vertical-align: center;"|5
|  style="text-align: center;vertical-align: center;"|1,000
|  style="text-align: center;vertical-align: top;"|-2,000
|  style="text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: center;vertical-align: center;"|6
|  style="text-align: center;vertical-align: center;"|0,250
|  style="text-align: center;vertical-align: top;"|-2,000
|  style="text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: center;vertical-align: center;"|7
|  style="text-align: center;vertical-align: center;"|0,250
|  style="text-align: center;vertical-align: top;"|-2,000
|  style="text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: center;vertical-align: center;"|8
|  style="text-align: center;vertical-align: center;"|1,000
|  style="text-align: center;vertical-align: top;"|-2,000
|  style="text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: center;vertical-align: center;"|9
|  style="text-align: center;vertical-align: center;"|3,000
|  style="text-align: center;vertical-align: top;"|-0,500
|  style="text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: center;vertical-align: center;"|10
|  style="text-align: center;vertical-align: center;"|0,500
|  style="text-align: center;vertical-align: top;"|-0,500
|  style="text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: center;vertical-align: center;"|11
|  style="text-align: center;vertical-align: center;"|0,500
|  style="text-align: center;vertical-align: top;"|-0,500
|  style="text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: center;vertical-align: center;"|12
|  style="text-align: center;vertical-align: center;"|3,000
|  style="text-align: center;vertical-align: top;"|-0,500
|  style="text-align: center;vertical-align: top;"|0,000
|-
|  style="text-align: center;vertical-align: center;"|13
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: top;"|-1,000
|  style="text-align: center;vertical-align: top;"|0,000
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|14
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|0,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-1,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|}


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.

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]].

<span id='_Ref533152504'></span>
{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
|-
|style="padding:10px;"| [[Image:Review_967755629029_8143_test-image11.png|350px]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" style="padding:10px;"| '''Figura 10'''. Pórtico em domo - Argyris et al.
|}


As propriedades do caso são apresentadas nas <span id='cite-_Ref534577723'></span>[[#_Ref534577723|Tabela 15]] a <span id='cite-_Ref534577726'></span>[[#_Ref534577726|Tabela 18]].


<div class="center" style="font-size: 75%;"><span id='_Ref534577723'></span>'''Tabela 15'''. Coordenadas do pórtico em domo - Argyris et al.
</div>

{| style="width: 45%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''Nó'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''X'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''Y'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|'''Z'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: center;"|
|-
|  style="text-align: center;vertical-align: center;border-bottom: 1pt solid black;"|
|  style="text-align: center;border-bottom: 1pt solid black;"|('''cm''')
|  style="text-align: center;border-bottom: 1pt solid black;"|('''cm''')
|  style="text-align: center;border-bottom: 1pt solid black;"|('''cm''')
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: center;"|610,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|628,500
|  style="text-align: center;vertical-align: center;"|455,000
|  style="text-align: center;vertical-align: center;"|1088,500
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|1257,000
|  style="text-align: center;vertical-align: center;"|455,000
|  style="text-align: center;vertical-align: center;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|4
|  style="text-align: center;vertical-align: top;"|628,500
|  style="text-align: center;vertical-align: top;"|455,000
|  style="text-align: center;vertical-align: top;"|-1088,500
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|5
|  style="text-align: center;vertical-align: top;"|-628,500
|  style="text-align: center;vertical-align: top;"|455,000
|  style="text-align: center;vertical-align: top;"|-1088,500
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|6
|  style="text-align: center;vertical-align: top;"|-1257,000
|  style="text-align: center;vertical-align: top;"|455,000
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|7
|  style="text-align: center;vertical-align: top;"|-628,500
|  style="text-align: center;vertical-align: top;"|455,000
|  style="text-align: center;vertical-align: top;"|1088,500
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|8
|  style="text-align: center;vertical-align: top;"|1219,000
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|2111,500
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|9
|  style="text-align: center;vertical-align: top;"|2438,000
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|10
|  style="text-align: center;vertical-align: top;"|1219,000
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|-2111,500
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|11
|  style="text-align: center;vertical-align: top;"|-1219,000
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|-2111,500
|  style="text-align: center;vertical-align: top;"|
|-
|  style="text-align: right;vertical-align: center;"|12
|  style="text-align: center;vertical-align: top;"|-2438,000
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|0,000
|  style="text-align: center;vertical-align: top;"|
|-
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: center;"|13
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-1219,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|2111,500
|  style="text-align: center;vertical-align: top;"|
|}


<div class="center" style="font-size: 75%;">'''Tabela 16'''. Características físicas do pórtico em domo - Argyris et al.
</div>

{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;text-align: center;"|''''Elem.'''
|  style="border-top: 1pt solid black;text-align: center;"|'''No I'''
|  style="border-top: 1pt solid black;text-align: center;"|'''No F'''
|  style="border-top: 1pt solid black;text-align: center;"|'''E'''
|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|<math>\boldsymbol{\sigma }_{{y}}</math>
|  colspan='1'  style="border-top: 1pt solid black;text-align: center;"|<math>\boldsymbol{\nu }</math>
|-
|  style="text-align: center;border-bottom: 1pt solid black;"|
|  style="text-align: center;border-bottom: 1pt solid black;"|
|  style="text-align: center;border-bottom: 1pt solid black;"|
|  style="border-bottom: 1pt solid black;text-align: center;"|('''kN/cm'''<math>^{\bf 2}</math>)
|  colspan='1'  style="border-bottom: 1pt solid black;text-align: center;"|('''kN/cm'''<math>^{\bf 2}</math>)
|  colspan='1'  style="border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|2
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|2068,0
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|8,0
|  rowspan='18' style="text-align: center;border-bottom: 1pt solid black;"|0,1716
|-
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|3
|-
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|4
|-
|  style="text-align: center;vertical-align: center;"|4
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|5
|-
|  style="text-align: center;vertical-align: center;"|5
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|6
|-
|  style="text-align: center;vertical-align: center;"|6
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|7
|-
|  style="text-align: center;vertical-align: center;"|7
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|3
|-
|  style="text-align: center;vertical-align: center;"|8
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|4
|-
|  style="text-align: center;vertical-align: center;"|9
|  style="text-align: center;vertical-align: center;"|5
|  style="text-align: center;vertical-align: center;"|4
|-
|  style="text-align: center;vertical-align: center;"|10
|  style="text-align: center;vertical-align: center;"|5
|  style="text-align: center;vertical-align: center;"|6
|-
|  style="text-align: center;vertical-align: center;"|11
|  style="text-align: center;vertical-align: center;"|6
|  style="text-align: center;vertical-align: center;"|7
|-
|  style="text-align: center;vertical-align: center;"|12
|  style="text-align: center;vertical-align: center;"|7
|  style="text-align: center;vertical-align: center;"|2
|-
|  style="text-align: center;vertical-align: center;"|13
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|8
|-
|  style="text-align: center;vertical-align: center;"|14
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|9
|-
|  style="text-align: center;vertical-align: center;"|15
|  style="text-align: center;vertical-align: center;"|4
|  style="text-align: center;vertical-align: center;"|10
|-
|  style="text-align: center;vertical-align: center;"|16
|  style="text-align: center;vertical-align: center;"|5
|  style="text-align: center;vertical-align: center;"|11
|-
|  style="text-align: center;vertical-align: center;"|17
|  style="text-align: center;vertical-align: center;"|6
|  style="text-align: center;vertical-align: center;"|12
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|18
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|7
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|13
|}


<div class="center" style="font-size: 75%;">'''Tabela 17'''. Propriedades do material do pórtico em domo - Argyris et al.
</div>

{| style="width: 40%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|'''Elemento'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|'''Seção transversal'''
|-
|  rowspan='11' style="text-align: center;border-bottom: 1pt solid black;"|1 a 18
|  style="text-align: center;"|<math>A=9272,000\, c{m}^{2}</math>
|-
|  style="text-align: center;"|<math display="inline">b=76\, cm</math> e <math display="inline">h=</math><math>122\, cm</math>
|-
|  style="text-align: center;"|<math>{I}_{z}=1,150\times {10}^{7}\, c{m}^{4}</math>
|-
|  style="text-align: center;"|<math>{I}_{x}=1,596\times {10}^{7}\, c{m}^{4}</math>
|-
|  style="text-align: center;"|<math>{I}_{y}=4,463\times {10}^{6}\, c{m}^{4}</math>
|-
|  style="text-align: center;"|<math>{M}_{zp}=2,262\times {10}^{6}\, kN\times cm</math>
|-
|  style="text-align: center;"|<math>{M}_{xp}=6,533\times {10}^{9}\, kN\times cm</math>
|-
|  style="text-align: center;"|<math>{M}_{yp}=1,409\times {10}^{6}\, kN\times cm</math>
|-
|  style="text-align: center;"|<math>{F}_{zp}=4,282\times {10}^{4}\, kN</math>
|-
|  style="text-align: center;"|<math>{F}_{xp}=7,416\times {10}^{4}\, kN</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;"|<math>{F}_{yp}=4,282\times {10}^{4}\, kN</math>
|}


As cargas aplicadas são as seguintes:

<span id='_Ref534577726'></span>
<div class="center" style="font-size: 75%;">'''Tabela 18'''. Cargas aplicadas do pórtico em domo - Argyris et al.
</div>

{| style="width: 50%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Nó'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{{\boldsymbol{F}}}_{{\boldsymbol{x}}}</math> '''(kN)'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math display="inline">{{\boldsymbol{F}}}_{{\boldsymbol{y}}}</math> '''(kN)'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{{\boldsymbol{F}}}_{{\boldsymbol{z}}}</math> '''(kN)'''
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|1
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|0,000
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|-1,000
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|0,000
|}


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.

==3. Resultados e discussão==

Os resultados e discussões dos estudos de caso 1 a 3 são apresentados.

===3.1 Caso 1===

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]].

<div class="center" style="font-size: 75%;">'''Tabela 19''' – Rótulas plásticas – pórtico plano - Thai e Kim.
</div>

{| style="width: 60%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span id='_Hlk4522464'></span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|Função <math display="inline">{f}_{4}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Elemento'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Nó'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
|-
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|<math>0,108246\times {10}^{5}</math>
|  rowspan='5' style="text-align: center;border-bottom: 1pt solid black;"|<math>309,146</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|<math>0,841207\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|<math>0,227389\times {10}^{2}</math>
|-
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|<math>0,844737\times {10}^{4}</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|4
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>0,108410\times {10}^{5}</math>
|}


{| style="width: 60%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|Função <math display="inline">{f}_{5}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Elemento'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Nó'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
|-
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|<math>0,106273\times {10}^{5}</math>
|  rowspan='5' style="border-bottom: 1pt solid black;text-align: center;"|<math>300,431</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|<math>0,208395\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|<math>0,626480\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|<math>0,836829\times {10}^{4}</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|4
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>0,106452\times {10}^{5}</math>
|-
|-
|}


{| style="width: 60%;margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{6}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Elemento'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|'''Nó'''
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
|-
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|1
|  style="text-align: center;vertical-align: center;"|<math>0,114845\times {10}^{5}</math>
|  rowspan='5' style="text-align: center;border-bottom: 1pt solid black;"|<math>318,103</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|<math>0,254762\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|2
|  style="text-align: center;vertical-align: center;"|<math>0,604007\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|3
|  style="text-align: center;vertical-align: center;"|<math>0,899557\times {10}^{4}</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|4
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>0,117705\times {10}^{5}</math>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Review_967755629029_7962_test-image12.png|480px]] </div>

<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.

{| style="width: 100%;border-collapse: collapse;" 
|-
|  style="text-align: center;vertical-align: top;"|
|}

===3.2 Caso 2===

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''']].

<span id='_Ref4571218'></span><span id='_Ref500765287'></span>'''Tabela 20''' - Rótulas plásticas – pórtico espacial - Thai e Kim

{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{1}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Nó'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,345728\times {10}^{4}</math>
|  rowspan='12' style="text-align: center;border-bottom: 1pt solid black;"|<math>141,886</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<math>0,353734\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<math>0,348667\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<math>0,350863\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<math>0,308873\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="text-align: center;vertical-align: center;"|<math>0,307316\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
|  style="text-align: center;vertical-align: center;"|<math>0,306402\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">8</span>
|  style="text-align: center;vertical-align: center;"|<math>0,309575\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">9</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">9</span>
|  style="text-align: center;vertical-align: center;"|<math>0,899973\times {10}^{3}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
|  style="text-align: center;vertical-align: center;"|<math>0,933775\times {10}^{3}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">11</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">11</span>
|  style="text-align: center;vertical-align: center;"|<math>0,984856\times {10}^{3}</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>0,849066\times {10}^{3}</math>
|}


{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{2}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Nó'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,321786\times {10}^{4}</math>
|  rowspan='12' style="border-bottom: 1pt solid black;text-align: center;"|<math>134,077</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<math>0,326692\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<math>0,326692\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<math>0,321786\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<math>0,299500\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="text-align: center;vertical-align: center;"|<math>0,298730\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
|  style="text-align: center;vertical-align: center;"|<math>0,298730\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">8</span>
|  style="text-align: center;vertical-align: center;"|<math>0,299500\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">9</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">9</span>
|  style="text-align: center;vertical-align: center;"|<math>0,106747\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
|  style="text-align: center;vertical-align: center;"|<math>0,107793\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">11</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">11</span>
|  style="text-align: center;vertical-align: center;"|<math>0,107793\times {10}^{4}</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>0,106747\times {10}^{4}</math>
|}


{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{3}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Nó'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (kN)}}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,345110\times {10}^{4}</math>
|  rowspan='5' style="text-align: center;"|<math>141,900</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<math>0,353048\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<math>0,348312\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<math>0,349939\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<math>0,309236\times {10}^{4}</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="text-align: center;vertical-align: center;"|<math>0,307839\times {10}^{4}</math>
|  style="text-align: right;vertical-align: center;"|
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
|  style="text-align: center;vertical-align: center;"|<math>0,306698\times {10}^{4}</math>
|  style="text-align: right;vertical-align: center;"|
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">8</span>
|  style="text-align: center;vertical-align: center;"|<math>0,309736\times {10}^{4}</math>
|  style="text-align: right;vertical-align: center;"|
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">9</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">9</span>
|  style="text-align: center;vertical-align: center;"|<math>0,905438\times {10}^{3}</math>
|  style="text-align: right;vertical-align: center;"|
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">10</span>
|  style="text-align: center;vertical-align: center;"|<math>0,938623\times {10}^{3}</math>
|  style="text-align: right;vertical-align: center;"|
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">11</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">11</span>
|  style="text-align: center;vertical-align: center;"|<math>0,983798\times {10}^{3}</math>
|  style="text-align: right;vertical-align: center;"|
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">12</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>0,855577\times {10}^{3}</math>
|  style="border-bottom: 1pt solid black;text-align: right;vertical-align: center;"|
|}


<div id="_Ref457419942" class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Review_967755629029_9261_test-image13.png|480px]] </div>

<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.

===3.3 Caso 3===

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''']].

<span id='_Ref4573319'></span>'''Tabela 21''' - Rótulas plásticas – pórtico espacial - pórtico espacial - Argyris et al.

{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{1}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Nó'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (MN)}}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,549351</math>
|  rowspan='12' style="border-bottom: 1pt solid black;text-align: center;"|<math>51,0815</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<math>0,891167</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,552499</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<math>0,861987</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,564206</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<math>0,883022</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,569621</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<math>0,885590</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,556237</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="text-align: center;vertical-align: center;"|<math>0,862806</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,541887</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>0,934919</math>
|-
|-
|}


{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{2}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Nó'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (MN)}}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>2,63534</math>
|  rowspan='12' style="border-bottom: 1pt solid black;text-align: center;"|<math>54,4497</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<math>2,62747</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>2,64188</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<math>2,63063</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>2,64561</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<math>2,63270</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>2,64226</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<math>2,63100</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>2,63496</math>
|-
|  style="text-align: center;vertical-align: center;"|
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="text-align: center;vertical-align: center;"|<math>2,62711</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>2,63079</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">7</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>2,62451</math>
|-
|}


{| style="width: 70%;margin: 1em auto 0.1em auto;border-collapse: collapse;" 
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|função <math display="inline">{f}_{3}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|
|-
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Elemento'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<span style="text-align: center; font-size: 75%;">'''Nó'''</span>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{\lambda }}</math>
|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>\mathit{\boldsymbol{Carga\, limite\, (MN)}}</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,227607</math>
|  rowspan='6' style="border-bottom: 1pt solid black;text-align: center;"|<math>48,0914</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">2</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,227118</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">3</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,226262</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">4</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,224668</math>
|-
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">5</span>
|  style="text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="text-align: center;vertical-align: center;"|<math>0,224136</math>
|-
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">6</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<span style="text-align: center; font-size: 75%;">1</span>
|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: center;"|<math>0,225236</math>
|}


<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
 [[Image:Review_967755629029_6932_test-image14.png|480px]] </div>

<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.

==4. Conclusões==

'''* '''As superfícies de escoamento em resultantes de tensões aplicadas para pórticos planos e espaciais obtiveram resultados satisfatórios tendo em vista os exemplos apresentados;

'''* '''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;

'''* '''O método apresentado permitiu o uso do modelo de dano de viga de Timoshenko 3D com bons resultados para estruturas de aço;

'''* '''A regressão linear múltipla se apresenta como uma solução viável para obter funções por análises numéricas e/ou experimentais;

'''* '''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.

==Agradecimentos==

À UFOB, CIMNE/UPC, PECC/UnB e a CAPES.

==Referências==

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