MESTRADO EM CONSTRUÇÃO METÁLICA

DEPARTAMENTO DE ENGENHARIA CIVIL

ESCOLA DE

UNIVERSIDADE FEDERAL DE OURO PRETO

ALEXANDRE DA SILVA GALVÂO

DISSERTAÇÃO DE MESTRADO

Orientadores: Ricardo Azoubel da Mota Silveira

Convênio USIMINAS/UFOP/FUNDAÇÃO Ouro Preto, março de 2000

Formulações Não-Lineares de Elementos Finitos para Análise

UNIVERSIDADE FEDERAL DE OURO PRETO - ESCOLA DE MINASDEPARTAMENTO DE ENGENHARIA CIVIL

PROGRAMA DE PÓS – GRADUAÇÃO EM ENGENHARIA CIVIL

Formulações Não-Lineares de Elementos Finitos para Análisede Sistemas Estruturais Metálicos Reticulados Planos

AUTOR: ALEXANDRE DA SILVA GALVÃO

ORIENTADOR: Prof. Dr. Ricardo Azoubel da Mota Silveira

Dissertação apresentada ao Programa de Pós-Graduação do Departamento de EngenhariaCivil da Escola de Minas da UniversidadeFederal de Ouro Preto, como parte integrantedos requisitos para obtenção do título deMestre em Engenharia Civil, área deconcentração: Construções Metálicas.

Ouro Preto, março de 2000.

III

“...A ciência é a potência do homem, e, o amor, a sua força;o homem só se torna homem pela inteligência, mas só é homempelo coração.

Saber, amar e poder; eis a vida completa.”

Henri-Frédéric Amiel (1821-1891)

A minha família.

IV

MEUS AGRADECIMENTOS

“Dê-me uma alavanca e um ponto de apoio,e eu faço mover o mundo.”

Arquimedes (287AC-217AC)

• Aos meus pais e ao meu irmão, por serem o meu mundo.

• Ao meu professor e orientador Ricardo Azoubel da Mota Silveira, por ter

sido a alavanca e o apoio dessa nossa empreitada.

• À Escola de Minas, por ser parte da minha vida.

• Aos meus colegas, pelo prazer da convivência nesses dois anos.

• À CAPES e à USIMINAS, pela ajuda financeira.

V

RESUMO

Este trabalho tem como principais objetivos o estudo e a implementação computacional

de formulações geometricamente não-lineares para elementos finitos reticulados planos

encontradas na literatura recente. Essas formulações, além de permitir a determinação da matriz

de rigidez e do vetor de forças internas de forma direta, podem ser acopladas com relativa

facilidade a várias estratégias de solução não-linear.

Procurando fornecer diferentes opções de modelagem de problemas de instabilidade

usando esses elementos finitos reticulados planos, foram implementadas as seguintes

formulações geometricamente não-lineares: (i) formulações definidas por Alves (1993b) e

Torkamani et al. (1997), implementadas aqui com procedimentos distintos de se avaliar o vetor

de forças internas: forma total e forma incremental; (ii) formulações propostas por Yang e Kuo

(1994), que se basearam em modelos linearizado, linearizado-simplificado e com termos de

ordem elevada; foram ainda introduzidas por esses autores duas abordagens diferentes,

implementadas neste trabalho, de obtenção do vetor de forças internas: deslocamentos naturais

incrementais e rigidez externa; e (iii) formulações em referencial Lagrangiano total, propostas

por Pacoste e Eriksson (1997), baseadas em diferentes relações cinemáticas e definições de

deformações; cinco formulações foram sugeridas por esses pesquisadores, onde todas foram

testadas no presente trabalho.

Essas formulações foram adaptadas à metodologia de solução não-linear que usa o

método de Newton-Raphson (Silveira, 1995), acoplado às diferentes estratégias de incremento

de carga e de iteração que permitem a ultrapassagem de pontos críticos (bifurcação e limite) que

possam existir ao longo da trajetória de equilíbrio.

A avaliação da eficiência computacional dessas formulações é feita no final do trabalho

através da análise de problemas estruturais fortemente não-lineares encontrados na literatura.

• Alves, R.V. (1993b). Formulação para Análise Não-Linear Geométrica em Referencial LagrangianoAtualizado. 3o Seminário de Doutorado, COPPE/UFRJ.• Pacoste, C. e Eriksson, A. (1997). Beam elements in instability problems. Comput. Methods Appl.Mech. Engrg., No 144, p. 163-197.• Silveira, R.A.M. (1995). Análise de Elementos Estruturais Esbeltos com Restrições Unilaterais deContato. Tese de Doutorado. PUC-RJ, Rio de Janeiro, RJ.• Torkamani, M.A.M., Sonmez, M. e Cao, J. (1997). Second-Order Elastic Plane-Frame Analysis UsingFinite-Element Method. Journal of Structural Engineering, Vol 12, No 9, p. 1225-1235.• Yang, Y.B. e Kuo, S.B. (1994). Theory & Analysis of Nonlinear Framed Structures, Prentice Hall.

VI

ABSTRACT

The main objectives of this work are the computational implementation and study of

geometrically non-linear formulations for two dimensional frame elements. In the formulations

here studied, the stiffness matrix and the internal forces vector can be obtained directly, and

they can be easily coupled to different non-linear solution strategies.

In order to give different options for the solutions of instability problems these two

dimensional frame elements, the following geometrically non-linear formulations were

implemented: (i) Alves (1993b) e Torkamani et al. (1997) formulations, where two different

procedures to obtain the internal forces vector (total and incremental approaches) were tested;

(ii) Yang and Kuo (1994) formulations, where a simplified, a linear-simplified and a higher

order planar frame element were used; these authors introduced two methodologies to obtain the

internal load vector: natural deformation and external stiffness approaches; (iii) Pacoste and

Eriksson (1997) formulations, where a total reference frame (total Lagrangian formulation) was

adopted, and different kinematic assumptions and strain definitions were used; five different

formulations were presented and tested in the present work.

These formulations were coupled to the non-linear solution methodology implemented

initially by Silveira (1995), which solves the resulting non-linear equations and obtains the non-

linear equilibrium paths through the Newton-Raphson method together with path following

techniques, such as the arc-length schemes proposed by Crisfield and orthogonal residual

procedures derived by Krenk.

The performance and capacity of these formulations are illustrated by means of several

numerical examples.

• Alves, R.V. (1993b). Formulação para Análise Não-Linear Geométrica em Referencial LagrangianoAtualizado. 3o Seminário de Doutorado, COPPE/UFRJ.• Pacoste, C. and Eriksson, A. (1997). Beam elements in instability problems. Comput. Methods Appl.Mech. Engrg., No 144, p. 163-197.• Silveira, R.A.M. (1995). Análise de Elementos Estruturais Esbeltos com Restrições Unilaterais deContato. Tese de Doutorado. PUC-RJ, Rio de Janeiro, RJ.• Torkamani, M.A.M., Sonmez, M. and Cao, J. (1997). Second-Order Elastic Plane-Frame AnalysisUsing Finite-Element Method. Journal of Structural Engineering, Vol 12, No 9, p. 1225-1235.• Yang, Y.B. and Kuo, S.B. (1994). Theory & Analysis of Nonlinear Framed Structures, Prentice Hall.

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SUMÁRIO

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Capítulo 1#INTRODUÇÃO

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!+!,-,./"'0%1)20/,30%1"/

O desenvolvimento de novos materiais e técnicas construtivas, bem como os

recursos computacionais disponíveis hoje, têm levado ao emprego de elementos

estruturais cada vez mais esbeltos. A medida que se aumenta a esbeltez de um dado

elemento estrutural, este torna-se cada vez mais susceptível a sofrer grandes deflexões

laterais, que tendem a ocorrer antes da ruptura física. Para se projetar esse tipo de

estrutura deve-se usar o chamado critério de estabilidade. O estudo da não-linearidade

geométrica desses elementos estruturais torna-se, portanto, cada vez mais importante,

possibilitando em certos casos o aparecimento de múltiplas configurações de equilíbrio

(estáveis e instáveis), e a existência de pontos críticos (pontos limites e pontos de

bifurcação) ao longo do caminho não-linear de equilíbrio onde a estrutura pode exibir

saltos dinâmicos.

Portanto, o conhecimento do comportamento não-linear de elementos estruturais

esbeltos, tais como colunas, arcos, anéis, placas e cascas, é de fundamental importância

na solução de problemas de estabilidade/instabilidade local e/ou global de sistemas

estruturais complexos, pois na maioria das aplicações de engenharia esses sistemas são

formadas a partir da união desses elementos.

A análise da estabilidade de sistemas estruturais esbeltos através do Método dos

Elementos Finitos (MEF) envolve invariavelmente a solução de um sistema de equações

algébricas não-lineares. Como relatado no artigo de Riks (1979) e também destacado

em Silveira (1995), existem basicamente duas classes de solução desse sistema de

equações:

2

1. Adaptação computacional do método de perturbação desenvolvido por Koiter

(1970). Durante muitos anos o método de Koiter foi considerado inadequado ao

contexto de elementos finitos, entretanto, trabalhos recentes como os

desenvolvidos por Salerno e Lanzo (1997) e Wu e Wang (1997) provam o

contrário, renovando o interesse dos pesquisadores por esse método;

2. Métodos que procuram resolver as equações não-lineares passo a passo. Incluídos

nessa segunda classe estão os métodos puramente incrementais, as técnicas

baseadas em relações de rigidez secante e os esquemas que combinam

procedimentos incrementais e iterativos, que são atualmente considerados os mais

eficientes e que serão utilizado no presente trabalho.

Recentemente, muitos pesquisadores têm desenvolvido formulações

geometricamente não-lineares para elementos finitos (Alves 1993a-b; Crisfield, 1991;

Yang e Kuo, 1994; Pacoste e Eriksson, 1997; Torkamani et al., 1997; Neuenhofen e

Fillippou, 1997 e 1998), que têm sido adequadamente empregadas na modelagem de

vários sistemas estruturais esbeltos. Essas formulações permitem a determinação da

matriz de rigidez e do vetor de forças internas de forma direta e podem ser acopladas

com relativa facilidade às várias estratégias de solução não-linear.

Essas considerações motivaram a adoção desse tema para a presente dissertação

de mestrado.

3

!+4,-,&560$"7&,0,'0/.%")*&,'&,$%15189&

O presente trabalho é parte integrante das seguintes linhas de pesquisa do

Mestrado em Construção Metálica (Deciv/EM/UFOP):

• Análise Numérica e Computacional em Engenharia: tem como objetivo a

aplicação de métodos numéricos, como o método dos elementos finitos (MEF) e/ou o

método dos elementos de contorno (MEC), na determinação de respostas de sistemas de

engenharia;

• Instabilidade das Estruturas: objetiva o estudo do equilíbrio e estabilidade de

elementos estruturais esbeltos (colunas, arcos, anéis, placas e cascas) submetidos a

carregamentos diversos.

O principal objetivo deste trabalho é o estudo e implementação computacional de

formulações geometricamente não-lineares, para elementos finitos reticulados planos.

Essas formulações serão integradas à metodologia de solução numérica implementada

por Silveira (1995) e expandida por Rocha (2000), que implementou com sucesso

algumas estratégias de solução não-linear encontradas recentemente na literatura.

A seguir, na Seção (1.3), é feita uma revisão bibliográfica onde atenção especial é

dada aos trabalhos que tratam diretamente de formulações geometricamente não-

lineares.

No Capítulo 2 é feita uma explanação geral sobre a metodologia de solução não-

linear adotada, apresentando de maneira resumida as estratégias de incremento de carga

e iterações usadas no presente trabalho.

Os Capítulos 3, 4 e 5 pretendem apresentar de forma detalhada o desenvolvimento

teórico das formulações de elementos finitos não-lineares, que é o principal objeto de

estudo desta dissertação. Na descrição dessas formulações merecem destaque as

relações deformação-deslocamento, as deduções das equações de equilíbrio, o tipo de

elemento finito e as funções de interpolação utilizadas e, finalmente, a obtenção dos

vetores de forças internas e das matrizes de rigidez.

O Capítulo 6 apresenta de forma resumida os procedimentos adotados na

implementação computacional das formulações apresentadas nos Capítulos 3, 4 e 5,

onde atenção especial é dada à montagem da matriz de rigidez e do vetor de forças

internas.

4

As formulações apresentadas nos Capítulos 3, 4 e 5 são analisadas no Capítulo 7,

que apresenta exemplos de problemas estruturais encontrados na literatura. A Seção

(7.2) fornece cinco exemplos clássicos que, por serem mais simples, têm soluções

analíticas (exatas) encontradas na literatura. Em função da confiabilidade dessas

análises, seus resultados têm o objetivo de avaliar a qualidade dos resultados produzidos

pelas diferentes formulações. Com o intuito de validar as observações feitas na Seção

(7.2), são abordados na Seção (7.3) cinco problemas estruturais de estabilidade elástica

fortemente não-lineares, cujas soluções, obtidas numericamente por diversos

pesquisadores, são encontradas na literatura.

Finalmente, no Capítulo 8, são apresentadas as conclusões sobre o emprego das

diversas formulações analisadas nos exemplos do Capítulo 7. São fornecidas também

algumas sugestões para o desenvolvimento de trabalhos futuros.

!+:,-,%07"/*&,5"58"&3%;

5

comparativo entre as formulações de Argyris (1964), Martin (1965), Jennings (1968),

Mallet e Marçal (1968) e Powell (1969).

Epstein e Murray (1976) desenvolveram uma formulação para elementos de

pórticos planos análoga à formulação para elementos de casca, para grandes

deformações, apresentada por Budiansky (1968).

Recentemente, formulações em referenciais Lagrangianos (RLT e RLA) foram

apresentadas por vários pesquisadores, dos quais pode-se citar: Wen et al. (1983);

Chajes et al. (1987); Goto et al. (1987); Wong e Loi (1990); Alves (1993a-b) e

Torkamani et al. (1997). Yang e Kuo (1994) sugeriram uma forma incremental de se

calcular o vetor de forças internas com duas abordagens diferentes para os

deslocamentos nodais: deslocamentos naturais incrementais e rigidez externa. Pacoste e

Eriksson (1995 e 1997) introduziram formulações em RLT baseadas em relações

deformação-deslocamento denominadas ‘relações melhoradas’, com a não-linearidade

expressa por funções trigonométricas.

Singh e Singh (1992) propuseram um modelo com matriz de rigidez e vetor de

forças internas modificáveis de acordo com a natureza das forças axiais.

Um procedimento geral para a obtenção de matrizes de rigidez simétricas foi

proposto por Moran et al. (1998). Neuenhofer e Filippou (1998) propuseram uma

formulação baseada no método da flexibilidade.

Crisfield (1991) afirmou em seu livro que o termo corrotacional tem sido

utilizado na literatura em diferentes contextos, sendo portanto, como afirmam ainda

Pacoste e Eriksson (1997), uma denominação inconsistente. A idéia central desse tipo

de formulação é o cálculo da matriz rigidez e do vetor de forças internas no campo dos

deslocamentos naturais (ou locais), que são os deslocamentos referidos a um sistema de

coordenadas que é atualizado a cada passo de carga, acompanhando a rotação sofrida

pelo elemento. Das formulações com abordagem corrotacional publicadas recentemente

pode-se destacar a formulação em RLA proposta por Crisfield (1991) e as formulações

desenvolvidas em RLT por Pacoste e Eriksson (1997) e Xu e Mirmiran (1997).

Ao contrário da teoria de vigas de Bernoulli, na teoria de vigas de Timoshenko os

efeitos devidos às deformações cisalhantes na seção transversal não são desprezados no

cálculo da rigidez da estrutura. Formulações baseadas na teoria de vigas de Timoshenko

6

foram propostas por Iwakuma (1990), Lee et al. (1994), Petrolito (1995) e Pacoste e

Eriksson (1997).

Os sistemas estruturais formados por barras curvas podem ser analisados com as

formulações de elementos finitos de vigas retas. Porém, com o intuito de melhorar a

eficiência dessas análises foram desenvolvidas formulações de elementos finitos curvos.

Entre os trabalhos recentes, merecem destaque as formulações de elementos curvos

propostas por Marquist e Wang (1989), Kim e Kim (1998), e Raveendranath et al.

(1999); e as formulações de elementos parabólicos propostas por Litewka e Rakowski

(1997 e 1998).

Paralelamente aos elementos para análises bidimensionais, têm-se desenvolvido o

estudo de elementos finitos para análise geometricamente não-linear de pórticos

tridimensionais. Sabe-se de antemão que uma formulação não-linear tridimensional não

é uma simples extensão de uma formulação bidimensional porque as rotações finitas

tridimensionais não são quantidades vetoriais. Recentemente, vários pesquisadores têm

publicado formulações não-lineares para análise estática de pórticos tridimensionais.

Entre eles pode-se destacar Yang e Kuo (1994), que propôs formulações para elementos

tridimensionais retos e curvos; Choi e Lim (1995), com uma formulação de elemento

curvo; al.et coviIbrahimbeg ′ (1996); Ammar et al. (1996), com formulações para

elementos de vigas esbeltas e não-esbeltas; Matsununga (1996), com uma formulação

para pilares não-esbeltos; Pacoste e Eriksson (1997), que propuseram formulações com

abordagem total e corrotacional; Rhim e Lee (1998), que desenvolveram uma

formulação dando um tratamento vetorial à geometria do problema; e Li (1998), que

elaborou uma formulação aplicando a teoria de rotações finitas.

Tem-se produzido também um grande número de formulações geometricamente

não-lineares incluindo análise elasto-plástica de sistemas estruturais reticulados. Essas

formulações têm como principal característica a adotação do critério de escoamento

plástico na determinação das tensões. Nessa linha de pesquisa pode-se destacar os

seguintes trabalhos já publicados: Hsiao et al. (1988), Lee (1988), Meek e Loganathan

(1990), Chang-New Chen (1996), Park e Lee (1996), Ovunc e Ren (1996), Saje et al.

(1997), Saje et al. (1998), Waszczyszyn e Michalska (1998).

7

Nos últimos 20 anos, os avanços tecnológicos e as exigências do mercado de

engenharia, que introduziram maior complexidade e eficiência aos cálculos estruturais,

levaram os pesquisadores a procurarem metodologias de solução que ao mesmo tempo

produzissem resultados precisos e fossem de rápido processamento. Juntamente com as

pesquisas relativas ao desenvolvimento de formulações não-lineares, muitos trabalhos

têm sido produzidos com a finalidade de se determinar a melhor estratégia de solução

não-linear. Os métodos que têm mostrado maior eficiência são os que combinam

procedimentos incrementais e iterativos. Como trabalhos pioneiros podem ser citados os

desenvolvidos por: Argyris (1964), com a aplicação de um método incremental para

solução não-linear; Mallet e Marçal (1968), que utilizaram iterações do tipo Newton

para contornarem os possíveis erros nas aproximações incrementais; Zienkiewicz

(1971), que apresentou uma modificação no método de Newton-Raphson, fazendo com

que a matriz de rigidez só fosse atualizada a cada passo de carga.

Diversos trabalhos têm sido publicados apresentando diferentes estratégias de

controle automático do processo incremental, bem como diferentes estratégias de

iteração. Utilizando um ‘parâmetro de rigidez corrente’ como indicador do grau de não-

linearidade do sistema, Bergan et al. (1978) e Bergan (1980) suprimiram as iterações de

equilíbrio nas zonas críticas da trajetória, até os pontos limites serem atravessados; os

trabalhos de Bergan et al. (1978) e Heijer e Rheinbold (1981) forneceram diferentes

estratégias de incremento de carga.

Batoz e Dhatt (1979) apresentaram uma técnica na qual o ciclo iterativo é

realizado não à carga constante, mas a deslocamento constante, o que permite se obter

os pontos limites de carga mas não os de deslocamento; Riks (1979) apresentou um

método, baseado no parâmetro comprimento de arco ∆l, capaz de calcular pontos limites

de carga e de deslocamento com a introdução de um parâmetro que controla o progresso

dos cálculos ao longo do caminho de equilíbrio; Meek e Tan (1984) apresentaram um

resumo das principais técnicas para se ultrapassar os pontos limites, das quais a técnica

do comprimento de arco foi reconhecida como uma das mais eficientes. Contribuíram

com essa técnica: Riks (1972 e 1979), Ramm (1981), Schweizerhof e Wriggers (1986) e

Crisfield (1981, 1991 e 1997).

Yang e Kuo (1994) introduziram estratégias de incremento de carga e iteração

baseadas em relações de restrição para as quais é definido um parâmetro generalizado,

8

Krenk (1993 e 1995) elaborou uma nova estratégia de iteração, introduzindo duas

condição de ortogonalidade: a primeira entre o vetor de cargas residuais e o incremento

de deslocamento e a outra entre o incremento de forças internas e o vetor de

deslocamentos iterativos.

Crisfield (1997) introduziu procedimentos numéricos que permitem avaliar com

precisão os pontos críticos existentes, e obter as trajetórias de equilíbrio secundárias.

Silveira et al. (1999a-b) forneceram uma metodologia geral de solução de

sistemas de equações algébricas não-lineares que, num contexto computacional pode ser

resumida em duas partes: (i) a partir de uma dada configuração de equilíbrio da

estrutura é calculada uma solução incremental inicial; (ii) em seguida, essa solução é

corrigida com iterações do tipo Newton até ser atingida a nova configuração de

equilíbrio. Nesses dois trabalhos diversas estratégias de iteração e incremento de carga

foram testadas. Utilizando a mesma metodologia, Rocha (2000), em sua Dissertação de

Mestrado, vem realizando um estudo comparativo de diversas estratégias de iteração e

incremento de carga através da análise de vários exemplos numéricos de sistemas

estruturais.

2SOLUÇÃO NÃO-LINEAR

2.1 – INTRODUÇÃO

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2.5.2 – Iteração usando Deslocamento Generalizado

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rr

gr

uu

uu

δδ

δδ−=δλ 6K

j6#jKj

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2.5.3 – Iteração usando Resíduo Ortogonal

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ML#QU[LHOKJL#FXL@UOFIQH#NHLNLRKL#NLH#eHIFj$

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LbKIFWXL# MLR# MIRULGQJIFKL# I# NQHTJIKHL# MI# GQH[Q# FQ# GLF^O[ZHQWXL# K##K ∆+ a# LZ# RIpQi

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GQH[Q# rFAλ∆ #I#HIRLU`IFML@RI#L#RORKIJQ#MI#InZQWgIR#QbQOfLi

rFuKAA λ∆=∆ ############################################################################################_3$3Ec

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MLR#MIRULGQJIFKLR#OKIHQKO`LR#δu$

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