William Taylor Matias Silva

Performance of Composite Implicit Time Integration Scheme for Nonlinear Dynamic Analysis

This paper presents a simple implicit time integration scheme for transient response solution of structures under large deformations and long-time durations. The authors focus on a practical method using implicit time integration scheme applied to structural dynamic analyses in which the widely used Newmark time integration procedure is unstable, and not energy-momentum conserving. In this integration scheme, the time step is divided in two substeps. For too large time steps, the method is stable but shows excessive numerical dissipation. The influence of different substep sizes on the numerical dissipation of the method is studied throughout three practical examples. The method shows good performance and may be considered good for nonlinear transient response of structures.

Uma abordagem analítica para detecção de pontos limites e de bifurcação

RESUMO: Neste trabalho descreve-se analiticamente de maneira detalhada a deteccao e a classificacao de pontos criticos na trajetoria primaria de equilibrio de sistemas estruturais. Utiliza-se a Formulacao Lagrangiana Total para descrever a cinematica de um elemento de barra biarticulado 2D. Atraves desta formulacao obtem-se o vetor de forcas internas e a matriz de rigidez tangente que levam em conta os efeitos da nao linearidade geometrica. Assume-se um modelo constitutivo linear elastico para o estado uniaxial de tensao-deformacao, usando a deformacao de Green-Lagrange e a tensao axial do segundo tensor de Piola-Kirchhoff que sao energeticamente conjugados. Como estudo de caso apresenta-se uma trelica plana hiperestatica composta com 3 elementos biarticulados 2D e com dois graus de liberdade. Por fim, determinam-se as condicoes geometricas e fisicas para a coalescencia entre os pontos limites e de bifurcacao. A principal contribuicao deste trabalho e demonstrar a necessidade de compreender melhor os fenomenos nao lineares para projetar sistemas estruturais mais seguros. ABSTRACT: Using an analytical this paper describes in detail the detection and classification of critical points in the primary equilibrium path of structural systems. The Total Lagrangian formulation is employed to describe the kinematics of a 2D bar element. With this formulation, the internal force vector and the tangent stiffness matrix including the geometric nonlinearity effects are obtained. An elastic linear constitutive model is assumed for the uniaxial stress-strain state. Such model uses the Green-Lagrange strain tensor and the second Piola-Kirchhoff axial stress tensor which are energetically conjugate tensors. As a study case, the article presents a statically inderminate plane truss discretized with three 2D bar elements. Finally, the geometrical and physical conditions for the coalescence between limit and bifurcation points are determined. The main contribution of this work is to demonstrate the need to better understanding the non linear phenomena. Such understanding is necessary for designing safer structural systems.

OBTENÇÃO DE UM ELEMENTO DE VIGA UNIFICADO COM A UTILIZAÇÃO DA FORMULAÇÃO CORROTACIONAL E O PRINCÍPIO DOS TRABALHOS VIRTUAIS COMPLEMENTARES

In this work is described a shear flexible beam element model without shear locking and integrating Euler-Bernoulli’s and Timoshenko’s beam theories. The unified stiffness matrix of this beam is obtained by using the co-rotational formulation that is based on the separation of displacement fields into rigid body modes and deformation modes. The principle of complementary virtual work is also used for finding the unified stiffness matrix. The stiffness matrix is then implemented in a finite element software for linear analysis of plane frames. Some examples are presented and numerical results demonstrate that a shear flexible beam element do not show shear locking effects. Keywords: Co-rotational approach, Complementary virtual work, Shear flexible beam element.