Assessment of solutions from the consistently linearized eigenproblem by means of finite difference approximations

X Jia,H.A Mang

doi:10.1016/j.compstruc.2015.01.016

X Jia, H.A Mang

Open Access

https://doi.org/10.1016/j.compstruc.2015.01.016

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Journal: Computers & Structures	Publication Date: Feb 11, 2015
Citations: 2	License type: cc-by

Affiliation: TU Wien, Tongji University

Abstract

The consistently linearized eigenproblem (CLE) plays an important role in stability analysis of structures. Solution of the CLE requires computation of the tangent stiffness matrix K∼T and of its first derivative with respect to a dimensionless load parameter λ, denoted as K∼̇T. In this paper, three approaches of computation of K∼̇T are discussed. They are based on (a) an analytical expression for the derivative of the element tangent stiffness matrix K∼Te, (b) a load-based finite difference approximation (LBFDA), and (c) a displacement-based finite difference approximation (DBFDA). The convergence rate, the accuracy, and the computing time of the LBFDA and the DBFDA are compared, using the analytical solution as the benchmark result. The numerical investigation consists of the analysis of a circular arch subjected to a vertical point load at the vertex, and of a thrust-line arch under a uniformly distributed load. The main conclusion drawn from this work is that the DBFDA is superior to the LBFDA.

Highlights

Buckling is one of the most important causes of loss of the integrity of structures
As follows from (42) and (43). (44) indicates that the error of kÃ1ð0Þ, resulting from the displacement-based finite difference approximation (DBFDA) of k_Tð0Þ, is larger than that based on the load-based finite difference approximation (LBFDA)
The curve has a minimum at B, which indicates that the prebuckling state involves both membrane and bending stresses

Summary

Introduction

Buckling is one of the most important causes of loss of the integrity of structures. the investigation of this phenomenon is very important in structural analysis and design. The socalled consistently linearized eigenproblem (CLE), originally proposed in [1,2], was initially used for ab initio estimates of stability limits by means of the Finite Element Method (FEM) [3]. A circular arch subjected to a vertical point load at the vertex, and a thrustline arch subjected to a uniformly distributed load will be investigated with special emphasis on loss of stability.

Basic mathematical properties of the consistently linearized eigenproblem

Circular arch

Thrust-line arch

Conclusions