Linear and Nonlinear Stability Analysis of Trusses

This paper presents a comparison of critical loads for trusses, calculated by linear and nonlinear stability analysis. Analyses are provided by using finite elements method. In linear case, critical load is extracted from derived algebraic eigenvalue problem. In case of nonlinear analysis, critical load is determined by construction of postbuckling equilibrium path. Numerical examples for characteristics trusses are given. It is shown that in the case of some perfect trusses, linear approach may produce significant error in the calculation of critical load, and nonlinear analysis should be introduced. The conclusions about conditions for using linear and nonlinear approach to critical load calculation for trusses are derived. 1. INTRODUCTION Stability analysis of engineering constructions requires calculation of buckling load and corresponding buckling shape. Stability problems are almost simplified by neglecting pre-buckling deformation, and considering construction with no imperfection (perfectly straight beams, etc.). This assumptions enables deriving eigenvalue problem, which solutions are critical (buckling) load and corresponding buckling shape. It is known that in case of beam stability analysis, these assumptions are correct and theoretical value of buckling load, if imperfections are sufficiently small, may be in practice closely obtained [1,2]. In case of beams, axial deformation does not change straight-line state of the beam, but, in case of some perfect trusses, prebuckling deformation may change distribution of forces in constitutive bars, and also acts as imperfection. Truss-like structures are widely used as load bearing structures, because of their relatively high stiffness related to low mass. In this paper is considered problem of calculation of critical load of truss structure. Both linear and nonlinear calculation is done using finite element method. Linear stability analysis is provided by solving linear algebraic eigenvalue problem, which derivation is also presented. In linear approach, prebuckling deformations are neglected. Nonlinear analysis is performed using linear expressions for constitutive matrices in equilibrium equation. Because of possible large displacements, analysis is done iteratively, checking does equilibrium of forces at every node is satisfied. Residual forces are used as additional nodal forces, until it reaches sufficiently small value. On this approach to nonlinear analysis, prebuckling deformation are taken into account. Results are compared for the specific two bar truss, commonly used in demonstration of numerical methods [5]. It is shown that in case of trusses, linear approach may lead to large overestimation of critical load, and that control of results using nonlinear analysis should be done.