Complementarity eigenvalue problems for nonlinear matrix pencils

Highlights

• Two algorithms for the solution of nonlinear complementarity eigenvalue problems of the non-polynomial type are proposed.

• Numerical experiments with randomly generated test examples for quadratic or trigonometric matrix pencils are presented.

• The computation of exact critical loads and buckling modes of columns with unilateral obstacles is efficiently addressed.

Abstract

This work deals with a class of nonlinear complementarity eigenvalue problems that, from a mathematical point of view, can be written as an equilibrium model

$$\begin{array}{c}\hfill \left[\begin{array}{cc}A\left(\lambda \right)& B\left(\lambda \right)\\ C\left(\lambda \right)& D\left(\lambda \right)\end{array}\right]\left[\begin{array}{c}u\\ w\end{array}\right]=\left[\begin{array}{c}v\\ 0\end{array}\right],u\ge 0,v\ge 0,u^{T}v=0,\end{array}$$

where the vectors u and v are subject to complementarity constraints. The block structured matrix appearing in this partially constrained equilibrium model depends continuously on a real scalar λ ∈ Λ. Such a scalar plays the role of a non-dimensional load parameter, but it may have also other physical meanings. The symbol Λ stands for a given bounded interval, possibly non-closed. The numerical problem at hand is to find all the values of λ (and, in particular, the smallest one) for which the above equilibrium model admits a nontrivial solution. By using the so-called Facial Reduction Technique, we solve efficiently such a numerical problem in various randomly generated test examples and in two mechanical examples of unilateral buckling of columns.

Introduction

This work concerns a well known phenomenon in engineering: confined buckling of elastic columns. The expressions “confined buckling”, “unilateral buckling” and “constrained buckling” are different designations for the same phenomenon: the static instabilization of slender columns or plates in the presence of physical obstacles that are reactive only if pushed. The instability of columns in the presence of obstacles is relevant in several branches of engineering: among many applications one may cite the buckling of railway tracks due to thermal effects [9], or the buckling of reinforcement bars in reinforced concrete members [12].

The class of problems that motivates this article may be described in simple terms by the structure represented in Fig. 1. It consists of a linear elastic slender column made of a material with a modulus of elasticity equal to E and a uniform cross section with a central second area moment of inertia equal to I, submitted to a positive compressive force P. The product EI is a measure of the cross section’s flexural stiffness. In the generic example represented in that figure the column has a fixed support with an articulation on the left end and an horizontally movable support with an articulation on the right end (those supports have a bilateral character since they resist to forces without restriction of their sign). The column has also a system of punctual obstacles (four, in the represented case), each one preventing a cross section of the set {S₁, S₂, S₃, S₄} to move in one of the two senses of the transverse direction. The column’s cross sections S₁ and S₃ cannot move upwards and sections S₂ and S₄ cannot move downwards. For a sufficiently slender column and a sufficiently large value of the compressive force P (above a certain critical value P_cr), the column will abandon the straight configuration because it is no longer stable for such intensity of P. The column then finds a buckled (or curved) configuration. Fig. 2 schematically represents the buckled configuration of the column’s axis: for the represented buckled configuration the three punctual constraints on the left are active and the rightmost unilateral constraint is inactive.

The problem of computation of the critical load P_cr for which a buckled configuration occurs for the first time in a monotonically increasing loading may be reduced to the computation of the smallest positive real λ for which the system

$$\begin{array}{cc}\hfill \left[\begin{array}{cc}{M}_{c,c}\left(\lambda \right)& M_{c,f}\left(\lambda \right)\\ M_{f,c}\left(\lambda \right)& M_{f,f}\left(\lambda \right)\end{array}\right]\left[\begin{array}{c}{x}_c\\ x_f\end{array}\right]=\left[\begin{array}{c}{y}_c\\ 0\end{array}\right],& \end{array}$$

$$\begin{array}{c}\hfill 0\le x_c\perp y_c\ge 0\end{array}$$

holds for some nonzero triplet (x_c, x_f, y_c). Condition (2) is a short way of saying that x_c and y_c are non negative vectors satisfying the orthogonality condition $x_c^T{y}_c=0$. In order to obtain the above formulation one has to discretize the column in a set of “column finite elements” as explained, for instance, in [5]. In that discretization, the finite elements’ nodes have to coincide with the bilateral end supports or with the intermediate unilateral supports materialized by the punctual obstacles. The subscript f refers to the set of degrees of freedom that are not kinematically constrained, while the subscript c refers to the set of degrees of freedom that are unilaterally constrained. The vector $x={(x_c,x_f)}^T$ in the complementarity system (1) and (2) groups the generalized coordinates, i.e., nodal transverse displacements and rotations, while the vector $y={(y_c,0)}^T,$ of which only the subvector y_c is non-vanishing, groups the generalized reactions. The block structured matrix

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{M}_{c,c}\left(\lambda \right)& M_{c,f}\left(\lambda \right)\\ M_{f,c}\left(\lambda \right)& M_{f,f}\left(\lambda \right)\end{array}\right]\end{array}$$

has the physical meaning of a total stiffness matrix encompassing the elastic stiffness and the geometrical stiffness. The precise form of each entry of matrix (3) depends on the specific mechanical problem under consideration. Usually, matrix (3) depends both polynomially and trigonometrically on the variable λ, which in turn is a simple function of the load parameter P. In fact, as it will be seen in the examples treated in Section 4, the entries of matrix (3) depend on the so-called stability functions that, for a canonical finite element of length L, flexural stiffness EI and submitted to a compressive force P, are defined by

$$\begin{array}{ccc}\hfill {\varphi }_1\left(\lambda \right)& =& \frac{\lambda \cos \lambda }{\sin \lambda }{\varphi }_2\left(\lambda \right),\hfill \\ \hfill {\varphi }_2\left(\lambda \right)& =& \frac{{\lambda }^2}{3}{[1-\frac{\lambda \cos \lambda }{\sin \lambda }]}^{-1},\hfill \\ \hfill {\varphi }_3\left(\lambda \right)& =& \frac{3}{4}{\varphi }_2\left(\lambda \right)+\frac{1}{4}\frac{\lambda \cos \lambda }{\sin \lambda },\hfill \\ \hfill {\varphi }_4\left(\lambda \right)& =& \frac{3}{2}{\varphi }_2\left(\lambda \right)-\frac{1}{2}\frac{\lambda \cos \lambda }{\sin \lambda },\hfill \end{array}$$

where $\lambda =(L/2)\sqrt{P/EI}$ plays the role of a non-dimensional load parameter. For general information on stability functions, see [11] or [5]. A detailed discussion on the mechanical problem (1) and (2) is postponed to Section 4.