Complementarity eigenvalue problems for nonlinear matrix pencils
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 Pcr), 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 Pcr 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 holds for some nonzero triplet (xc, xf, yc). Condition (2) is a short way of saying that xc and yc are non negative vectors satisfying the orthogonality condition . 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 in the complementarity system (1) and (2) groups the generalized coordinates, i.e., nodal transverse displacements and rotations, while the vector of which only the subvector yc is non-vanishing, groups the generalized reactions. The block structured matrix 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 where 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.
Section snippets
Mathematical background
In this section we alleviate the notation and analyze the system (1) and (2) from a purely mathematical point of view. Let denote the linear space of square matrices of order n. For economy of language, we refer to a matrix-valued function as a pencil. An eigenvalue of a pencil M is understood as a scalar such that the system has a nonzero solution . Such a nonzero vector x is called an eigenvector of M. Some authors refer to a pencil as a lambda-matrix and to an
Numerical experiments with the FRT
We use the Facial Reduction Technique to compute numerically all the critical inputs of a pencil M on a bounded interval Λ. We suppose that the hypotheses and hold. The computational work is divided into three phases:
- (I)
Computation of degenerate critical inputs. One finds all the eigenvalues of the subpencil D on the interval Λ. If σ(D, Λ) is empty, then one goes directly to Phase II. For each λ ∈ σ(D, Λ), one checks whether the system (14) admits a nonzero solution . If the answer
Analysis of the buckling of confined columns
This section deals with two examples motivated by the study of the conditions under which a slender straight confined column may buckle. From a numerical point of view, both examples can be handled with the Facial Reduction Technique explained in Section 2.
Conclusions
The FRT is well suited to handle a nonlinear complementarity eigenvalue problem described by the partially constrained equilibrium model (6) and (7). The FRT not only serves to compute all the critical inputs of a block structured pencil M on a given bounded interval Λ, but it serves also to clarify whether a particular critical mode is degenerate, of interior type, or of boundary type. From a conceptual point of view, the FRT has a clear geometric interpretation: one chooses a face of the
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