Nonlinear analysis of structural elements under unilateral contact constraints by a Ritz type approach

Abstract

A nonlinear modal solution methodology capable of solving equilibrium and stability problems of uni-dimensional structural elements (beams, columns and arches) with unilateral contact constraints is presented in this work. The contact constraints are imposed by an elastic foundation of the Winkler type, where special attention is given to the case in which the foundation reacts in compression only, characterizing the contact as unilateral. A Ritz type approach with moveable boundaries, where the coordinates defining the limits of the contact regions are considered as additional variables of the problem, is proposed to solve this class of unilateral contact problems. The methodology is illustrated by particular problems involving beams, beam-columns and arches, and the results are compared with available results obtained by finite element and mathematical programming techniques. It is concluded that the Ritz type approach proposed is particularly suited for the analysis of structural problems where the number, but not the length, of the contact regions between the bodies are known a priori. Therefore, it can substitute in these cases finite element applications and be used as a benchmark for more general and complex formulations as well.

Nonlinear Equilibrium and Stability Analysis of Axially Loaded Piles Under Bilateral Contact Constraints

“…Such differences result in the remaining contact area being subjected to high stress concentrations. Researchers have, since the 70's, actively studied the unilateral contact problem (Silveira et al, 2013;Celep et al, 2011;Sapountzakis and Kampitsis, 2010;Silveira et al, 2008;Holanda and Gonçalves, 2003;Silva et al, 2001;Silva, 1998;Silveira, 1995;Stein and Wriggers, 1984;Weitsman, 1970). A separate type of problem is the bilateral contact problem (Yu et al, 2013;Maciel, 2012;Shen, 2011 andKien, 2004;Naidu and Rao, 1995).…”

Nonlinear analysis of shear deformable beam-columns partially supported on tensionless three-parameter foundation

“…Recently, Silveira et al [12] presented a nonlinear analysis of Bernoulli structural elements under unilateral contact constraints employing a Ritz type approach, while Tsiatas [13] demonstrated a boundary integral equation solution to the nonlinear problem of nonuniform Bernoulli beams resting on a nonlinear triparametric elastic foundation. In these research efforts, the shear deformation effect is ignored.…”

The unilateral contact buckling problem of continuous beams in the presence of initial geometric imperfections: An analytical approach based on the theory of elastic stability

“…The unilateral contact buckling problem of delaminated plates in a composite member, the possible contact buckling phenomena in metal forming processes, the buckling of the compressive plates of the steel sheeting in a composite slab, are some of the many structural and mechanical problems where the beginning and the evolution of the buckling phenomena are strongly affected by unilateral contact conditions. Problems of that type are usually handled using computational techniques based on variational formulations of the governing differential equations ( [1], [2], [3], [4], [5], [6], [7], [8]). For the solution of some specific problems in the area of civil engineering, simplified mathematical models have also been developed by Ma et al [9], [10].…”