# Complementarity Problems and Unilateral Constraints

## Abstract

A simple formalism is proposed that leads to the unified treatment of many problems in Structural Mechanics. Such are the self-adjoint problems of statics of structures made of linear elastic, piecewise-linear elastic, elastic-plastic or rigid-plastic material. The linear part of the complete set of governing relations for each of those problems has symmetric matrix of coefficients with positive/negative definite submatrices along the diagonal. Hence, such a set can be replaced by a minimax problem, which in turn is equivalent to a pair of dual Quadratic Programming or Linear Programming problems. If inequalities are absent, then the dual problems are further reducable to the sets of equations corresponding to the Direct Stiffness (Displacement) Method and Flexibility (Force) Method. Otherwise no reduction is possible and either the dual problems must be solved directly (Direct Energy Approach) or the clasical methods should be applied in an iterative manner. In any case the proposed methodology allows us to establish easily the existence and uniqueness properties of the solution.

## Keywords

Complementarity Problem Dual Problem Collapse Mechanism Energy Principle Unilateral Constraint## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Maier, G.: Quadratic programming and theory of elastic-perfectly plastic structures, Meccanica, 3 (1968), 265–273.MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Maier, G.: A matrix structural theory of piecewise linear elastoplasticity with interacting yield planes, Meccanica, 5 (1970), 54–66.CrossRefzbMATHGoogle Scholar
- 3.Capurso, M. and G. Maier: Incremental elastoplastic analysis and quadratic optimization, Meccanica, 5 (1970), 107–116.CrossRefzbMATHGoogle Scholar
- 4.De Donato, O.: Sul calcolo della strutture nonlineari mediante programmazione quadratica, Rend. Ist. Lomb., Cl. Sci., Vol. 103, 1969.Google Scholar
- 5.Cohn, M. Z. and G. Maier, Eds.: Engineering Plasticity by Mathematical Programming, Proc. NATO ASI, Waterloo 1977, Pergamon Press, New York 1979.Google Scholar
- 6.Borkowski, A.: Statics of Elastic and Elastoplastic Skeletal Structures, Elsevier-PWN, Warsaw 1987.Google Scholar
- 7.Cottle, R. W.: Symmetric dual quadratic programs, Quarterly of Applied Mathematics, 21 (1963) 237.MathSciNetzbMATHGoogle Scholar
- 8.Mangasarian, O. L.: Nonlinear Programming, McGraw-Hill, New York 1969.zbMATHGoogle Scholar
- 9.Gill, P. E., Murray, W. and M.A. Saunders: User’s guide for SOL/QPSOL: a FORTRAN package for quadratic programming, Report SOL 83–7, Department of Operations Research, Stanford University, California 1983.CrossRefGoogle Scholar
- 10.Powell, M. J. D.: Report of DAMTP 1983/NA 17, Department of Applied Mechanics, University of Cambridge, Cambridge 1983.Google Scholar
- 11.Borkowski, A., Siemitkowska, B., Weigl, M.: User’s guide for “MATRIX”: a matrix interpretation system for linear algebra and mathematical programming, Laboratory of Adaptive Systems, Institute of Fundamental Technological Research, Warsaw 1985.Google Scholar