Keywords: rigid-plastic beam, nodal description, kinematic indeterminacy, impulsive loading, linear complementarity problem, bending-shear interaction.

Sahlit [1] expressed the dynamic response of a rigid-plastic structure through the solution of a linear complementarity problem (LCP). The current work sets out solutions for a rigid-plastic beam under impulsive loading as a vehicle for presenting extensions to Sahlit's work. Although of little significance for ductile beams under quasi-static loading, plastic shear deformation can be highly relevant for beams under impulsive loading. The interaction of bending and shear is considered through a simple rectangular yield-surface. The paper also suggests that the solution process can be made more robust for the semi-definite form of the LCP associated with rigid-plastic dynamic problems by adapting the algorithm of Lemke [2,3] and by incorporating a lexicographic pivot selection rule.

The beam is divided into rigid finite elements that have two different forms of mass distribution: mass lumped at the two ends of each element or mass uniformly distributed; and two different forms of plastic deformation concentrated at the two ends of each element; plastic bending rotations only and interacting bending and shear deformations. Plastic deformations are considered to be rate-insensitive and the beam to undergo only small displacements. Ordinary differential equations are obtained by combining the laws of kinetics, kinematics and plasticity for the assembly of elements, but they are complicated by the presence of complementary variables. These equations are converted into an LCP form through application of Newmark's time-integration scheme. Recurrent solution of the LCP then traces out the evolution of beam's motion. Once the numerical solution by LCP is initiated, it proceeds to identify and track automatically the resulting sequence of different plastic deformation mechanisms.

A comparison is made between a theoretical solution for the impulsively loaded beam, when only plastic bending deformation is permitted, and solutions given by the LCP for both lumped mass and continuous mass elements. An accurate solution is provided by lumped mass modelling, while that for continuous mass modelling proved to be numerically sensitive, less accurate and required some finessing of the initial velocity profile.

A more realistic representation of the impulsively loaded beam allows plastic shear deformation to develop at the simple supports at the commencement of the motion. For this situation, the continuous mass modelling now provides the LCP with a comparatively accurate solution, showing none of the numerical sensitivity evident in the previous solution.

1

C.L.M. Sahlit, "Mathematical Programming Methods for Dynamically Loaded Rigid Plastic Frame Structures", Ph.D. thesis, Civil Engineering Department, Imperial College, University of London, 1992.

2

C.E. Lemke, "Bimatrix Equilibrium Points and Mathematical Programming", Management Science, 11, 681-689, 1965.

3

C.E. Lemke, "Some Pivot schemes for the Linear Complementarity Problem", Mathematical Programming Study, 7, 15-35, 1978.

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