A modified Lemke Algorithm for dynamic rigid plastic response of skeletal structures

Highlights

• Modified Lemke Algorithm is used for dynamic rigid plastic response.

• Finite element formulation is in the form of Linear Complementarity Problem.

• The algorithm is adapted to handle strongly nonlinear rigid-plastic systems.

• The algorithm output for a pulse loaded simply supported beam is investigated.

Abstract

This paper proposes a modified Lemke algorithm to determine the non-holonomic response of rigid plastic skeletal structures subjected to extreme dynamic loading. The basic formulation for the dynamic rigid-plastic response has a mathematical form of linear complementarity problem (LCP), and, equivalently, a pair of dual quadratic programs (QP). Previous attempts using the Wolfe type LCP solver exhibited very pronounced sensitivity with semidefinite LCPs, which appear in this class of dynamics. Therefore, the current study offers a numerically robust Lemke algorithm, which is considerably adapted to deal effectively with the presence of unrestricted variables and the semidefinite characteristics of structural matrices. In addition, degeneracy in the LCP solution of rigid plastic structures is efficiently handled by using the Lexicographic Minimum Ratio test. To validate the formulation and the algorithm developed, bench tests are conducted, which involve a simply supported beam and a portal frame subjected to a uniformly distributed rectangular pulse loading.

Introduction

Mathematical programming has a wider application in various specialised fields of engineering such as robotics [1], fluid simulation [2], and agriculture [3]. As a discrete variable mathematical system founded on the finite element modelling of a structure, mathematical programming also has the potential for providing systematic numerical solutions to various problems of structural analysis. Sixty years have now passed since these virtues were envisaged and their demonstration first began. The historic development of mathematical programming in structural plasticity has been traced in the surveys of Maier and Munro [4], Maier [5], and Maier and Lloyd Smith [6].

The survey of published works on the dynamic plastic behavior of structures reveals a multiplicity of different approaches for solving the problem [7], [8]. These practical problems of structures under extreme dynamic loading often involve large displacements, viscoplastic behaviour and elastic–plastic interaction. Concerning the effects of elastic–plastic interaction, it appears reasonable to neglect the material elasticity when the total dynamic energy imparted to the structure is significantly larger than the maximum amount of strain energy that can be absorbed elastically [8], [9]. Thus capatilising on this notion, the rigid-plastic material behavior is characterised by two distinct ranges, a path independent range termed as rigid, and a path dependent range termed plastic. The rigid-plastic theory has proved useful in solving many practical extreme dynamic problems and in supplying a conceptual framework that is founded upon the fundamental principles of mechanics. Furthermore, this simple theory is numerically efficient in dealing with the ductile structures subjected to short duration, high-intensity dynamic loads [9]. However, experiments have indicated that the rigid-plastic theory may require corrections for the dependence of the yield stress on strain rate, for the effects of large displacements and other parameters in the structural and material modelling [10].

An important and fundamental development in the theory of the dynamic behavior of rigid, perfectly plastic continua is offered by the kinematic minimum principle of Tamuzh [11]. This principle has been fruitfully formulated into a quadratic programming problem by Capurso [12] to rigid plastic planar framed structures subjected to impulsive loads or to constant pulse loads of short duration. In recent times, Patsios et-al [13] have applied a mathematical programming method for the analysis of planar and spatial structural frames. In the context of dynamic problems, although the reported applications have been few, they show promising results [14], [15]. It is of note that, quadratic programming has also been successfully applied to the rigid plastic modelling of masonry walls subjected to impact load [16]. Kaliszky and Logo [17] have obtained some interesting developments in layout optimization of rigid-plastic structures under high intensity short-time dynamic pressure through the use of quadratic programming.

A re-examination and elaboration of the theory of structural plasticity for quasi-static loading was undertaken by Lloyd Smith [18] in the context of mathematical programming. It provided the impetus for a series of researches by Al-Samara [19], Sahlit [20], and Khan [21], relating to elasto-plastic and rigid-plastic framed structures under general pulse loads, impulsive loads, and impact.

For rigid-plastic structures under general pulse or impulsive loads, Sahlit [20] employed nodal and mesh descriptions for small-displacement kinetic and kinematic structural laws, and, in combination with appropriate plasticity relations, led to the formulation of the governing system as an LCP by means of Newmark’s scheme for numerical integration. It was confirmed that the LCP can provide the complete dynamic response of a structure, determining automatically the sequence of mechanisms or velocity profiles through which the response must pass [20], [22]. However, the algorithm used for solving the LCP, based on Wolfe’s simplex method for quadratic programs, was not always able to deal with the semi-definite nature of the LCP involved in the rigid-plastic dynamics of framed structures.

It is worth re-emphasizing that, for structures subjected to extreme dynamic loads, the simple rigid-plastic theory has played an important role in offering simplified procedures that exhibit low computational complexity and reduced number of mechanical parameters [23], [24]. Extensive literature on the theoretical and experimental studies of structures under extreme dynamic load is available [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], which has been influential in the development of the current work. These available closed-form theoretical solutions for dynamic response, required its deviser to make an intuitive and inspired proposal of the sequence of velocity profiles that gives rise to the evolving displaced configuration, each profile being checked a posteriori to confirm its correctness. The essential advantages of the LCP solution [34], [35], [36], [37], [38], [39], [40], [41], [42] are that the sequence of velocity profiles is determined automatically, without need for intuition, and that the procedure is applicable to structural forms more general than beams.

This current work, a continuation of the previous LCP investigation [14] has three aims. To make this work self-contained, the first aim highlights the mathematical apparatus developed previously and reviews some basic concepts in the theory of quadratic programming and linear complementarity. The second aim replaces Sahlit’s adaptation [20] of Wolfe’s Algorithm, which exhibited very pronounced sensitivity with semidefinite LCPs, with the more robust Lemke’s Algorithm [34], [35]. Therefore, the important feature of the current work is the development of a Lemke Algorithm and its adaptation to accommodating the local unstressing of plastified sections. Effort is specifically directed to identifying, investigating and resolving the numerical difficulties in the Lemke algorithm caused by the semi-definite nature of the LCP. The third aim tests the predictive ability of the LCP formulation for rigid plastic dynamics obtained by the LCP using a beam and a frame problem. The first example of beam problem compares the LCP results with accurate solutions that have previously been obtained for a rectangular pulse loading on beams. The second example utililises a single storey portal frame which is subject to triangular pulse load. This time the LCP validation is executed by making a comparison with the finite element method through ABAQUS.

The paper is organised as follows. Section 2 presents the governing linear complementarity problem which can be written in the form of a primal and dual quadratic system following Karush-Kuhn-Tucker equivalence. Section 3 describes the development of the Lemke LCP solver which deals effectively with the behavior of dynamically loaded rigid-plastic structural systems. Section 4 reports the validation of the LCP results using beam and frame response. Section 5 summarises the work and draws conclusions.