J. B. Martin

Applications of mathematical programming concepts to incremental elastic-plastic analysis☆

Abstract As a conceptual framework for studying problems in elastic-plastic structural mechanics, internal variable and mathematical programming formulations have provided valuable insights. This paper attempts to extend these insights. It is argued, from an internal variable formulation, that the appropriate method of analysis of elastic-plastic problems is by means of an incremental holonomic constitutive equation which relates total stresses to internal variable changes. The relationship between Newton-Raphson type iteration schemes and the solution of the programming problem associated with the incremental holonomic problem is also explored.

Internal Variable Formulations of Problems in Elastoplasticity: Constitutive and Algorithmic Aspects

This work surveys a broad range of related issues in quasistatic elastoplasticity, beginning with a development of an internal variable constitutive theory. The initial-boundary value problem is then considered, and the remainder of the work is concerned with the properties of the time-discrete problem. It is shown how this discrete problem has associated with it a holonomic constitutive law (that is, one relating stress to strain or strain increment), and this holonomic law in turn forms the basis of a solution algorithm. Conditions for the convergence of the algorithm are discussed. The entire treatment applies to the spatially continuous problem.

Consistent predictors and the solution of the piecewise holonomic incremental problem in elasto-plasticity

Abstract Algorithms for the solution of the classical incremental problems in static elasto-plasticity are an important feature in large-scale finite element programs for nonlinear stress analysis. The algorithms are based on the Newton-Raphson method, where a linearized iteration is used. In each iteration a predictor step is employed to compute a new, improved estimate of the kinematic variables, and a corrector step recomputes the stresses and the internal forces associated with the improved solution. Backward difference methods have been shown empirically to provide an effective and a stable means of implementing the corrector step. Very recently it has also been shown that so-called consistent predictors should be used together with the backward difference correctors in order to ensure the quadratic rate of convergence normally expected with the Newton-Raphson method. Very considerable success has been achieved with the use of consistent predictors, both in terms of stability and rate of convergence. In this paper we present a rational formulation of the incremental problem. It is shown that the adoption of a backward difference corrector is equivalent to the reformulation of the problem as a piecewise holonomic problem. This permits the problem to be written as an unconstrained convex nonlinear programming problem. It is further shown that the consistent predictor can easily be identified within this formulation, and as a consequence it can be interpreted as a rational step in the algorithm for the solution of the nonlinear programming problem.

Internal variable formulations for the plastic analysis of plane frames

Abstract There are a number of methods available for the plastic analysis of plane frames; of these the most widely used are those which are rooted in mathematical programming techniques. This paper presents a formulation which utilizes the thermodynamically based internal variable approach where plastic hinges are characterized as internal variables. The algebraic equations resulting from the formulation are versatile and it is illustrated how they can be incorporated in an algorithm to solve incremental problems. A number of examples illustrate the application and effectiveness of the formulation.

Evaluation of the inelastic spectrum design method for two degree-of-freedom structures under earthquake loading

Abstract The design spectrum method is commonly applied to the design of structures to resist earthquake loading. It is essentially an elastic method, depending on a superposition of the elastic modes of the structure. It is impractical, however, to design any but the most sensitive structures to resist a major earthquake elastically. Plastic deformation in the structure absorbs energy, and attenuates the most severe effects of strong ground motion. It is thus highly desirable to incorporate inelastic behaviour into the design method. Newmark and Hall have suggested an extension of the design spectrum method, making use of the concept of ductility under earthquake loading. Depending on the anticipated ductility of the structure, the elastic design spectrum is modified to be less severe, and the structure is designed elastically on the basis of the modified spectrum. This method is acceptable for one degree-of-freedom structures, but cannot be rigorously established for multi-degree-of-freedom structures. Several studies of tall steel frames have been carried out, and show that caution must be exercised in the application of the method. These studies are particular, however, and give little information on the way in which the unconservatism introduced varies with the parameters of the problem, or how it compares with other sources of error. This paper seeks to provide such information: a parametric study of two degree-of-freedom systems was carried out, and the results were plotted in such a way that the range of relative errors can be readily appreciated.