F. Tin-Loi

Limit analysis of frictional block assemblies as a mathematical program with complementarity constraints

The computation of the collapse loads of discrete rigid block systems, characterized by frictional (nonassociative) and tensionless contact interfaces, is formulated and solved as a special constrained optimization problem known as a Mathematical Program with Equilibrium Constraints (MPEC). In the present instance, some of the essential constraints are defined by a complementarity system involving the orthogonality of two sign-constrained vectors. Due to its intrinsic complexity, MPECs are computationally very hard to solve. In this paper, we investigate a simple numerical scheme, involving appropriate relaxation of the complementarity term, to solve this nonstandard limit analysis problem. Some computational results are presented to illustrate potentialities of the method.

Nonholonomic elastoplastic analysis involving unilateral frictionless contact as a mixed complementarity problem

This paper deals with the elastoplastic analysis of suitably discretized elastoplastic structures under frictionless unilateral contact conditions. Whilst the mathematical programming formulation we adopt can cater equally for hardening as well as softening path-dependent (nonholonomic) plasticity, the state problem involving softening is by far the more difficult to solve in view of the inherent nonconvex nature of the underlying mathematical program. We propose a novel description of a fairly general class of piecewise linear softening laws and investigate use of an approximate stepwise holonomic (path-independent) approach for capturing the evolution of the structure with softening and/or hardening constitutive laws. This finite-step problem is formulated in a specific way to take advantage of the use of a robust and industry-standard complementarity solver named PATH from within the GAMS modeling environment. We illustrate its application on truss-like structures.

Efficient computation of multiple solutions in quasibrittle fracture analysis

We propose a novel and efficient computational scheme for capturing the multiplicity of solutions that may exist at any loading step of a quasibrittle fracture analysis formulated as a linear complementarity problem (LCP). The algorithm proposed is based on successively augmenting the current LCP by complementarity constraints that effectively remove previously found solutions from the solution set. We focus on the single-step or so-called holonomic analysis for mode I behavior. The performance of the proposed approach is illustrated by means of physically meaningful benchmark problems.

Parameter identification of quasibrittle materials as a mathematical program with equilibrium constraints

The identification of the cohesive fracture parameters of quasibrittle materials forms the focus of the present work. This important and challenging task is formulated as an inverse problem, requiring the solution of a special type of constrained optimization problem known as a Mathematical Program with Equilibrium Constraints (MPEC), a key feature of which is the presence of complementarity conditions involving the orthogonality of two nonnegative vectors. We propose a simple scheme, based on transforming the MPEC into a standard nonlinear programming problem through appropriate smoothing of the complementarity constraints, for solving the parameter identification problem. Some computational results, using actual experimentally obtained data, are presented to assess the effectiveness of the approach.

Numerical evaluation of cohesive fracture parameters from a wedge splitting test

Abstract Direct determination of the parameters describing the popular cohesive crack model from experimental results, such as would be obtained from standard three-point bending or wedge splitting tests, poses a challenging problem. This paper describes a numerical method for obtaining key fracture parameters describing the softening behavior of quasibrittle materials in mode I failure. The identification problem is formulated as a special type of constrained optimization problem known in the mathematical programming literature as a mathematical program with equilibrium constraints. Experimental results of a wedge splitting test recently conducted at the Laboratory of Construction Materials, Swiss Federal Institute of Technology (LMC/EPFL) are used to illustrate applicability of the proposed method.

on the optimal plastic synthesis of frames

This paper concerns the first-order optimal plastic synthesis (design) of frames under combined axial force and bending caused by a single load condition. A two phase solution scheme is presented whereby both the restrictions of an initially assumed linear variation in element plastic capacities and of fixed, possibly inaccurate, piecewise linear yield polygons are removed. The first phase implements the so-called nonlinear growth laws while the second phase automatically refines the yield curve linearizations. A simple frame example is used to illustrate these concepts.

Plastic limit analysis of plane frames and grids using GAMS

Abstract This paper describes the implementation and solution on a microcomputer of the plastic limit analysis problem for the class of plane frame and grid structures with the help of the mathematical programming language GAMS. The formulation used is based on the well-known static theorem of limit analysis applied to a suitably discretized structure. It leads to a problem in constrained optimization which becomes one of linear programming only if the yield conditions are linear. After a review of the main features of GAMS, the limit analysis model is described and applied to solve frame and grid problems involving a variety of yield constraints. An important aspect of the work described is that is serves to show that a modeling system can efficiently remove many of the barriers associated with the practical application of mathematical programming for solving structural plasticity problems. In particular, capability for large model construction, generation of detailed reports, and ease of model maintenance and modification are provided in addition to automatic interfacing with a variety of powerful solvers.

A Yield Surface Linearization Procedure in Limit Analysis

ABSTRACT Within the framework of discretized structural models and piecewise linear yield polyhedra, a heuristic is presented to iteratively improve the limit load estimation by approaching locally the plastic admissibility domain. For the two-dimensional active stress situation, the scheme requires the generation of at the most three vertices, in addition to the initial four, to describe the improved inscribed yield polygon. Two examples are given to illustrate the procedure and to clarify various aspects of the algorithm presented.

HOLONOMIC SOFTENING: MODELS AND ANALYSIS*

This paper deals with a special class of holonomic (path-independent) structural analysis problems involving nonlinear or piecewise linear softening. In particular, the formulation takes the form of a complementarity problem, an important class of mathematical problems characterized by the orthogonality of two sign-constrained vectors. A feature and difficulty associated with softening, which violates Druckers stability postulate, is multiplicity of solutions. The main aims of this paper are to give a precise mathematical description of a wide class of softening models. This is achieved via a theoretically and computationally advantageous complementarity format. Second, key ideas underlying a recently developed complementarity solver, PATH, which has the potential of capturing any multiplicity of solutions or to show that none exists, are outlined. Two examples concerning discretized truss structures—a prototype of other more advanced finite element based structural models—are given for illustrative purposes.

Determination of quasibrittle fracture law for cohesive crack models

Abstract A basic and necessary ingredient of cohesive crack models, so widely used for numerical quasibrittle fracture simulations, is the softening function which characterizes the property of concrete-like materials. In this paper, a simple procedure to determine the softening parameters for such functions is proposed. In addition to knowledge of the tensile strength—typically obtainable within about 20% from a cylinder splitting strength (Brazilian) test—the only other requirement is knowledge of the load-displacement curve, as would be obtained from a standard three-point bending test. The method is particularly suitable for materials, such as fiber reinforced concrete, which exhibit a long tail in their load-displacement response. Although suitable for identifying both nonlinear and piece-wise linear relations, we illustrate applications of the scheme using a bilinear law.