Static Limit Analysis Research Articles

Lower‐bound limit analysis by using the EFG method and non‐linear programming

AbstractIntended to avoid the complicated computations of elasto‐plastic incremental analysis, limit analysis is an appealing direct method for determining the load‐carrying capacity of structures. On the basis of the static limit analysis theorem, a solution procedure for lower‐bound limit analysis is presented firstly, making use of the element‐free Galerkin (EFG) method rather than traditional numerical methods such as the finite element method and boundary element method. The numerical implementation is very simple and convenient because it is only necessary to construct an array of nodes in the domain under consideration. The reduced‐basis technique is adopted to solve the mathematical programming iteratively in a sequence of reduced self‐equilibrium stress subspaces with very low dimensions. The self‐equilibrium stress field is expressed by a linear combination of several self‐equilibrium stress basis vectors with parameters to be determined. These self‐equilibrium stress basis vectors are generated by performing an equilibrium iteration procedure during elasto‐plastic incremental analysis. The Complex method is used to solve these non‐linear programming sub‐problems and determine the maximal load amplifier. Numerical examples show that it is feasible and effective to solve the problems of limit analysis by using the EFG method and non‐linear programming. Copyright © 2007 John Wiley & Sons, Ltd.

Numerical determination of limit loads for three-dimensional structures using boundary element method

Based on the static limit analysis theorem, a solution procedure for limit analysis of three-dimensional (3-D) structures is established making use of conventional boundary element method (BEM) rather than of finite element method (FEM). The self-equilibrium stress fields are expressed by linear combination of several basic self-equilibrium stress fields with parameters to be determined. These basic self-equilibrium stress fields are elastic responses of the body to imposed permanent strains obtained through elastic-plastic incremental analysis by three-dimensional boundary element method (3-D BEM). The Complex method is used to directly solve the resulting nonlinear programming problem and to determine the maximal load amplifier. The numerical results show that it is efficient and accurate to solve 3-D limit analysis problems by using BEM and the Complex method.

A numerical method for lower bound limit analysis of 3-D structures with multi-loading systems

The determination of lower bound limit load of 3-D structures is by no means an easy task, especially for complex configurations and loading systems. In our previous work, a numerical method of upper bound limit analysis for 3-D structures with multi-loading systems was proposed. This method combines FEM and mathematical programming technique in an iterative procedure. In the present article, on the basis of the nature of the iterative procedure for upper bound limit analysis, the statically admissible stress fields, which satisfies the equilibrium equation and boundary conditions, are constructed using some intermediate variables obtained by upper bound limit analysis procedure. Moreover, a mathematical programming formulation is set up for the static limit analysis of 3-D structures under multi-loading systems and a direct iterative algorithm used to determine the lower bound limit load multiplier is proposed, which depends on the static theorem of plasticity. The numerical examples are given to demonstrate the applicability of the procedure.

Explicit solutions of limit analysis problems as interval linear programs

The paper is concerned with the application of interval linear programming to limit analysis of discrete structures. The static limit analysis problem is formulated as an interval linear program, i.e. a linear programming problem with two-sided constraints. A new method to obtain an explicit representation of the set of optimal solutions to this problem, using the generalised inverses of matrices, is proposed in the paper. The method is illustrated by simple examples of truss, beam and frame.

The determination of mode shapes for dynamically loaded rigid-plastic structures

Mode solutions in rigid-plastic structures subjected to fixed external loads are dynamic solutions which are products of separate functions of space and time. If the small displacement assumptions are adopted, there will exist at least one mode shape for any structure subjected to given fixed external load, and possibly a multiplicity of modes. The time function is a simple linear function, and is easily determined once the mode shape is known. The paper puts forward, a simple in physical terms, algorithm for the determination of mode shapes. An iterative procedure is established in which each iteration is equivalent to the solution of a static limit analysis problem. Convergence is proved.

Simulated Earthquake Tests of R/C Frames

The inelastic response of reinforced concrete multistory frames was investigated by subjecting small-scale one-bay three-story reinforced concrete frames to a series of strong base motions simulating one horizontal component of representative earthquake acceleration records. The test results were evaluated from the perspective of analytical methods routinely and economically available to structural-design office: linear dynamic response analysis and static limit analysis based on elasto-plastic response. Linear dynamic analyses provided a good qualitative understanding of the observed inelastic behavior. The observed maximum base shears and overturning moments were much higher than the values predicted by the limit analysis due to strain-hardening of the steel. It is concluded that strain-hardening characteristics of the reinforcing steel must be explicitly considered in the earthquake-resistant design of reinforced concrete structures, especially in relation to design of web reinforcement, foundations, and columns.