Force method solution of finite element equilibrium models for plane continua

T.F. van Heerden

Abstract:

Diffusive finite element models provide a viable alternative to the traditional displacement (conforming) elements. In this dissertation, a new approach to stress-based equilibrium models is developing for use with the force method of analysis.

The principal innovation is the choice of a set of orthogonal shape functions as stress and displacement modes. This reduces the amount of numerical integration required by either the force or stiffness methods. In addition, a relationship between orthogonal functions and integration points is utilised, which enables the method to be applied in a routine manner to elements of any order. In this Integration Point Method, the orthogonal nature of load and displacement shape functions leads to considerable simplifications and provides a better understanding of the incompatible displacement fields encountered in equilibrium models.

Equilibrium models are quite attractive but they have proven to be more costly than their displacement-based counterpart. Three optimisation schemes are proposed. Equilibrium models are shown to be suitable for conventional finite element meshes, not requiring the construction of macro elements as previously suggested. A condensation technique is introduced as the natural counterpart of the existing substructuring technique in the stiffness method, resulting in reduced dimensions for the equilibrium matrix, which is shown to be independent of the node numbering scheme and requires no additional memory for the calculation of the full set of states of self-stresses.

Equilibrium models are known to provide lower bounds to the collapse load in plasticity calculations, but are generally thought to be relatively expensive. With the condensation technique and reduced storage schemes developed in this thesis, considerable computational savings are achieved. Furthermore, equilibrium models are used to ensure that the yield criterion is never violated, a requirement not met by displacement models. A new method of analysis, the Tangent Equation Approach, is developed for linear-elastic perfectly-plastic analyses.

The models and techniques developed in this dissertation are applied to a range of standard problems, and shown to behave well.