# 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.

This page is maintained by rcb@eng.cam.ac.uk (last update 10 November
2003)