Curious convergence with hypostatic hybrid equilibrium models

Abstract

The paper describes an unexpected type of convergence behaviour occurring for a single, variable degree, primitive-type equilibrium element. For this element the number of independent stress fields is less than the number of independent boundary displacement variables that do not correspond to rigid element modes of displacement. This leads to the conclusion that the element is hypostatic and that, in the absence of self-stressing modes, no convergence can occur. Such ‘conventional’ counting procedures do not, however, reveal the whole picture, and numerical determination of the rank of the coefficient matrix of the equilibrium equations shows that, in practice, self-stressing modes can and do exist in a model which would conventionally be described as hypostatic. The rank deficiency in the coefficient matrix is shown to be due to the fact that, upon transformation, independent stress fields do not necessarily lead to independent boundary tractions. Generalization to conventionally iso- and hyperstatic models demonstrates that, whenever the coefficient matrix is rank-deficient, spurious kinematic modes coexist with self-stressing modes. The problem which reveals the curious convergence characteristics for the primitive-type element is resolved using a macro-type element, and it is seen that, with the larger degree of hyperstaticity available to this element, strictly monotonic convergence characteristics are observed. © 1997 John Wiley & Sons, Ltd.