Pergamon Compwers & Srrucrures Vol. 52. No. I, pp. 35-39. 1994 0045-7949(94)EOO19-X zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR Copyright Q 1994 Elsevicr Science Ltd Printed in Great Britain. All rights nservcd 004s7949/94 57.00 + 0.00 LOCKING, RANK AND SINGULARITY OF PENALTY-LINKED STIFFNESS MATRIX AND CONSISTENCY OF STRAIN-FIELD G. PRATHAP National Aeronautical Laboratory, Bangalore- 017 and Jawaharlal Nehru Centre for Advanced Scientific Research, 11% Campus, Bangalore- 012, India (Received 27 February 1993) Abstract-The poor behaviour of conventionally formulated displacement type Co finite elements (i.e., with exactly integrated stiffness matrices) is often attributed to the high rank and non-singularity of the penalty-linked stiffness matrix. In this paper, we show that the correct rank and non-singularity required emerges directly from the consistency of the discretized strain field approximations. A simple study using Ritz-type approximations of the Timoshenko beam problem shows how these aspects are all linked. This investigation therefore unifies many of the statements made about such problems and provides a single, consistent viewpoint. 1. INTRODUCTION The locking phenomenon in the Co finite element formulation of beam, plate, shell and plane stress or 3D flexure and near imcompressible plane strain or 3D problems has been a very intriguing challenge. By locking, we mean that finite element solutions vanish quickly to zero as the penalty multipliers become very large. Most published literature, including all text-books, associate locking with the rank or non-singularity of the stiffness matrix linked to the penalty term (e.g. the shear stiffness matrix in a Timoshenko beam element which becomes very large as the beam be- comes very thin). However, on reflection, it is obvious that these are symptoms of the problem and not the cause. The high rank and non-singularity is the outcome of certain assumptions made (or not made, i.e. leaving certain unanticipated requirements un- satisfied) during the discretization process. It is there- fore necessary to trace this to the origin. An explanation offered by Prathap and co-workers [I-3] is promising-they have argued that it is necessary in such problems to discretize the penalty-linked strain fields in a consistent way so that only physically meaningful constraints appear. the way in which its terms are chosen. An ‘inconsist- ent’ choice of parameters in a low order approxi- mation leads to a full-rank non-singular penalty stiffness matrix that causes the approximate solution to lock. By making it ‘consistent’, locking can be eliminated. In higher order approximations, ‘incon- sistency’ does not lead to locked solutions but instead produces poorer convergence than would otherwise be expected of the higher order of approxi- mation involved. It is again demonstrated that a Ritz approximation that ensures an ab initio consistent definition will produce the expected rate of convergence--a simple example will illustrate this. It is hoped that these simple examples demonstrate sufficiently the importance of the ‘consistency’ re- quirement in the formulation of displacement type finite elements for the many problems where the locking phenomenon was seen. It also follows that the popular methods of constraint counting, etc., of which much has been made earlier is not really required to understand the locking behaviour in terms of the rank or non-singularity of the penalty- linked matrices. 2. THE CLASSICAL BEAM THEORY In this paper, we do not enter into a formal finite Consider the transverse deflection of a thin element discretization but instead, illustrate the con- cantilever beam of length L under a uniformly dis- cepts involved using a simple Ritz-type variational tributed transverse load of intensity q per unit length method of approximation of the beam problem via of the beam. We know from simple structural mech- both classical and Timoshenko beam theory. It is anical concepts that this should produce a linear possible to show how the Timoshenko beam can be shear force distribution increasing from 0 at the free reduced to the classical thin beam theory by using a end to qL at the fixed end and correspondingly, a penalty function interpretation and in doing so, show bending moment that varies from 0 to qL2/2. Using how the Ritz approximate solution is very sensitive to what is called the classical or Euler-Bernoulli theory, 35