Implantation of a nonlinear capability on a linear software system

Abstract

The present paper proposes a simple procedure which extends with minor additions an existing linear elastic program system (ASKA) to geometrically nonlinear applications involving large displacements but small strains. Such an attempt is of course not new. Numerous other techniques related to this subject are documented in a large list of publications which have appeared over the last twenty-five years or so. For the present purpose it suffices to quote in place of an extensive review a few pertinent papers [l--8]. At the same time we propose to discuss in what follows certain general principles of a geometrically nonlinear finite element analysis and their relation to the present work. Clearly, an extension of a linear computational capability to nonlinear applications demands in the first instance a suitable incrementation procedure. Pertinent equations determining the so-called incremental equilibrium are well known. Inherent to the concept of an incremental transition from a given equilibrium state to another is the notion of a tangential stiffness matrix of the structure. In this context, the elastic (material) stiffness matrix already established for a linear analysis must be supplemented by the geometrical stiffness matrix which expresses the influence of the existing stress state on the incremental condition of equilibrium of the structure. Simple incrementation of the nonlinear response leads to a desired linearisation of the computational process but also to inevitable accumulating deviations from the true equilibrium solution. In a first attempt to remove the error imposed by the linearisation higher-order incremental approximations may be employed. In the limit, this approach leads to a fully nonlinear formulation of the problem for large deflections and entails, as a rule, software concepts (LARSTRAN) different from those valid in a linear analysis. Also important in this context is the development and implementation of numerical methods for the solution of the nonlinear governing equations. The literature on the subject is still growing and partial overlapping of the methods is unavoidable. The treatment of geometrically nonlinear problems is based on the equilibrium condition between internal stresses and applied loads for the unknown deflected configuration of the structure. Elastic structures do not require a priori an incremental application of the loading which proves, however, advantageous in the numerical process. On the other hand, incremen-