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SUMMARYStarting with a mathematical statement of the convergence requirements for an element stiffness matrix, the paper discusses displacement shape functions that may be used in connection with the potential energy principle. In short, these functions must be force orthogonal and energy orthogonal, but they need not be conforming (satisfy interelement compatibility). It is shown that the requirements to the displacement functions may be greatly relaxed through slight modifications of the coupling stiffness between fundamental and higher order displacement modes. Several alternative formulations are examined. In particular, a new 'free formulation' is suggested. Using this form, which is very simple, the only requirement to the displacement patterns used is that they should contain the fundamental deformation modes and be linearly independent. Applications of the theory to triangular and rectangular plate bending elements are shown; the simple stiffness matrix for the latter is given explicitly. The numerical results compare favourably with other types of finite elements.

The paper presents a classification of mathematical formulations commonly encountered in connection with solution of non-linear finite element problems. The principal methods for numerical solution of the non-linear equations are surveyed and discussed. Special emphasis is placed upon the description of an automatic load incrementation procedure with equilibrium iterations. It is shown how this algorithm can be adapted for solving problems involving instabilities, snap-through and snap-back. A simple scalar quantity denoted the current stiffness parameter is suggested; this parameter is used to characterize the overall behaviour of non-linear problems. It can also be used as a steering parameter in the solution process. The use of the present technique is illustrated by several examples.A crucial factor in the development of finite element computer programs for non-linear analysis is the proper selection of solution algorithms. Non-linear problems, in general, require the solution of a set of non-linear algebraic equations, which, in itself, is a formidable task. In addifion, the non-hear prob\ems encountered in structural mechanics may be path-dependent (e.g. plasticity, non-conservative loading) or they may possess multiple solutions (e.g. snap-through buckling). Thus, the quest for reliable solutions to non-linear structural problems is indeed very demanding.Solution procedures for non-linear problems have been discussed by several authors.'-' As opposed to linear problems, it is extremely difficult, if not impossible, to develop one single method of general validity that can be used in a routine manner. Several of the existing solution procedures are either limited to certain classes of non-linear problems, or certain requirements must be satisfied in order to ensure convergence to the correct solution. Very often, the particular problem at hand will require special consideration and it may be necessary to modify the available solution algorithms. For these reasons, it is believed that a computer program for non-linear analysis should possess several alternative algorithms for the solution of the non-linear system. These procedures should also allow for the possibility of an extensive control over the solution process by parameters that are input to the program. Such a scheme would lead to increased flexibility, and the experienced user has the possibility of obtaining improved reliability and efficiency for the solution of a particular problem.t Associate Professor. f dr. ing. 5 Lecturer, dr. ing.

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