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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The present paper describes a hybrid stress finite element formulation for geometrically non-linear analysis of thin shell structures. The element properties are derived from an incremental form of Hellinger-Reissner's variational principle in which all quantities are referred to the current configuration of the shell. From this multi-field variational principle, a hybrid stress finite element model is derived using standard matrix notation. Very simple flat triangular and quadrilateral elements are employed in the present study. The resulting non-linear equations are solved by applying the load in finite increments and restoring equilibrium by Newton-Raphson iteration. Numerical examples presented in the paper include complete snap-through buckling of cylindrical and spherical shells. It turns out that the present procedure is computationally efficient and accurate for non-linear shell problems of high complexity.

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