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The paper shows a comprehensive analysis of geometrically non linear structural problems by the finite element method. The theoretical approach is based on a variational principle stating the incremental equilibrium through the stationarity of a functional that could be defined as the incremental total potential energy. The analysis is carried out in two distinct phases: first a prediction of the behaviour of the structure subjected to an increment of load then a correction by means of Newton Raphson method of the results obtained in the previous incremental step. The approach makes it possible to determine the complete load deflection curve either in the prebuckling region or in the postbuckling one and to find out the critical load taking into account the deflection prior to buckling (non linearized buckling analysis).

The procedure for extending the finite-element displacement method to include geometric nonlinearity is outlined and a consistent formulation applicable to arbitrary plate and shallow shell elements is derived. Examples illustrating the application of a rectangular element to a plate, a shallow cylindrical shell, and a shallow hyperbolic paraboloid are presented and the results are compared with other theoretical and experimental results. Close agreement is obtained even for fairly coarse mesh spacing.

The application of the finite element displacement method to a shallow shell is described, and a stiffness matrix for an element of rectangular plan is presented. Convergence studies are carried out for a shallow clamped hyperbolic paraboloid and the results are compared with a finite difference solution using a fine mesh spacing. Agreement to within 4% is obtained for an element size equal to twice the finite difference mesh spacing.