Geometrically non-linear analysis of plane frames composed of prismatic and tapered sections

Abstract

The geometrically non-linear analysis of plane frames is studied using the finite element technique. Stiffness matrices for a constituent prismatic member are first formulated from both derived and approximate displacement functions, the axial force being considered both constant and flexurally dependent. Both direct and Newton-Raphson type iteration methods are invoked as solution methods, the latter being used in the tangential stiffness matrix approach. The development is then extended to the study of non-prismatic frames. The use of derived functions in this case proved intractable and formulation was based on the derived functions for the geometrically linear behaviour of a non-prismatic member, this being an approximation to the true non-linear descriptions. Again direct and Newton-Raphson iteration techniques are used for solution. In addition to nodal loading, the effect of distributed and non-nodal loading is described, this being reduced to the application of equivalent fixed-end forces. Supporting experimental work is presented for frames composed of both prismatic and tapered members to assess the accuracy of the theoretical solutions with respect to both deflections and bending moments, these tests indicating generally that axial forces should be considered as flexurally dependent. It can be noted that the literature survey showed very little experimental work of this nature. Several examples are presented to demonstrate the range of problems appertaining to the theoretical developments, of particular note being those describing the behaviour of frames containing initial imperfections. Further recommendations for study are given, these including the examination of plastic behaviour in conjunction with geometrically non-linear effects.