Formulation of quasi‐axisymmetric boundary value problems for finite element analysis

In the finite element method two options are currently available for dealing with problems involving an axisymmetric geometry but in which loads, displacements or other boundary conditions do not have rotational symmetry. The first involves a full three‐dimensional solution whereas the second is to use a two‐dimensional (2D) axisymmetric formulation and to express the non‐symmetric loads/displacements as Fourier series in the circumferential direction. There are some cases, however, that, in spite of not being truly axisymmetric, it can be shown that the non‐zero components of the full strain tensor number four or less. In this paper it is shown that such problems may be solved using simple 2D finite element formulations and two alternative solution methods are presented. One of these involves a modification to the matrix relating strains to displacements and the second employs conventional 2D formulations with tied degrees of freedom. The solution procedures are applied to three examples which have some geotechnical interest, namely the behaviour of a long rigid pile under either torsional or vertical loading and the behaviour of a hollow cylinder sample subjected to torsion. In all three cases the soil is modelled by means of an elastoplastic constitutive law of the Cam‐clay type.