Mixed finite element methods for the circular arch problem

Abstract The boundary-value problem for linear elastic circular arches is studied. The governing equations are based on the Timoshenko-Mindlin-Reissner assumptions. The problem is formulated in both the standard and mixed variational forms, and after establishing the existence and uniqueness of solutions to these equivalent problems, the corresponding discrete problems are studied. Finite element approximations of the mixed are shown to be stable and convergent, and selective reduced integration applied to the standard discrete problem renders it equivalent to the mixed problem. The results of numerical experiments are presented; these confirm the convergent behaviour of the mixed problem. For the standard problem with full integration, convergence is suboptimal or nonexistent for small values of the thickness parameter, while for the mixed or reduced integration problem, the numerical rates of convergence coincide with those predicted by the theory.