Mixed finite element analysis for arbitrarily curved beams

Abstract

A convergence of a mixed finite element method for three-dimensional curved beams with arbitary geometry is investigated. First, the governing equations are derived for linear elastic curved beams with uniformly loaded based on the Timoshenko-Reissner-Mindlin hypotheses. Then, standard and mixed variational problems are formulated. A new norm, equivalent to H¹- type norm, is introduced. By making use of this norm, sufficient conditions for existence and uniqueness of the solutions of the above problems are established for both continuous and discrete cases. The estimates of the optimal order and minimal regularity are then derived for errors in the generalised displacement vector and the internal force vector. These analytical findings are compared with numerical results, verifying the role of reduced integration and the accuracy of the methods.