Analysis of a bilinear finite element for shallow shells.: II:: Consistency error

We consider a bilinear reduced-strain finite element of the MITC family for a shallow Reissner-Naghdi type shell. We estimate the consistency error of the element in both membrane- and bending-dominated states of deformation. We prove that in the membrane- dominated case, under severe assumptions on the domain, the finite element mesh and the regularity of the solution, an error bound O(h + t(-1) h(1+s)) can be obtained if the contribution of transverse shear is neglected. Here t is the thickness of the shell, h the mesh spacing, and s a smoothness parameter. In the bending-dominated case, the uniformly optimal bound O( h) is achievable but requires that membrane and transverse shear strains are of order O(t(2)) as t --> 0. In this case we also show that under sufficient regularity assumptions the asymptotic consistency error has the bound O(h).