Non-linear strain–displacement equations exactly representing large rigid-body motions. Part I Timoshenko–Mindlin shell theory Original Research Article

The precise representation of arbitrarily large rigid-body motions in the displacement patterns of curved Timoshenko–Mindlin (TM) shell elements is considered. This consideration requires the development of the strain–displacement relationships of the finite deformation TM shell theory with regard to their consistency with the large rigid-body motions. For this purpose a refined TM theory of multilayered anisotropic shells undergoing finite deformations is elaborated. The transverse shear and transverse normal deformation response and bending–extension coupling are included. The fundamental unknowns consist of six displacements and 11 strains of the bottom and top surfaces of the shell, and 11 stress resultants. On the basis of this theory the simple and efficient mixed models are developed by using the incremental total Lagrangian formulation in conjunction with the Newton–Raphson method. The elemental arrays are derived applying the Hu–Washizu mixed variational principle. Numerical results are presented to demonstrate the high accuracy and effectiveness of the developed four-node shell elements and to compare their performance with other non-linear finite elements reported in the literature.