Updated Lagrangian Formulation of a Flat Triangular Element for Thin Laminated Shells

Abstract

An updated Lagrangian formulation of a three-node flat triangular shell element is presented for geometrically nonlinear analysis of laminated plates and shells. The flat shell element is obtained by combining the discrete Kirchhoff theory plate bending element and a membrane element that is similar to the Allman element but a derivative of the linear strain triangular element. Results are presented for large-rotation static response analysis of a cantilever beam under end moment, cylindrical shell under pinching and stretching loads, a hemispherical shell under pinching and stretching loads, and a ring plate under a line load; for dynamic response analysis of a cylindrical panel; and for thermal postbuckling analysis of an imperfect square plate and a cylindrical panel. To estimate the accuracy of the present formulation in predicting the nonlinear response of large flexible structures, static analysis of an apex-loaded circular arch is performed. The arch is a building block of a large inflatable structure. The results are in good agreement with those available in the existing literature and those obtained using the commercial finite element software ABAQUS, demonstrating the accuracy of the present formulation. HE two most widely adopted approaches in the finite element analysis of shells are use of curved shell elements based on a suitable shell theory and approximation of the curved structure by an assemblage of flat shell elements in which the membranebending coupling is brought about as a result of material anisotropy and transformation of the element stiffness matrices computed in a local coordinate system to the global coordinate system prior to assembly. The curved shell elements can be computationally very expensive, especially in the case of nonlinear analysis, because of the complexity of the formulation and the need to compute the curvature information. Flat shell elements are more attractive because of their simplicity and the ease with which they can be built from alreadyexisting familiar membrane and plate bending elements. Though a large number of elements are required to accurately model curved structures, the analysis is computational ly less expensive because of extremely simple formulation. Updated Lagrangian formulations have been predominantly used in the flat shell formulations available in the existing literature. In an updated Lagrangian formulation, all of the variables are referred to a known configuration, the reference configuration, which is updated continuously during the deformation process. If the rigid-body modes are removed from the total or incremental displacements, the resulting deformational translations and rotations are very small and hence a linearized incremental formulation1 can be used. In a linearized incremental formulation, the stresses are computed using linear strain-displacement relations. The tangent stiffness matrix contains only the linear stiffness matrix and the initial stress matrix. The stiffness matrices resulting from the nonlinear terms in the strain-displacement relations are neglected, resulting in very economical analysis. If the rigid-body modes are not removed, all nonlinear terms in the strain-displacement relations will have to be considered for computing the stresses and the tangent stiffness matrix. A three-node flat triangular shell element was introduced by Argyris et al.2 for nonlinear elastic stability problems. This formula