The geometric stiffness of triangular composite-materials shell elements

This paper is concerned with the development of the geometric stiffness matrix for Newton type large rotation analysis of composite thin shell structures. The geometric stiffness matrix is derived from load perturbation of the discrete equilibrium equations of a given linear finite element formulation. The geometric stiffness matrix is extracted from the gradient, in global coordinates, of the element nodal force vector when stresses are kept fixed. In order to overcome the difficulties in taking derivatives of the rotation matrix with respect to the nodal coordinates, gradient evaluations are performed in the local coordinate system to result in an in-plane geometric stiffness matrix. An out-of-plane geometric stiffness matrix is then introduced to account for the effect of rigid body rotations on member forces. A unique procedure is used for the removal of rigid body displacements and rotations that enables stress recovery via linear, kinematic and constitutive, relationships. The geometric stiffness matrix derived was used to study several examples whose results compare well with the literature.