TRIC: a simple but sophisticated 3-node triangular element based on 6 rigid-body and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells

Abstract

TRIC is a simple but sophisticated 3-node shear-deformable isotropic and composite flat shell element suitable for large-scale linear and nonlinear engineering computations of thin and thick anisotropic plate and complex shell structures. Its stiffness matrix is based on 12 straining modes but essentially requires the computation of a sparse 9 by 9 matrix. The element formulation departs from conventional Cartesian mechanics as well as previously adopted physical lumping procedures and contains a completely new implementation of the transverse shear deformation; it naturally circumvents all previously imposed constraints. The methodology is based on physical inspirations of the Natural-Mode finite element method (NM-FEM) formalized through appropriate geometrical, trigonometrical and engineering mathematical relations and it involves only exact integrations; its stiffness, mass and geometrical matrices are all explicitly derived. The kinematics of the element are hierarchically decomposed into 6 rigid-body and 12 straining modes of deformation. A simple congruent matrix operation transforms the elemental natural stiffness matrix to the local and global Cartesian coordinates. The modes show explicitly how the element deforms in axial straining, symmetrical and antisymmetrical bending as well as in transverse shearing; the latter has only become clear in the formulation presented here and has brought about a completion of the understanding of natural modes as they apply to the triangular shell element. A wide range of numerical examples substantiate the conception and purpose of the element TRIC; fast convergence is observed in many examples.