Geometrically nonlinear formulation of a 48 D.O.F. quadrilateral shell element with rational B‐spline geometry

Abstract

A geometrically nonlinear formulation for a 48 degrees-of-freedom (d.0.f.) quadrilateral shell element is presented. Each of the three displacement functions of u, u and w is based on the bicubic Hermitian polynomial. The surface of the element is modelled using linearized rational €3-spline functions which may be linked to the geometric data bases generated by the computer-aided design system. The displacement functions and the 48 d.0.f. are expressed in both curvilinear and Cartesian forms, whereas the strain-displacement equations are expressed in curvilinear co-ordinates. The use of Cartesian displacement functions and degrees-of-freedom allow for the proper representation of the six rigid body motions even for the element with nonzero Gaussian curvature, such as a bellows. Two examples are demonstrated: snap-through buckling of a spherical cap and large deflection of a semi-toroidal bellows shell. Results compare well with alternative solutions.

Geometrically Nonlinear Finite Element Analysis of Imperfect Laminated Shells

“…A curved thin shell element previously developed by Yang, Moore, and Anderson [17] for geometrically nonlinear analysis of isotropic shell structures, which was later extended to include the effect of initial geometric imperfections of complex isotropic shell structures by Kapania and Yang [1] is further extended in this paper for large displacement behavior of laminated anisotropic imperfect shell structures. The geometric imperfections considered are the small deviations of the actual fabricated shell from the intended or the designed shape.…”

Formulation of an imperfect quadrilateral doubly curved shell element for postbuckling analysis

“…The tangential strain measure then becomes (3) The effect of imperfections on the curvature-displacement relations are ignored. The expression for strain and curvature tensors in terms of Cartesian displacement components are given in the full paper.…”

“…The element has 12 dof at each node: u l \ w f M ; t/', 2 ; an d u',i2 0 = 1»2,3). The interpolation functions are the same as those used by Yang et al 3 The nonlinear effects due to large displacements and imperfections were included using an incremental formulation based on Lagrangian mode of description. If [q e ] is the vector of the nodal displacements at a given load configuration, then the incremental strain [e] due to an incremental nodal displacement vector {q e } can be written as where the strain-displacement matrix [B 0 ] is the same as that used in linear analysis, [B f ] is the matrix due to geometric imperfections, and [B L (q e )\ the matrix due to nonlinear terms.…”