Optimal structural membranes .1. Formulation of a curved quadrilateral element for surface definition

Based on a finite element discretisation, the numerical form-finding of geometrically non-linear surfaces spanning arbitrary boundaries and subjected to an initial prestress is presented. A 24 d.f. quadrilateral finite element is formulated to represent a general curved elastic (or inelastic) geometrically non-linear surface. The proposed isoparametric element is C-0 continuous, of constant thickness, and assumes a plane-stress criterion. A rigorous derivation of the expressions describing the strains within a curved surface is offered, while the element equations are written with special consideration of the effects of large strains and large displacements. Assuming small incremental displacements, expressions are derived to explicitly include the adequate representation of rigid body rotations in the element geometric stiffness matrix. Fundamental tests of element quality are applied to the proposed formulation. Spurious zero energy modes arising from particular forms of the element stiffness matrices and from the use of reduced integration are shown to be suppressed by pre-conditioning. Conditional satisfaction of the eigenvalue test and the patch test is then achieved. Application of the formulation to the form-finding of various minimal surfaces is presented in Part II. Copyright (C) 1996 Elsevier Science Ltd