Hyperbolic paraboloid shell analysis via mixed finite element formulation

Abstract

AbstractAn isoparametric rectangular mixed finite element is developed for the analysis of hypars. The theory of shallow thin hyperbolic paraboloid shells is based on Kirchhoff–Love's hypothesis and a new functional is obtained using the Gâteaux differential. This functional is written in operator form and is shown to be a potential. Proper dynamic and geometric boundary conditions are obtained. Applying variational methods to this functional, the HYP9 finite element matrix is obtained in an explicit form. Since only first‐order derivatives occur in the functional, linear shape functions are used and a C0 conforming shell element is presented. Variation of the thickness is also included into the formulation without spoiling the simplicity. The formulation is applicable to any boundary and loading condition. The HYP9 element has four nodes with nine Degrees Of Freedom (DOF) per node—three displacements, three inplane forces and two bending, one torsional moment (4 × 9). The performance of this simple, and elegant shell element, is verified by applying it to some test problems existing in the literature. Since the element matrix is obtained explicitly, there is an important save of computer time.