An exact sinusoidal beam finite element

Abstract

AbstractThe purpose of the paper is to derive an efficient sinusoidal thick beam finite element for the static analysis of 2D structures. A two–node, 6–DOF curved, sine–shape element of a constant cross–section is considered. Effects of flexural, axial and shear deformations are taken into account. Contrary to commonly used curvilinear co–ordinates, a rectangular co–ordinates system is used in the present analysis. First, an auxiliary problem is solved: a symmetric clamped–clamped sinusoidal arch subjected to unit nodal displacements of both supports is considered using the flexibility method. The exact stiffness matrix for the shear–flexible and compressible element is derived. Introduction of two parameters “n” and “t” enables the identification of shear and membrane influences in the element stiffness matrix. Basing on the principle of virtual work a full set of 18 shape functions related to unit support displacements is derived (total rotations of cross–sections, tangential and normal displacements along the element). The functions are found analytically in the closed form. They are functions of one linear dimensionless coordinate of x–axis and depend on one geometrical parameter of sinusoidal arch, height/span ratio “c” and on physical and geometrical properties of the element cross–section. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)