Unified finite elements based on the classical and shear deformation theories of beams and axisymmetric circular plates

TLDR: In this paper, a unified finite element model that contains the Euler-Bernoulli, Timoshenko and simplified Reddy third-order beam theories as special cases is presented, and a stiffness matrix based on the exact analytical form of the solution of the first-order theory of circular plates is derived.

Abstract: In this paper a unified finite element model that contains the Euler-Bernoulli, Timoshenko and simplified Reddy third-order beam theories as special cases is presented. The element has only four degrees of freedom, namely deflection and rotation at each of its two nodes. Depending on the choice of the element type, the general stiffness matrix can be specialized to any of the three theories by merely assigning proper values to parameters introduced in the development. The element does not experience shear locking, and gives exact generalized nodal displacements for Euler-Bernoulli and Timoshenko beam theories when the beam is homogeneous and has constant geometric properties. While the Timoshenko beam theory requires a shear correction factor, the third-order beam theory does not require specification of a shear correction factor. An extension of the work to axisymmetric bending of circular plates is also presented. A stiffness matrix based on the exact analytical form of the solution of the first-order theory of circular plates is derived.