Unified Finite Elements Based on the Classical and Shear Deformation Theories of Beams and Axisymmetric Circular Plates

Abstract

SUMMARYIn this paper a uni®ed ®nite element model that contains the Euler±Bernoulli, Timoshenko and simpli®ed Reddy third-order beam theories as special cases is presented. The element has only four degrees of freedom, namely de¯ection and rotation at each of its two nodes. Depending on the choice of the element type, the general stiness 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 speci®cation of a shear correction factor. An extension of the work to axisymmetric bending of circular plates is also presented. A stiness matrix based on the exact analytical form of the solution of the ®rst-order theory of circular plates is derived.

On the numerical properties of high‐order spectral (Euler‐Bernoulli) beam elements

“…elements are capable of providing a higher inter-element continuity (up to C 𝑝−1 ) depending on the polynomial degree 𝑝 of the shape functions [109], no optical branches are encountered 8 , which is in contrast to the proposed SBEs. Nevertheless, the observed behavior is still an improvement compared to classical C 0 -continuous displacement-based finite elements (see also discussions in Ref.…”

Exact elasticity-based finite element for circular plates

“…The general interior elasticity solution can be used as the basis for the derivation of an exact annular plate finite element (FE). The force-based derivation to be presented below is similar to that presented by Reddy et al [4] for an annular Mindlin plate (except for the distributed load p 0 added here). As the point of departure in terms of methodology, we develop the plate element also by the principle of minimum total potential energy.…”

Exact microstructure-dependent Timoshenko beam element

“…Due to the linearity of the problem, the particular solution by Eqs. (29) and (30) is valid for any distributed load which can be expressed as a Maclaurin series…”

“…For nodes i = 1, 2, we have transverse displacements w i and rotations φ i and θ i . In the classical case, there is only one rotation [29]. Using the solution (18), we obtain for nodes 1 and 2 the following six equations…”