On locking-free shear deformable beam finite elements

On applications of generalized functions to beam bending problems

Free vibration analysis of Timoshenko beams has been presented. Discrete singular convolution method is used for numerical solution of equation of motion of Timoshenko beam. Clamped, pinned and sliding boundary conditions and their combinations are taken into account. Typical results are presented for different parameters and boundary conditions. Numerical results are presented and compared with that available in the literature. It is shown that very good results are obtained. This method is very effective for the study of vibration problems of Timoshenko beam. Copyright © 2009 John Wiley & Sons, Ltd.

Free vibration analysis of Timoshenko beams by DSC method

An overview of the relationships between solutions of the classical and shear deformation plate theories

An efficient and accurate locking-free corotational beam finite element for the analysis of large displacements and small-strain problems is developed in this paper. Three different finite element models based on three different beam theories, namely, the Euler–Bernoulli, Timoshenko, and simplified Reddy theories are presented. In order to develop a single corotational finite element that incorporates the kinematics of all three theories, the unified linear finite element model of beams developed by Reddy (Comm. Numer. Meth. Eng. 1997; 13:495–510) is included in the formulation. An incremental iterative technique based on the Newton–Raphson method is employed for the solution of the non-linear equilibrium equations. Numerical examples that demonstrate the efficiency and large rotation capability of the corotational formulation are presented. The element is validated by comparisons with exact and/or approximate solutions available in the literature. Very good agreement is found in all cases. Copyright © 2005 John Wiley & Sons, Ltd.

A corotational finite element formulation for the analysis of planar beams

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.

K. Y. Lam

Papers

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