On the theory of a Cosserat point and sheer locking in thin beams

It is well known that with the assumption of constant strain elements, the Galerkin approach yields a numerical solution of the equilibrium equations of a beam with shear deformation which exhibits the unphysical feature of s hear locking. Here, it is shown that the numerical solution that is based o n the theory of a Cosserat point converges to the exact solution of the bea m theory even when the kinematics are consistent with the constant strain a ssumption and the beam thickness approaches zero. The main difference betwe en these two numerical approaches is the way they each determine the consti tutive equations (or stiffnesses) of the elements. The Galerkin approach de termines the stiffnesses of each element by integrating the constitutive eq uations for the beam assuming that the kinematic approximation is valid poi ntwise. In contrast, the constitutive equations of the Cosserat point are r elated to derivatives of a strain energy function and the constitutive cons tants are determined using a physical approach which matches the responses to simple shear and pure bending. The results indicate that the Cosserat ap proach takes full advantage of the reduced number of degrees of freedom use d to describe the beam. Copyright (C) 2001 John Wiley & Sons, Ltd.