Accuracy of Finite Element Solutions for Flexible Beams Using Corotational Formulation

Abstract

According to the polar decomposition theorem, the total deformation is decomposed into the rigid body deformation and the relative deformation. On the basis of this theorem, the relative deformation is described by using the moving coordinate system, referred to as the corotational formulation. This formulation has often been used in the finite element analysis for geometrically nonlinear problems of flexible beams. The use of the corotational formulation is motivated by the assumption of small strains in the beam. A linear theory or a beam-column theory has often been introduced for describing the relative deformation so that a simple expression for the strain energy function is obtained. In spite of using the small-strain assumption, satisfactory numerical results have been obtained by increasing the number of elements. The accuracy of finite element solutions, however, has not fully been discussed from a theoretical point of view. Goto, Hasegawa and Nishino (1984), Goto, Kasugai and Nishino (1987) and Iura (1994) have discussed the accuracy of finite element solutions for Bernoulli-Euler’s beam. Iura (1994) has pointed out that even if the linear theory is used for describing the relative deformation, the numerical solutions obtained converge to the solutions of the finite-strain beam theory as the number of elements is increased. In the case of Timoshenko’s beam, however, there have been few studies to ascertain the accuracy of finite element solutions.