Journal of Sound and < ibration (2001) 243(3), 565}576 doi:10.1006/jsvi.2000.3416, available online at http://www.idealibrary.com on A TWO-DIMENSIONAL SHEAR DEFORMABLE BEAM FOR LARGE ROTATION AND DEFORMATION PROBLEMS M. A. OMAR AND A. A. SHABANA Department of Mechanical Engineering, ;niversity of Illinois at Chicago, 842 = est ¹ aylor Street, Chicago, I¸ 60607-7022, ;.S.A. (Received 13 July 2000, and in ,nal form 30 October 2000) 1. INTRODUCTION In Euler}Bernoulli beam theory it is assumed that the beam cross-section remains rigid and perpendicular to the neutral axis of the beam [1, 2]. In this theory, the e!ect of the shear deformation is neglected. In Timoshenko beam theory on the other hand, the cross-section does not remain perpendicular to the beam neutral axis. Nonetheless, the cross-section remains rigid in many models. A shear coe$cient is introduced in order to account for the shear deformation [3]. In this investigation, a two-dimensional shear-deformable beam element based on the non-incremental absolute nodal co-ordinate formulation [4] is developed. In this approach, only absolute co-ordinates and global slopes are used to de"ne the element nodal co-ordinates without the need for using in"nitesimal or "nite rotations. Using this co-ordinate representation with the appropriate element shape function, exact modelling of the rigid body dynamics can be achieved. Using the non-incremental absolute nodal co-ordinate formulation, the resulting mass matrix of the "nite element is a constant matrix and the centrifugal and Coriolis forces are identically equal to zero. A problem encountered in the implementation of the non-incremental absolute nodal co-ordinate formulation is the formulation of the elastic forces. Shabana et al. [4}8] proposed two methods for formulating the elastic forces of the two-dimensional beam element. In the "rst method, a local element co-ordinate system is introduced for the convenience of describing the element deformation. This approach leads to a complex expression for the elastic forces even when a linear elastic model is used. In the second method [9] a continuum mechanics approach is used to obtain the elastic forces without introducing the local element co-ordinate system. In this continuum mechanics approach, non-linear strain}displacement relationships must be used in order to obtain zero strain under an arbitrary rigid body motion. Nonetheless, the previous models developed using the continuum mechanics approach are based on Euler}Bernoulli beam theory which does not account for the shear deformation. It is the objective of this investigation to develop a model for the elastic forces for two-dimensional beam elements that accounts for shear deformation by using a general continuum mechanics approach without introducing a local element co-ordinate system. This new model relaxes the assumptions of Euler}Bernoulli and Timoshenko beam models and does not require the use of a shear coe$cient. 0022-460X/01/230565#12 $35.00/0  2001 Academic Press