Analysis of large displacements and large rotations of three-dimensional beams by using small strains and unit vectors

Abstract

SUMMARYThe large displacements and large rotations of three-dimensional beams are analysed by using small strain theory and unit vectors of the cross-sections without using complicated three-dimensional rotation vectors. The accurate directions of the unit vectors of the cross-sections are computed by monotonically reducing the moment constraint error by the iterative scheme. The computation at each iteration consists of two basic steps. The ®rst step is to compute the unit vectors of the cross-sections with the previous moment constraint error, and the second step is to compute the displacements and moment constraint error by solving the linear equations of equilibrium formulated with the given unit vectors of the cross-sections. The convergence of the iterative scheme is analysed, and the accuracy and eciency of the solution are demonstrated by the numerical examples. #

A mixed co‐rotational formulation of 2D beam element using vectorial rotational variables

“…The deformed shapes of the cantilever at different end moment levels are depicted in Figure 4, the cantilever experiences large displacement and large rotation, and the end rotation arrives at 2 , however, the proposed procedure still demonstrates satisfying efficiency and reliability. Urthaler and Reddy [23] and Lee [27] also solved similar problems (but they did not present the geometry and material properties of the problems), and Lee's procedure [27] seems to be quite efficient in solving similar problem, however, it cannot cope with buckling and post-buckling problem, and will run into computational difficulty once locking phenomena in thin beam element occur. This cantilever belongs to a thin beam, while, the proposed finite element procedure demonstrates satisfying efficiency in overcoming membrane/shear locking problems.…”

“…where u 1 (24) and (26)), respectively; k keeps the same value in ith loading increment. The incremental equations are the same as Equations (22) and (25)- (27). 56 Z. X. LI In the external force vector P of Equation (22), the components with respect to vectorial rotational variables are not moment, so in coping with a concentrated moment at a node, it will be transformed into an equivalent load with respect to the vectorial rotational variable in advance.…”

A Mixed Co-Rotational 3d Beam Element Formulation for Arbitrarily Large Rotations

“…Urthaler & Reddy [13] and Lee [26] had also solved a similar problem, but they did not present the geometry and material properties of the cantilever beam. To illuminate the computational efficiency and accuracy of the present beam element using assumed membrane strains and shear strains (for convenience, it is abbreviated as AM+AS element), 4 cantilever beams with the same width (b=0.5) and different thickness values (h=0.2, 0.1, 0.05, 0.01) are solved respectively.…”

A co-rotational formulation for 3D beam element using vectorial rotational variables

“…5, it experiences large displacement and large rotation, and its end rotation arrives at 2π under M = 2π EI L , however, the proposed procedure still demonstrates satisfying efficiency and reliability. Urthaler and Reddy (2005) and Lee (1997) had also solved similar problems (but they did not present the geometry and material properties of the problems), and Lee's procedure (Lee 1997) seems to be quite efficient in solving similar problems, however, it can not cope with buckling and post-buckling problem, and will run into computational difficulty once locking phenomena in thin beam element occur.…”