A corotational finite element formulation for the analysis of planar beams

Abstract

SUMMARYAn e cient and accurate locking-free corotational beam ÿnite element for the analysis of large displacements and small-strain problems is developed in this paper. Three di erent ÿnite element models based on three di erent beam theories, namely, the Euler-Bernoulli, Timoshenko, and simpliÿed Reddy theories are presented. In order to develop a single corotational ÿnite element that incorporates the kinematics of all three theories, the uniÿed linear ÿnite 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 e ciency 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.

Nonlinear quasi‐static finite element formulations for viscoelastic Euler–Bernoulli and Timoshenko beams

“…In addition, it may also prove useful to extend the current formulation such that more pronounced geometric nonlinearities can be captured. In particular, the current formulation may be modified in the context of a corotational [36] finite element formulation.…”

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.…”

“…Surveys of the existing co-rotational finite element formulations were presented respectively by Stolarski et al [10], Crisfield and Moita [11], Yang et al [3], and Felippa and Haugen [12]. Recently, Urthaler and Reddy [13] developed three locking-free co-rotational planar beam element formulations by adopting respectively the Euler-Bernoulli, Timoshenko, and simplified Reddy theories in modelling of the element kinematic behaviour. Galvaneito and Crisfield [14] proposed an energy-conserving procedure for the implicit non-linear dynamic analysis of planar beam structures by using a form of co-rotational technique.…”

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

“…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.…”