A Spatial Euler-Bernoulli Beam Element for Rigid-Flexible Coupling Dynamic Analysis of Flexible Structures

Abstract

A two-node spatial beam element with the Euler-Bernoulli assumption is developed for the nonlinear dynamic analysis of slender beams undergoing arbitrary rigid motions and large deformations. During the analysis, the global displacement and rotation vectors with six degrees of freedom are selected as the nodal coordinates. In addition, the “shear locking” problem is avoided successfully since the beam cross-sections are always perpendicular to the current neutral axes by employing a special coupled interpolation of the centroid position and the cross-section orientation. Then a scheme is presented where the original transient strains representing the nodal forces are replaced by proposed average strains over a small time interval. Thus all the high frequencies can be filtered out and a corresponding equivalent internal damping will be produced in this new formulation, which can improve the computation performance of the proposed element for solving the stiff problem and evaluate the governing equations even by using the nonstiff ordinary differential equation solver. Finally, several numerical examples are carried out to verify the validation and efficiency of this proposed formulation by comparison with the analytical solutions and other research works.

Geometrically Exact Finite Element Formulations for Slender Beams: Kirchhoff–Love Theory Versus Simo–Reissner Theory

“…This limitation simplifies the theory considerably, and already the modeling of simple piecewise straight frames is difficult since no variables are available that determine the cross-section orientation required for kinematic constraints at beam-to-beam joints. The more recent contributions of Zhang and Zhao [154,155] allow for anisotropic cross-sections but still focus on initially straight beams (denoted as straight case in the second column of Table 2). However, later it will become clear that the first-order twist angle interpolation underlying these formulations might in general not allow for optimal spatial convergence rates considering the employed third-order centerline interpolation.…”

Coupling dynamic behaviors of spatial flexible beam with weak damping

“…It has been mentioned that the dynamic model (12) presented for the spatial flexible damping beam in Section 2 can be decoupled approximately into two sub-systems, Equations (13) and (14). The main purpose of the decoupling process is to reduce the computational work in the subsequent numerical simulation.…”

“…In our previous job [18], the transverse vibration of a damping continuous beam subjected to the moving load was investigated by the generalized multi-symplectic method [15]. Following the outline of our previous job, a generalized multi-symplectic scheme can be constructed for sub-system (14), as follows.…”

“…To reveal the influence mechanism of the damping on the dynamic behaviors of the spatial flexible beam, the transverse vibration of the flexible damping beam without the plane motion (that is, =̇= 0rad∕s in model (14) and the beam is not a spatial beam at all) is simulated (during this simulation, the cross section fixed located at the midpoint x = 0 and the same initial vibration speed obtained in the numerical experiment on the Case 1 is considered). Figure 3 shows the transverse displacement of the beam-end with c = 0.1 (to make the figure clearly readable, we select the data with time interval 200s), in which, the blue scatter diagram is the transverse displacement of the beamend when the plane motion is considered (Case 1) and the red curve is the transverse displacement of the beam-end without the plane motion.…”

“…During the simulation of the plane motion presented in Section 4.2, the transverse vibration process of the spatial flexible beam can be obtained simultaneously. Firstly, Case 1 and Case 2 are considered when the damping effect is neglected to further investigate the structure-preserving properties of the numerical scheme (18) from the view of energy because the energy transfer is the essence of the coupling between sub-system (13) and sub-system (14). For the absence of the damping, scheme (18) is multi-symplectic that satisfies the discrete multi-symplectic conservation law [22,24].…”