Dynamic multi-patch isogeometric analysis of planar Euler–Bernoulli beams

Abstract

This study presents a novel isogeometric Euler-Bernoulli beam formulation for in-plane dynamic analysis of multi-patch beam structures. The kinematic descriptions involve only displacements of the beam axis, which are approximated by non-uniform rational B-spline (NURBS) curves. Translational displacements of the control points are here considered as control variables. The motivation of this work is to propose a penalty-free method to handle in-plane dynamic analysis of multi-patch beam structures. A simple relation between cross-sectional rotations at the ends of the beams and control variables is derived, allowing the incorporation of the end rotations as degrees of freedom. This improved setting can straightforwardly tackle beam structures with many rigid multi-patch connections, a major challenging issue when using existing isogeometric Euler-Bernoulli beam formulations. Additionally, rotational boundary conditions are conveniently prescribed. Numerical examples with complicated beam structures such as circular arches and frames with kinks are considered to show the accuracy and performance of the developed formulation. The computed results are verified with those derived from the conventional finite element method, and the superior convergence properties of the proposed formulation are illustrated. We additionally discuss about the possible extension of the present approach to spatial beam structures.

Modal Analysis of Beam Oscillation

“…First, it is necessary to define several valid conditions and relationships, which were defined in publication by the authors [8]: -Moment of inertia and shear deformation are neglected.…”

“…The considered beam is bend in the transverse direction with a length 1 and a rectangular crosssection A(x) as shown in Figure 4. The flexural stiffness of the beam is EJ(x), where E is the Young's modulus and J(x) is the quadratic moment of the cross section around the y-axis [8]. From the Euler-Bernoulli theory of beams, the relation for the bending moment M(x, t) and the deflection in the direction of the z-axis w(x, t) is applied [8]: results are shown in Table 2.…”

“…By successive modifications of Equation 1, we finally obtain a relation for the analytical calculation of natural frequencies in the form [8]:…”

“…Thanks to this, it is possible to implement several system optimizations, such as geometric optimization, material optimization, shape optimization or addition of supporting parts, etc. The purpose of this optimization is to eliminate dangerous oscillations [8].…”

An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods

“…The G 1 implicit conforming formulation proposed by Greco and Cuomo [6, 7], Greco [8], and Greco et al [9, 10] is used in this work. A generalization of this idea to the IGA for the static non-linear case can be found in Vo et al’s study [55, 56] while for the dynamic linear case in another Vo et al’s study [55]. In order to avoid locking phenomena and to obtain a more efficient and robust non-linear numerical model, a mixed formulation based on the Hellinger–Reissner (HR) principle is considered.…”