High-Order Thin-Walled Curved Beam Elements With the Absolute Nodal Coordinate Formulation
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摘要: 薄壁曲梁的截面易出现畸变和翘曲现象，为了能够描述其在大转动、大变形情况下的截面变形行为，提出了一种基于绝对节点坐标法的高阶梁单元. 借鉴Taylor级数展开模式构建了全局位置矢量场，通过增加单元横向节点的方式，避免了高阶导数几何意义不明确带来的困扰. 基于非线性连续介质力学理论和坐标变换策略，推导了薄壁曲梁单元的广义弹性力表达式. 通过与有限元软件中的薄壳单元进行比较，验证了该薄壁梁单元的准确性.Abstract: The cross sections of thin-walled curved beams are susceptible to distortion and warping phenomena. To describe the cross-section deformation behaviors under large rotations and deformations, a high-order beam element based on the absolute nodal coordinate formulation was proposed. The global position vector field was constructed with the Taylor series expansion method, and the issue arising from the ambiguous geometrical significance of high-order derivatives was obviated with the increase of the number of transverse nodes in the element. Based upon the nonlinear continuum mechanics theory and the coordinate transformation strategy, the generalized elastic force expression for the thin-walled curved beam element was derived. The accuracy of the proposed thin-walled beam element was validated through comparison with a thin-shell element in ABAQUS.
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图 1 薄壁曲梁上任意一点的位置矢量和薄壁曲梁的正交曲线坐标系
Figure 1. The position vector of any point on a thin-walled curved beam and the orthogonal curve coordinate system of the thin-walled curved beam
图 2 前八阶梁单元的节点位置坐标
Figure 2. The nodal position coordinates of the first eight-order beam elements
图 3 薄壁矩形截面悬臂梁的形状示意图及其模态振型
Figure 3. Schematic diagram of a cantilever beam with a thin-walled rectangle cross-section and its modes
图 4 薄壁矩形截面悬臂梁的形状及其变形应力图
注 为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.
Figure 4. Shape of a cantilever beam with thin-walled rectangular cross-section and deformation stresses
图 5 集中力作用下梁截面的变形
Figure 5. The cross-sectional beam deformation under the concentrated force
图 6 三角形截面梁受力示意图及其面外翘曲变形
Figure 6. A beam with a triangular cross section under a force couple and its warping deformation
图 7 不同截面梁单摆分析图
Figure 7. The analytical diagram of a single pendulum featuring beams with varying cross-sectional geometry
图 8 正交各向异性薄壁曲梁的形状及其变形图
Figure 8. The orthogonal anisotropic thin-walled curved beam shape and the deformation diagram
表 1 薄壁矩形截面梁的固有频率(误差)(单位：Hz)
Table 1. Frequencies (errors) of the beam with a thin-walled rectangular cross-section (unit: Hz)
kth-order 1st sweeping 1st flapping 2nd sweeping 2nd flapping 1st torsion 2nd torsion 1 46.14(27.1%) 49.73(22.4%) 220.61(26.2%) 236.81(22.7%) 1 251.57(25.6%) 1 435.59(23.2%) 2 41.57(14.5%) 42.89(5.6%) 198.13(13.39%) 204.80(6.1%) 1 138.61(14.31%) 1 259.76(8.1%) 3 39.74(9.4%) 41.72(2.7%) 186.37(6.6%) 194.81(0.9%) 1 098.45(10.28%) 1 195.15(2.5%) 4 38.62(6.3%) 40.41(0.5%) 180.48(3.2%) 191.69(0.6%) 1 065.15(6.9%) 1 185.06(1.7%) ABAQUS 36.30 40.62 174.73 192.95 966.04 1 165.10 
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