NONLINEAR ANALYSIS OF TIMOSHENKO BEAM ELEMENT BASED ON IMPROVED COROTATIONAL FORMULATION
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摘要: 为提高空间Timoshenko梁单元非线性问题的计算精度，在共旋坐标法的基础上，提出了一种改进的Timoshenko梁单元几何非线性分析方法。利用虚功原理得到改进空间梁单元的刚度矩阵；使用有限质点法中的逆向运动思路计算单元局部坐标系下的刚体旋转矩阵；根据整体坐标系与局部坐标系之间旋转角度的转化以及微分关系，求得空间梁单元的切线刚度矩阵；编制了相应的有限元程序，对多个经典的大变形结构进行几何非线性分析。计算结果印证了该文所提出改进方法的正确性，同时与传统共旋坐标法相比，具有更高的精度。
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关键词:
- Timoshenko梁单元 /
- 共旋坐标法 /
- 逆向运动 /
- 非线性分析 /
- 切线刚度矩阵
Abstract: An improved geometrically nonlinear analysis method based on corotational formulation is proposed to improve the computation accuracy of nonlinear problems of spatial Timoshenko beam element. The stiffness matrix of the improved spatial beam element is obtained by using the principle of virtual work; The rigid-body rotation matrix of the element in local coordinate system is calculated by using the reverse motion approach in finite particle method; The tangent stiffness matrix of the spatial beam element is obtained according to the transformation and differential relationship of the rotation angle between the global coordinate system and the local coordinate system; A finite element program is made to analyze the geometric nonlinearity of several typical large deformation structures. The results show that the improved method proposed in this paper is correct and has higher accuracy than the traditional corotational formulation. -
图 3 利用改进共旋法进行静位移计算流程图
Figure 3. Flow chart of static displacement calculation using improved co-rotational formulation
表 1 悬臂梁承受集中荷载时的大挠度变形
Table 1. Cantilever's large deformation under concentrated load
$ P{L}^{2}/EI $ $ w/L $ $ u/L $ ${\theta }_{0}/{\rm rad}$ 解析解 本文 误差/(%) 解析解 本文 误差/(%) 解析解 本文 误差/(%) 0.2 0.0664 0.0665 0.1506 0.0027 0.0026 2.1852 0.0996 0.0996 0.0000 0.8 0.2495 0.2499 0.1603 0.0382 0.0381 0.2618 0.3791 0.3791 0.0000 1.6 0.4294 0.4305 0.2562 0.1186 0.1185 0.0843 0.6707 0.6711 0.0596 3.0 0.6033 0.6054 0.3481 0.2544 0.2544 0.0000 0.9860 0.9871 0.1116 4.0 0.6700 0.6728 0.4179 0.3289 0.3290 0.0304 1.1212 1.1227 0.1338 
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表 2 悬臂梁承受弯矩作用的大变形
Table 2. Cantilever’s large deformation under bending moment
$ k $ 水平位移
$ u $解析
解/m本文
解/mHAWC2-
30解/mHAWC2-
50解/m本文误
差/(%)HAWC2-
30误差/(%)HAWC2-
50误差/(%)竖直位移
$ v $解析
解/m本文
解/mHAWC2-
30解/mHAWC2-
50解/m本文误
差/(%)HAWC2-
30误差/(%)HAWC2-
50误差/(%)0.4 −2.4317 −2.4268 −2.4439 −2.4366 0.20 0.50 0.20 5.4987 5.5023 5.5042 5.4987 0.07 0.10 0.00 0.8 −7.6613 −7.6551 −7.7609 −7.6996 0.08 1.30 0.50 7.1978 7.2168 7.2194 7.2050 0.26 0.30 0.10 1.2 −11.5591 −11.5684 −11.6978 −11.6053 0.08 1.20 0.40 4.7986 4.8271 5.0145 4.8802 0.59 4.50 1.70 1.6 −11.8921 −11.9121 −12.0467 −11.9516 0.17 1.30 0.50 1.3747 1.3893 1.6868 1.5080 1.06 22.70 9.70 2.0 −10.0000 −10.0000 −10.5100 −10.2000 0.00 5.10 2.00 0.0000 0.0000 −0.0080 −0.0100 0.0000 −0.0080 −0.0100
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表 3 自由端位移和转角参数比较
Table 3. Comparison of tip displacements and rotations
方法对比及误差分析 解析解/m 经典共旋法解/m HAWC2解/m 本文解/m 共旋法误差/(%) 本文误差/(%) HAWC2误差/(%) x方向位移 −0.090 64 −0.090 13 −0.091 64 −0.090 09 0.56 0.61 1.10 y方向位移 −0.064 84 −0.063 20 −0.067 24 −0.064 76 2.53 0.12 3.70 z方向位移 1.229 98 1.229 50 1.229 98 1.229 18 0.04 0.06 0.00 x方向扭转角度 0.184 45 0.184 47 0.188 88 0.184 40 0.01 0.03 2.40 y方向旋转角度 −0.179 85 −0.179 87 −0.179 87 −0.179 84 0.01 0.01 0.01 z方向旋转角度 0.004 88 0.005 23 0.004 99 0.005 01 7.17 2.66 2.30
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表 4 自由端受力下曲梁端部位移比较
Table 4. Comparison of the curved beam tip displacements under a force applied at the free end
方法对比及
误差分析解析
解本文方
法解经典共
旋法解HAWC2
解本文
误差/(%)共旋法
误差/(%)HAWC2/
(%)$x/{\rm m}$ −23.7 −23.7 −23.8 −24.2 0.2 0.4 2.1 $y/{\rm m}$ −13.4 −13.6 −13.7 −13.8 1.3 2.2 3.1 ${\textit{z} }/{\rm m}$ 53.4 53.6 53.7 53.4 0.4 0.5 0.0
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