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| Comparison of solutions from different displacement boundary conditions at fixed end of cantilever beams |
| YANG Lian-zhi1,2, ZHANG Liang-liang2,3, YU Lian-ying2, SHANG Lan-ge2, GAO Yang2, WANG Min-zhong4 |
| 1. Civil and Environmental Engineering School, University of Science and Technology Beijing, Beijing 100083, China; 2. College of Science, China Agricultural University, Beijing 100083, China; 3. College of Engineering, China Agricultural University, Beijing 100083, China; 4. Department of Mechanics and Aerospace Engineering, Peking University, Beijing 100871, China |
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Abstract To obtain the influence of different displacement boundary conditions for the fixed end on analytical solutions of a cantilever beam, three load cases for a cantilever beam were investigated, which were a transverse shear force at the free end, a uniformly distributed load at the top surface, and a linearly distributed load at the top surface, respectively. Analytical solutions were given for Levinson theory, Timoshenko theory, and the elastic theory by using the conventional displacement boundary condition and the boundary condition through least squares method at the fixed end of the beam, and were compared with the solutions by finite element method. It is shown that the solutions from Timoshenko theory by using both the conventional displacement boundary condition and the condition through least squares method are the same; Levinson theory and the elastic theory by using the boundary condition through least squares method provide better results than those by using the conventional boundary condition. With an increase in the order of the load, the superiority becomes more and more obvious．
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Published: 01 November 2014
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悬臂梁固定端不同位移边界条件下解的对比
为了获得不同的悬臂梁固定端位移边界处理方式对结果的影响,针对悬臂梁承受3种载荷的情况：自由端受切向力,上表面受均布载荷和线性分布载荷,给出悬臂梁固定端利用传统边界条件和最小二乘法处理边界时,Timoshenko梁理论、Levinson梁理论和弹性力学理论的解析解,与有限元计算结果对比.结果表明,Timoshenko梁理论采用传统位移边界和最小二乘法处理边界的结果一致,采用最小二乘法处理边界获得的Levinson梁理论和弹性力学理论的解明显优于传统位移确定方法,且这种优势随着载荷阶次的增加而越加明显.
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| 1］ WANG C M, REDDY J N. Shear deformable beams and plates relations wiht classical solutions ［M］. New York: Elsevier, 2000: 117．
［2］ TIMOSHENKO S P. On the correction for shear of the differential equation for transverse vibration of prismatic bars \[J\]. Philosophical Magazine Series 6, 1921, 41: 744-746.
［3］ LEVINSON M. A new rectangular beam theory ［J］. Journal of Sound and Vibration, 1981, 74(1): 81-87.
［4］ TIMOSHENKO S P, GOODIER J C. Theory of elasticity ［M］. New York: McGraw-Hill, 1970: 44-45.
［5］ GERE J M, Goodno B J. Mechanics of Materials (8th edition) ［M］. USA: Cengage Learning, 2012: 730-735．
［6］ 于涛, 仲政. 均布载荷作用下功能梯度悬臂梁弯曲问题的解析解［J］.固体力学学报, 2006, 27(1): 15-20．
YU Tao, ZHONG Zheng. A general solution of a clamped functionally graded cantilever-beam under uniform loading ［J］. Acta Mechanica Solida Sinica, 2006, 27(1): 15-20．
［7］ 杨永波, 石志飞, 陈盈. 线性分布载荷作用下梯度功能压电悬臂梁的解［J］. 力学学报, 2004, 36(3): 305-310．
YANG Yong-bo, SHI Zhi-fei, CHEN Ying. Piezoelectric cantilever actuator subjected to a linearly distributed loading ［J］. Acta Mechanica Sinica, 2004, 36(3): 305-310．
［8］ SHI Zhi-Fei, CHEN Ying. Functionally graded piezoelectric cantilever beam ［J］. Archive of Applied Mechanics, 2004, 74: 237-247．
［9］ HUANG De-jin, DING Hao-Jiang, CHEN Wei-Qiu. Analysis of functionally graded and laminated piezoelectric cantilever actuators subjected to constant voltage［J］. Smart Materials and Structures, 2008, 17: 065002．
［10］ 王敏中. 关于“平面弹性悬臂梁剪切挠度问题［J］. 力学与实践,2004,26(6)：66-68．
WANG Min-zhong. About “Problems on the shear deflection of a planer elastic cantilever beam” ［J］. Mechanics in Engineering, 2004, 26(6): 66-68．
［11］ 黄文彬.平面弹性悬臂梁剪切挠度问题［J］.力学与实践,1997,19(2): 61-62．
HUANG Wen-bin. Problems on the shear deflection of a planer elastic cantilever beam ［J］. Mechanics in Engineering, 1997, 19(2): 61-62．
［12］ 黄文彬.平面弹性悬臂梁剪切挠度的进一步研究［J］.力学与实践, 1998, 20(5): 65．
HUANG Wen-bin. Further study on the shear deflection of a planer elastic cantilever beam ［J］ . Mechanics in Engineering, 1998, 20(5):65．
［13］ 唐玉花,王鑫伟.关于“平面弹性悬臂梁剪切挠度问题”的进一步研究［J］.力学与实践,2008,30:97-99．
TANG Yu-hua, WANG Xin-wei. Further study on “Problems on the shear deflection of a planer elastic cantilever” ［J］. Mechanics in Engineering, 2008, 30: 97-99．
［14］ COWPER G R. The shear coefficients in Timoshenkos beam theory ［J］. ASME Journal of Applied Mechanics, 1966, 33: 335-340. |
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