The Variational Principle for Multi-Layer Timoshenko Beam Systems Based on the Simplified

XU Xiao-jian; DENG Zi-chen

doi:10.3879/j.issn.1000-0887.2016.03.002

Volume 37 Issue 3

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XU Xiao-jian, DENG Zi-chen. The Variational Principle for Multi-Layer Timoshenko Beam Systems Based on the Simplified[J]. Applied Mathematics and Mechanics, 2016, 37(3): 235-244. doi: 10.3879/j.issn.1000-0887.2016.03.002

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The Variational Principle for Multi-Layer Timoshenko Beam Systems Based on the Simplified

doi: 10.3879/j.issn.1000-0887.2016.03.002

XU Xiao-jian^{1, 2},
DENG Zi-chen²

¹Key Laboratory for Special Area Highway Engineering of Ministry of Education, School of Highway, Chang’an University, Xi’an 710064, P.R.China;²Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an 710072, P.R.China

Funds: The National Natural Science Foundation of China(11372252;11502202)

Received Date: 2015-11-24
Rev Recd Date: 2015-12-29
Publish Date: 2016-03-15

Abstract

Abstract

The mechanical properties of member materials exhibit notable size effects when the characteristic sizes of the members are comparable to their instinct length parameters. A variational formulation of the nanosize multi-layer Timoshenko beam problem was developed via the semi-inverse method within the context of the simplified strain gradient theory. This method was fit for determining all the possible low-order and high-order boundary condtions directly from the governing equations of the system, according to the minimum total potential energy principle. In turn, the Rayleigh solutions of buckling load and free vibration frequencies of the simply supported beam system were given. The numerical simulations indicate the prominent effects of the instinct length parameters and aspect ratios on the free vibration frequencies of the double-layer beam systems. As a possible benchmark for the later numerical studies with the transfer matrix method or the finite element method, the present Rayleigh solutions of buckling load and free vibration frequencies of the multi-layer beam systems will make good sense.
- strain gradient theory,
- buckling,
- vibration,
- variational principle

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References(17)

References

[1]	Peddieson J, Buchanan G R, McNitt R P. Application of nonlocal continuum models to nanotechnology[J]. International Journal of Engineering Science,2003,41(3/5): 305-312.
[2]	Wang K F, Wang B L, Kitamura T. A review on the application of modified continuum models in modeling and simulation of nanostructures[J]. Acta Mechanica Sinica,2015: 1-18. doi: 10.1007/s10409-015-0508-4.
[3]	Adali S. Variational principles for transversely vibrating multiwalled carbon nanotubes based on nonlocal Euler-Bernoulli beam model[J]. Nano Letters,2009,9(5): 1737-1741.
[4]	Adali S. Variational formulation for buckling of multi-walled carbon nanotubes modelled as nonlocal Timoshenko beams[J]. Journal of Theoretical and Applied Mechanics,2012,50(1): 321-333.
[5]	Kucuk I, Sadek I S, Adali S. Variational principles for multiwalled carbon nanotubes undergoing vibrations based on nonlocal Timoshenko beam theory[J]. Journal of Nanomaterials,2010,2010(3): 461252. doi: 10.1155/2010/461252.
[6]	李明, 郑慧明. 小尺度对振动简支单层碳纳米管边界条件的影响[J]. 固体力学学报, 2014,35(S): 6-8.(LI Ming, ZHENG Hui-ming. Small scale effect on boundary conditions of vibrating simply supported single-walled carbon nanotubes[J]. Chinese Journal of Solid Mechanics,2014,35(S): 6-8.(in Chinese))
[7]	姚征, 郑长良. 积分形式非局部本构关系的界带分析方法[J]. 应用数学和力学, 2015,36(4): 362-370.(YAO Zheng, ZHENG Chang-liang. Inter-belt analysis of the integral-form nonlocal constitutive relation[J]. Applied Mathematics and Mechanics, 2015,36(4): 362-370.(in Chinese))
[8]	HE Ji-huan. Variational approach to (2+1)-dimensional dispersive long water equations[J]. Physics Letters A,2005,335(2/3): 182-184.
[9]	Kumar D, Heinrich C, Waas A M. Buckling analysis of carbon nanotubes modeled using nonlocal continuum theories[J]. Journal of Applied Physics,2008,103(7): 073521.
[10]	LI Xian-fang, WANG Bao-lin, LEE Kang-yong. Size effects of the bending stiffness of nanowires[J]. Journal of Applied Physics,2009,105(7): 074306.
[11]	WANG Bing-lei, ZHAO Jun-feng, ZHOU Shen-jie. A micro scale Timoshenko beam model based on strain gradient elasticity theory[J]. European Journal of Mechanics—A/Solids,2010,29(4): 591-599.
[12]	Nojoumian M A, Salarieh H. Comment on “A micro scale Timoshenko beam model based on strain gradient elasticity theory”[J]. European Journal of Mechanics—A/Solids,2013. doi: 10.1016/j.euromechsol.2013.12.003.
[13]	Challamel N. Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams[J]. Composite Structures,2013,105: 351-368.
[14]	XU Xiao-jian, DENG Zi-chen. Variational principles for the buckling and vibration of MWCNTs modeled by strain gradient theory[J]. Applied Mathematics and Mechanics(English Edition),2014,35(9): 1115-1128.
[15]	KONG Sheng-li, ZHOU Shen-jie, NIE Zhi-feng, WANG Kai. Static and dynamic analysis of micro beams based on strain gradient elasticity theory[J]. International Journal of Engineering Science,2009,47(4): 487-498.
[16]	Akgz B, Civalek . Analysis of micro-sized beams for various boundary conditions based on the strain gradient elasticity theory[J]. Archive of Applied Mechanics,2012,82(3): 423-443.
[17]	WANG Li-feng, GUO Wan-lin, HU Hai-yan. Group velocity of wave propagation in carbon nanotubes[J]. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science,2008,464(2094): 1423-1438.