Comparison of natural characters between buckling Timoshenko and Euler-Bernoulli beams under the axial force

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Journal of Vibration and Shock ›› 2022, Vol. 41 ›› Issue (24) : 33-40.

JING Jie，MAO Xiaoye，DING Hu，CHEN Liqun#br#

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Abstract

Natural characters of the buckling Timoshenko beam under the axial force is investigated comparing with the Euler-Bernoulli one for the first time. Firstly, the partial-differential-integral governing equation of the Timoshenko beam with simply supported boundaries is established. The critical buckling axial force and the buckling non-trivial configuration are derived analytically. The stability of the nontrivial configuration is discussed together with the Euler-Bernoulli beam. Meanwhile, the influence of the parameters on the buckling bifurcation is discussed. The Galerkin truncation method is applied to calculate the frequencies of the two models under their supercritical axial force. It is found that, after buckling, the frequency of the Timoshenko beam is always larger than that of the Euler-Bernoulli beam, for the critical axial force of the Timoshenko one is smaller. Part of the vibration energy of the Timoshenko beam is stored in the rotational deformation of the cross-section, which softens the bending stiffness. This softening effect is more obvious for higher order frequencies, which makes higher order frequencies more different than the fundamental frequency while comparing with the Euler-Bernoulli beam. Also, during the truncation, it can be found that higher order frequencies are no longer affected by the axial force in the supercritical region, except for the first order one. For beams with large slenderness ratio, the smaller the difference in the natural frequencies will be. This further illustrates the necessity of using the Euler-Bernoulli beam for slender structures and the Timoshenko beam for those short and thick.

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Timoshenko beam / buckling / axial force / natural frequency / truncation

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JING Jie，MAO Xiaoye，DING Hu，CHEN Liqun. Comparison of natural characters between buckling Timoshenko and Euler-Bernoulli beams under the axial force[J]. Journal of Vibration and Shock, 2022, 41(24): 33-40

References

[1] 陈红永, 李上明. 轴向运动梁在轴向载荷作用下的动力学特性研究[J]. 振动与冲击, 2016, 35(19): 75-80.
Chen hongyong, Li shangming. Dynamic characteristics of an axially moving Timoshenko beam under axial loads[J]. Journal of vibration and shock, 2016, 35(19): 75-80.
[2] 许醇义. 轴向力对弯曲振动的影响[J]. 苏州科技大学学报(工程技术版), 1994, 7(1): 21-5.
Xu Chunyi. The influence of axial force on bending vibration[J]. Journal of Suzhou University of Science and Technology (Engineering Technology Edition), 1994, 7(1): 21-5.
[3] 杜绍洪. 梁自由横振动方程的有限差分方法[J]. 重庆交通大学学报(自然科学版), 2012, 31(1): 6-10.
Du shaohong. Finite difference method of free transverse Vibration equations of beams[J]. Journal of Chongqing Jiaotong University( Natural Science), 2012, 31(1): 6-10.
[4] Ma Y, Chen G, Wu X. Modal and Fatigue Life Analysis on Beam with Multiple Cracks Subjected to Axial Force[J]. Transactions of Nanjing University of Aeronautics and Astronautics, 2018, 35(5): 874-82.
[5] Karimi A.H, Shadmani M. Nonlinear vibration analysis of a beam subjected to a random axial force[J]. Archive of Applied Mechanics, 2019, 89(1):385-402.
[6] Li X.Y, Wang X.H, Chen Y.Y, et al. Bending, Buckling and Free Vibration of an Axially Loaded Timoshenko Beam with Transition Parameter: Direction of Axial Force[J]. International Journal of Mechanical Sciences, 2020, 176(8): 105545.
[7] Liu H, Hong J, Zhang D. Bending and vibration of a discontinuous beam with a curvic coupling under different axial forces[J]. Frontiers of Mechanical Engineering, 2020, 15(1):417-29.
[8] 李世荣,杨静宁. 可伸长变截面杆的弹性过屈曲模型及其数值解[J]. 计算力学学报, 2000, 17(1): 114-8.
Li Shirong, Yang Jingning. Elastic overflexion model of an extensible variable-section rod and its numerical solution[J]. Journal of Computational Mechanics, 2000, 17(1): 114-8.
[9] Mao X-Y, Ding H, Lim C W, et al. Super-harmonic resonance and multi-frequency responses of a super-critical translating beam[J]. Journal of Sound and Vibration, 2016, 385: 267-83.
[10] 李世荣, 周又和, 滕兆春. 加热弹性梁在热过屈曲构形附近的自由振动[J]. 振动与冲击, 2005, (6): 43-7.
Li Shirong, Zhou Youhe, Teng Zhaochun. Free vibration of heated elastic beams in the vicinity of thermal overbending configurations[J]. Journal of vibration and shock, 2005, (6): 43-7.
[11] Li X-F and Lee K Y. Effects of engesser’s and Haringx’s Hypotheses on buckling of Timoshenko and Higher-Order Shear-Deformable columns[J]. Journal of Engineering Mechanics ,2018, 144(1): 04017150.
[12] Ziane N, Ruta G, Meftah S.A, et al. Instances of mixed buckling and post-buckling of steel RHS beams[J]. International Journal of Mechanical Sciences, 2020, 190: 106013.
[13] Einafshar N, Lezgy-Nazargah M, Beheshti-Aval S B. Buckling, post-buckling and geometrically nonlinear analysis of thin-walled beams using a hypothetical layered composite cross-sectional model[J]. Acta Mechanica, 2021, 232(7):2733-2750.
[14] Ferreira F, Tsavdaridis K.D, Martins C.H, et al. Buckling and Post-Buckling Analyses of Composite Cellular Beams[J]. Composite Structures, 2021, 262: 113616.
[15] 滕兆春, 李世荣. 轴向力作用下非对称支承Euler—Bernoulli梁在过屈曲前后的横向自由振动[J]. 振动与冲击, 2004, (4): 88-90.
Teng Zhaochun, Li Shirong. Pre-buckling and post-buckling transverse free vibration of an asymmetrically supported Euler-bernoulli beam subjected to axial force[J]. Journal of vibration and shock, 2004, (4): 88-90.
[16] 刘吉源, 戈新生, 陈立群. 轴向力作用下Timoshenko梁的横向振动[J]. 北京机械工业学院学报(综合版), 2000(03):56-59.
Liu Jiyuan, Ge Xinsheng, Chen Liqun. The crosswise vibration of Timoshenko beam under axial forces[J]. Journal of Beijing Institute of Machinery Industry(Comprehensive Edition), 2000(03):56-59.
[17] Ding H, Tan X, Dowell E.H. Natural frequencies of a super-critical transporting Timoshenko beam[J]. European Journal of Mechanics - A/Solids, 2017, 66: 79-93.
[18] 王乐, 余慕春. 轴力对自由边界Timoshenko梁横向动特性影响研究[J]. 兵器装备工程学报, 2018, (3): 36-9.
Wang Le, Yu Muchun. Effect of axial forces on the lateral vibration characteristics of Timoshenko beam under boundary condition[J]. Journal of Ordnance Equipment Engineering, 2018, (3): 36-9.
[19] Khodabakhsh R, Saidi A.R, Bahaadini R. An analytical solution for nonlinear vibration and post-buckling of functionally graded pipes conveying fluid considering the rotary inertia and shear deformation effects[J]. Applied Ocean Research, 2020, 101:102277.
[20] Mao X-Y, Ding H, Chen L-Q. Forced vibration of axially moving beam with internal resonance in the supercritical regime[J]. International Journal of Mechanical Sciences, 2017, 131-132: 81-94.
[21] Tan X, Mao X-Y, Ding H, et al. Vibration around non-trivial equilibrium of a supercritical Timoshenko pipe conveying fluid[J]. Journal of Sound and Vibration, 2018, 428: 104-18.