Structural Analysis of Unsymmetric Laminated Composite Timoshenko Beam Subjected to Moving Load

Document Type: English


Department of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran.


The structural analysis of an infinite unsymmetric laminated composite Timoshenko beam over Pasternak viscoelastic foundation under moving load is studied. The beam is subjected to a travelling concentrated load. Closed form steady state solutions, based on the first-order shear deformation theory (FSDT) are developed. In this analysis, the effect of bend-twist coupling is also evaluated. Selecting of an appropriate displacement field for deflection of the composite beam and using the principle of total minimum potential energy, the governing differential equations of motion are obtained and solved using complex infinite Fourier transformation method. The dynamic response of unsymmetric angle-ply laminated beam under moving load has been compared with existing results in the literature and a very good agreement is observed. The results for variation of the deflection, bending moment, shear force and bending stress are presented. In addition, the influences of the stiffness, shear layer viscosity of foundation, velocity of the moving load and also different thicknesses of the beam on the structural response are studied.


[1]        D. G. Duffy, "The response of an infinite railroad track to a moving, vibrating mass," Journal of Applied Mechanics, vol. 57, pp. 66-73, 1990.

[2]        C. Cai, Y. Cheung, and H. Chan, "Dynamic response of infinite continuous beams subjected to a moving force—an exact method," Journal of Sound and Vibration, vol. 123, pp. 461-472, 1988.

[3]        S. Mackertich, "The response of an elastically supported infinite Timoshenko beam to a moving vibrating mass," The Journal of the Acoustical Society of America, vol. 101, pp. 337-340, 1997.

[4]        V.-H. Nguyen and D. Duhamel, "Finite element procedures for nonlinear structures in moving coordinates. Part II: Infinite beam under moving harmonic loads," Computers & Structures, vol. 86, pp. 2056-2063, 2008.

[5]        S. P. Patil, "Response of infinite railroad track to vibrating mass," Journal of engineering mechanics, vol. 114, pp. 688-703, 1988.

[6]        R. U. A. Uzzal, R. B. Bhat, and W. Ahmed, "Dynamic response of a beam subjected to moving load and moving mass supported by Pasternak foundation," Shock and Vibration, vol. 19, pp. 205-220, 2012.

[7]        H. Ding, K.-L. Shi, L.-Q. Chen, and S.-P. Yang, "Dynamic response of an infinite Timoshenko beam on a nonlinear viscoelastic foundation to a moving load," Nonlinear Dynamics, vol. 73, pp. 285-298, 2013.

[8]        A. Mallik, S. Chandra, and A. B. Singh, "Steady-state response of an elastically supported infinite beam to a moving load," Journal of Sound and Vibration, vol. 291, pp. 1148-1169, 2006.

[9]        S. Lu and D. Xuejun, "Dynamic analysis to infinite beam under a moving line load with uniform velocity," Applied mathematics and mechanics, vol. 19, pp. 367-373, 1998.

[10]      A. D. Kerr, "Elastic and viscoelastic foundation models," Journal of Applied Mechanics, vol. 31, pp. 491-498, 1964.

[11]      Y.-H. Chen, Y.-H. Huang, and C.-T. Shih, "Response of an infinite Timoshenko beam on a viscoelastic foundation to a harmonic moving load," Journal of Sound and Vibration, vol. 241, pp. 809-824, 2001.

[12]      L. Sun, "A closed-form solution of a Bernoulli-Euler beam on a viscoelastic foundation under harmonic line loads," Journal of Sound and vibration, vol. 242, pp. 619-627, 2001.

[13]      S. Verichev and A. Metrikine, "Instability of a bogie moving on a flexibly supported Timoshenko beam," Journal of sound and vibration, vol. 253, pp. 653-668, 2002.

[14]      T. Liu and Q. Li, "Transient elastic wave propagation in an infinite Timoshenko beam on viscoelastic foundation," International journal of solids and structures, vol. 40, pp. 3211-3228, 2003.

[15]      M. Kargarnovin and D. Younesian, "Dynamics of Timoshenko beams on Pasternak foundation under moving load," Mechanics Research Communications, vol. 31, pp. 713-723, 2004.

[16]      M. Kargarnovin, D. Younesian, D. Thompson, and C. Jones, "Response of beams on nonlinear viscoelastic foundations to harmonic moving loads," Computers & Structures, vol. 83, pp. 1865-1877, 2005.

[17]      G. Muscolino and A. Palmeri, "Response of beams resting on viscoelastically damped foundation to moving oscillators," International Journal of Solids and Structures, vol. 44, pp. 1317-1336, 2007.

[18]      F. F. Çalım, "Dynamic analysis of beams on viscoelastic foundation," European Journal of Mechanics-A/Solids, vol. 28, pp. 469-476, 2009.

[19]      M. Kadivar and S. Mohebpour, "Finite element dynamic analysis of unsymmetric composite laminated beams with shear effect and rotary inertia under the action of moving loads," Finite elements in Analysis and Design, vol. 29, pp. 259-273, 1998.

[20]      M. Rezvani and K. M. Khorramabadi, "Dynamic analysis of a composite beam subjected to a moving load," Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, vol. 223, pp. 1543-1554, 2009.

[21]      M. J. Rezvani, M. H. Kargarnovin, and D. Younesian, "Dynamic analysis of composite beam subjected to harmonic moving load based on the third-order shear deformation theory," Frontiers of Mechanical Engineering, vol. 6, pp. 409-418, 2011.

[22]      H. Abramovich and A. Livshits, "Free vibrations of non-symmetric cross-ply laminated composite beams," Journal of sound and vibration, vol. 176, pp. 597-612, 1994.

[23]      J. N. Reddy, Mechanics of laminated composite plates and shells: theory and analysis: CRC press, 2004.

[24]      A. David Wunsch, "Complex Variables With Applications [M]," ed: Reading, Mass: Addison-Wesley Publishing Company, 1994.

[25]      L. Fryba, Vibration of solids and structures under moving loads: Thomas Telford, 1999.