Sensitivity Analysis of Vibration Response of Railway Structures to Velocity of Moving Load and Various Depth of Elastic Foundation

Document Type : Original Article

Authors

Department of Civil Engineering, University of Mohaghegh Ardabili, Ardabil, Iran

Abstract

Railway structures are one of the most important structures in transportation. So the lack of precise study of their dynamic behavior leads to irreparable damages. The significant factors contributing to the accurate analysis of the dynamic behavior of railways are the type of load and foundation used in it. In this study, an Euler-Bernoulli beam subjected to a moving load on a finite depth foundation is presumed. According to the feature of finite beams, just the dynamic equilibrium in the vertical direction is regarded. In this paper, by using equilibrium equations and considering the influence of soil and structure interaction, the physical problem is simulated and by using Fourier transform method the governing differential equations are obtained. Then, the mathematical model based on suggested models is expanded and verified. By assessing the efficiency of the recommended method, dynamic behavior of beam is specified and the deflection ratios for various foundations are illustrated. The sensitivity analysis is provided to study the influence of various parameters such as velocity of moving load, elastic foundation depth and damping. Eventually, by considering the sequences of shear waves, critical velocity, which is dependent on the mass ratio, and various kinds of damping, deflection shapes of the beam are attained for the different velocities of the moving load, and the effect of soil depth on the dynamic behavior of the beam is discussed. It is indicated that, foundation inertia leads to a considerable reduction in critical velocity and can also intensify the response of the beam.

Keywords


 
1. Timoshenko, S. "Method of analysis of statical and dynamical
stresses in rail." In Proceedings of the Second International
Congress for Applied Mechanics, Zurich Switzerland, (1926),
407-418. 
2. Kenney, J. T. "Steady-state vibrations of beam on elastic
foundation for moving load." Journal of Applied Mechanics,
Vol. 21, (1954), 359-364. 
3. Györgyi, József. "Frequency-dependent Geometrical Stiffness
Matrix for the Vibration Analysis of Beam Systems."
Periodica Polytechnica Civil Engineering, Vol. 25, No. 3-4
(1981), 151-163. 
4. Li, Wen L. "Free vibrations of beams with general boundary
conditions." Journal of Sound and Vibration, Vol. 237, No. 4
(2000), 709-725, Doi: 10.1006/jsvi.2000.3150 
5. Hilal, M. Abu, and H. S. Zibdeh. "Vibration analysis of beams
with general boundary conditions traversed by a moving
force." Journal of Sound and vibration, Vol. 229, No. 2
(2000), 377-388. 
6. Abu-Hilal, M. "Forced vibration of Euler–Bernoulli beams by
means of dynamic Green functions." Journal of sound and
vibration, Vol. 267, No. 2, (2003), 191-207. 
7. Kargarnovin, M. H., and D. Younesian. "Dynamics of
Timoshenko beams on Pasternak foundation under moving
load." Mechanics research communications, Vol. 31, No. 6,
(2004), 713-723. 
8. Ying, J., C. F. Lü, and W. Q. Chen. "Two-dimensional
elasticity solutions for functionally graded beams resting on
elastic foundations." Composite Structures, Vol. 84, No. 3,
(2008), 209-219. 
9. Mehri, B. A. H. M. A. N., A. Davar, and O. Rahmani.
"Dynamic Green function solution of beams under a moving
load with different boundary conditions." Scientia Iranica.
Transaction B Mechanical Engineering, (2009), 273-279. 
10. Gładysz, M., and P. Śniady. "Spectral density of the bridge
beam response with uncertain parameters under a random train
of moving forces." Archives of Civil and Mechanical
Engineering, Vol. 9, No. 3, (2009), 31-47. 
11. Balkaya, Müge, Metin O. Kaya, and Ahmet Sağlamer.
"Analysis of the vibration of an elastic beam supported on
elastic soil using the differential transform method." Archive
of Applied Mechanics, Vol. 79, No. 2, (2009), 135-146. 
12. Motaghian, S. E., M. Mofid, and P. Alanjari. "Exact solution to
free vibration of beams partially supported by an elastic
foundation." Scientia Iranica. Transaction A, Civil
Engineering Vol. 18, No. 4, (2011), 861. 
13. Barari, Amin, Hamed Dadashpour Kaliji, Mojtaba Ghadimi,
and G. Domairry. "Non-linear vibration of Euler-Bernoulli
beams." Latin American Journal of Solids and Structures,
Vol. 8, No. 2, (2011), 139-148.  
14. Bazehhour, Benyamin Gholami, Seyed Mahmoud Mousavi,
and Anoushiravan Farshidianfar. "Free vibration of high-speed
rotating Timoshenko shaft with various boundary conditions:
effect of centrifugally induced axial force." Archive of Applied
Mechanics, Vol. 84, No. 12, (2014), 1691-1700.  
15. Prokić, A., M. Bešević, and D. Lukić. "A numerical method for
free vibration analysis of beams."Latin American Journal of
Solids and Structures, Vol. 11, No. 8, (2014), 1432-1444.  
16. Yayli, Mustafa Özgür, Murat Aras, and Süleyman Aksoy. "An
efficient analytical method for vibration analysis of a beam on
elastic foundation with elastically restrained ends." Shock and
Vibration, Vol. 2014, (2014).  
17. Mohammadzadeh, Saeed, and Seyed Ali Mosayebi. "Dynamic
analysis of axially beam on visco-elastic foundation with
elastic supports under moving load." International Journal of
Transportation Engineering, Vol. 2, No. 4, (2015): 289-296.  
18. Zrnić, N. Đ., V. M. Gašić, and S. M. Bošnjak. "Dynamic
responses of a gantry crane system due to a moving body
considered as moving oscillator." Archives of Civil and
Mechanical Engineering, Vol. 15, No. 1 (2015): 243-250. 
19. Zakeri, Jabbarali, and Shahbabaei. "Investigation on effect of
elastic supports stiffness on natural frequencies and modes of
two span beams under free vibration." Journal of
Transportation Engineering, Vol. 7, No 1, (2015), 45-54. 
20. Roshandel, Davod, Massood Mofid, and Amin Ghannadiasl.
"Dynamic response of a non-uniform Timoshenko beam,
subjected to moving mass." Proceedings of the Institution of
Mechanical Engineers, Part C: Journal of Mechanical
Engineering Science, Vol. 229, No. 14, (2015), 2499-2513.  
21. Dimitrovová, Zuzana. "Critical velocity of a uniformly moving
load on a beam supported by a finite depth foundation."
Journal of Sound and Vibration, Vol. 366, (2016), 325-342.  
22. Shih, Jou-Yi, D. J. Thompson, and Antonios Zervos. "The
effect of boundary conditions, model size and damping models
in the finite element modelling of a moving load on a
track/ground system." Soil Dynamics and Earthquake
Engineering, Vol. 89, (2016), 12-27