Journal Menu
By: .
Ph.D. Scholar, Department of Mathematics, Federal
University, Lokoja, Kogi State, Nigeria
Abstract
The consequences of a uniform Bernoulli-Euler beam resting on a Pasternak foundation under the action of a focussed moving load with a damping term were investigated in this paper using the merely support, clamped-clamped, and clamped-free classical boundary conditions. The Laplace transform method, the extended finite integral transform, and Struble’s asymptotic methodology were used to solve the governing equation after the Dirac delta function was expressed as a Fourier cosine series. The dynamic responses of the beam and their corresponding resonance condition were obtained using the simply support, clamped-clamped, and clamped-free conditions after the equations defining the transverse displacement for the moving force and moving mass issues were obtained. The uniform beam’s deflection profiles were then calculated, and MATLAB was used to display the corresponding graphs for various damping coefficient, axial force, foundation modulus, and shear modulus values. The graphs for these boundary conditions showed that, for the moving force and moving mass problems, the response amplitude of the beam decreases as the axial force, foundation modulus, shear modulus, and damping term values rise. The influence of the damping term was however, more noticeable than other parameters. In addition, the transverse displacement of the simply support beam was generally higher for the moving force and moving mass problems, followed by the clamped-free condition with the clamped-clamped showing the least displacement. Furthermore, the highest deflection of the simply support condition concentrated at the mid-points of the graphs for the moving force problem whereas, the case of the moving mass is not significantly noticeable. On the other hand, the clamped-clamped and clamped-free conditions generally had their peak deflections at the ends of the graphs. Furthermore, the moving mass problem cannot be safely approximated using the moving force problem. These results are consistent with previous research.
Key Words: Bernoulli-Euler beam, Classical boundary conditions, Concentrated moving load, Damping term, Pasternak foundation.
Citation:
Refrences:
1. Celep Z, Güler K, Demir F. Response of a completely free beam on a tensionless Pasternak
foundation subjected to dynamic load. Struct Eng Mech. 2011;37(1):61–77. doi:
10.12989/sem.2011.37.1.061.
2. Morfidis K, Avramidis IE. Formulation of a generalized beam element on a two-parameter elastic
foundation with semi-rigid connections and rigid offsets. Comput Struct. 2002;80(25):1919–1934.
doi: 10.1016/S0045-7949(02)00226-2.
3. Abbas W, Bakr OK, Nassar MM, Abdeen AMM, Shabrawy M. Analysis of tapered Timoshenko
and Euler–Bernoulli beams on an elastic foundation with moving loads. Hindawi J Math.
2021;2021:1–13. doi: 10.1155/2021/6616707.
4. Oni ST, Awodola TO. Dynamic behaviour under moving concentrated masses of simply supported
rectangular plates resting on variable wrinkle elastic foundation. Lat Am J Solids Struct.
2010;8(4):373–392. doi: 10.1590/S1679-78252011000400001.
5. Misra RR. Free vibration analysis of isotrophical plate using multi-quadric radial basis function. Int
J Sci Environ Technol. 2012;1(2):99–107.
6. Mutman U. Free vibration analysis of an Euler-Bernoulli beam of variable width on the Winkler
foundation using homotopy perturbation method. Math Probl Eng. 2013;2013:1–9. doi:
10.1155/2013/721294.
7. Huang MS, Zhou XC, Yu J, Leung CF, Jorgin QWT. Estimating the effects of tunnelling on existing
jointed pipelines based on Winkler model. Tunn Undergr Space Technol. 2019;86:89–99.
8. Usman MA, Hammed FA, Daniel DO. Free vibrational analysis of non-uniform double Euler-
Bernoulli beams on a Winkler foundation using Laplace differential transform method. J Eng Sci.
2023;14(1):111–121. doi: 10.3329/jes.v14i1.67640.
9. Mirsaidov M, Mamasoliev Q. Contact problems of multilayer slabs interaction on an elastic
foundation. IOP Conf Ser Earth Environ Sci. 2020;614(1):012089. doi: 10.1088/1755-
1315/614/1/012089
10. Gholami M, Alizadeh MA. Quasi-3D modified strain gradient formulation for static bending of
functionally graded micro beams resting on Winkler-Pasternak elastic foundation. Sci Iran.
2022;29(1B):26–40. doi: 10.24200/sci.2021.55000.4019.
11. Oni ST, Jimoh A. Dynamic response to moving concentrated loads of non-uniform simply
supported prestressed Bernoulli-Euler beam resting on bi-parametric subgrades. Int J Sci Eng Res.
2016;7(9):584–600.
12. Hammed FA, Usman MA, Onitilo SA, Alade FA, Omoteso KA. Forced response vibration of
simply supported beams with an elastic Pasternak foundation under a distributed moving load.
FUDMA J Sci. 2020;4(2):1–7. doi: 10.33003/fjs-2020-0402-130.
13. Ogunbamike OK. A new approach on the response of non-uniform prestressed Timoshenko beams
on elastic foundation subjected to harmonic loads. Afr J Math Stat Stud. 2021;4(2):66–87. doi:
10.52589/AJMSS-NZXSUM3I.
14. Awodola TO, Awe BB, Jimoh SA. Vibration of non-uniform Bernoulli-Euler beam under moving
distributed masses resting on Pasternak elastic foundation subjected to variable magnitude. Afr J
Math Stat Stud. 2024;7(1):1–19. doi: 10.52589/AJMSS-ZQCKBA87.
15. Omolofe B, Adedowole A. Response characteristics of non-uniform beam under the actions of
travelling distributed masses. J Appl Math Comput Mech. 2017;16(2):77–99. doi:
10.17512/jamcm.2017.2.07.
16. Adeoye AS, Awodola TO. Dynamic response to moving distributed masses of pre-stressed uniform
Rayleigh beam resting on variable elastic Pasternak foundation. Edelweiss Appl Sci Tech.
2018;2(1):1–9. doi: 10.33805/2576.8484.106.
17. Krylov AN. Mathematical collection of papers. Petersburg: Academy of Sciences; 1905.
18. Ojih PB, Ibiejugba MA, Adejo BO. Dynamic response under moving concentrated loads of non-
uniform Rayleigh beam resting on Pasternak foundation. Adv Appl Sci Res. 2013;4(4):30–48.
19. Lee HP. Dynamic response of a beam with a moving mass. J Sound Vib. 1996;191:289–294.
20. Ozdemir O, Kaya MO. Flaps bending vibration analysis of a rotating tapered cantilever Bernoulli-
Euler beam by differential transform method. J Sound Vib. 2006;289(1–2):413–420. doi:
10.1016/j.jsv.2005.01.055.
21. Somia A. The rope wear analysis with the use of 3D vision system. Control Eng. 2013;60(5):48–
49.
22. Mirzaei MMH, Loghman A, Arefi M. Effect of temperature dependency on thermoelastic behavior
of rotating variable thickness FGM cantilever beam. J Solid Mech. 2019;11(3):657–669.
23. Mirzabeigy A, Madoliat R. Large amplitude free vibration of axially loaded beams resting on
variable elastic foundation. Alex Eng J. 2016;55(2):11–14. doi: 10.1016/j.aej.2016.03.021.
24. Kumar S. Free vibration analysis on axially graded beam resting on variable Pasternak foundation.
Dehradun: IOP Publishing Ltd; 2021. doi: 10.1088/1757-899X/1206/1/012016.
25. Olotu OT, Agboola OO, Gbadeyan JA. Free vibration analysis of non-uniform Rayleigh beams on
variable Winkler elastic foundation using differential transform method. Ilorin J Sci. 2021;8(1):1–
20. doi: 10.54908/iljs.2021.08.01.001.
26. Usman MA, Hammed FA, Olayiwola MO, Okusaga ST, Daniel DO. Free vibrational analysis of
beam with different support conditions. Trans Niger Assoc Math Phys. 2018;7:27–32.
27. Ahmadi A, Abedi M. Transient response of laminated composite curved beams with general
boundary conditions under moving force. AUT J Mech Eng. 2022;6(4):561–578. doi:
10.1016/j.istruc.2023.104960.
28. Chen W, Cai J, Huang J, Yang X, Ma J. Resistance behaviours of clamped HFR-LWC beam using
membrane approach. Int J Concr Struct Mater. 2024;18:1–13. doi: 10.1186/s40069-023-00652-x.
29. Jimoh A, Ajoge EO, Ezeofor CD. Comparative study of clamped-clamped and clamped-free elastic
beams resting on bi-parametric subgrades and subjected to concentrated moving loads. J Exp Appl
Mech. 2025;16(1):17–33.
30. Serdar H. Vibration analysis of systems subjected to moving loads. Izmir: Dokuz Eylül University;
2005.
31. Jitendra N. Dynamic response of beam and frame structure subjected to moving load. Unpublished
thesis. Rourkela: National Institute of Technology; 2014.
32. Mehmood A, Khan AA, Mehdi H. Vibration analysis of beam subjected to moving loads using
finite element method. IOSR J Eng. 2014;4(5):7–17. doi: 10.9790/3021-04510717.
33. Usman MA, Muibi BT. Effect of free vibration analysis on Euler-Bernoulli beam with different
boundary conditions. FUDMA J Sci. 2019;3(4):501–510.
34. Fryba L. Vibrations of Solids and Structures Under Moving Loads. Groningen: Noordhoff; 1972.
doi: 10.1680/vosasuml.35393.