This study presents an analytical investigation of the dynamic behavior of a simply supported uniform Bernoulli-Euler beam resting on both constant and variable Pasternak foundations subjected to concentrated moving load with damping. The governing equations of motion were formulated by incorporating the effects of foundation stiffness, shear interaction, damping, and load inertia. Analytical solutions were obtained for both moving-force and moving-mass problems. For the constant foundation case, the Generalized Finite Integral Transform technique (GFIT) combined with Laplace transformation and convolution theory is employed, while Galerkin’s method was utilized for the variable foundation case, in place of GFIT. Numerical simulations were performed to compare the responses of the beam under the different loading and foundation conditions. The results showed that moving-mass responses exhibit larger amplitudes than the corresponding moving-force responses and that beams resting on variable Pasternak foundations experience greater deflections than those on constant foundations. The study further demonstrates that the moving-force model cannot be considered a reliable approximation for the moving-mass problem and that constant Pasternak foundations provide a stiffer and safer structural response. The findings contribute to a better understanding of beam-foundation interaction and provide useful information for the design and analysis of engineering structures subjected to moving loads.

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1. Abbas, W., Bakr, O.K., Nassar, M. M, Abdeen, A. M. M. & Shabrawy, M. (2021). Analysis of tapered Timoshenko and Euler–Bernoulli beams on an elastic foundation with moving loads. Hindawi Journal of Mathematics, 1-13.
2. Adeoye, A. S. & Awodola, T. O. (2018). Dynamic response to moving distributed masses of pre-stressed uniform Rayleigh beam resting on variable elastic Pasternak foundation. Edelweiss Appli Sci Tech,2, 1-9.
3. Ayre, R. S., Jacobsen, L. S. & Hsu, C. S. (1951). Transverse vibration of one and of two-span beams under the action of a moving mass load, Proceedings of the first U.S National Congress of Applied Mechanics, 81-90.
4. Bechtold, J. E. & Riley, D. R. (1991). Application of beams on elastic foundation and B-spline solution methodologies to parametric analysis of intramedullary implant systems. Journal Biomechanics, 24(6), 441-448. https://dx.doi.org/10.1016/0021-9290(91)90032-I
5. Celep, Z., Güler, K & Demir, F. (2011). Response of a completely free beam on a tensionless Pasternak foundation subjected to dynamic load. Structural Engineering and Mechanics, 37 (1) 61-77. DOI: 10.12989/sem.2011.37.1.061
6. Dobromir, D. (2012). Analytical solution of beam on elastic foundation by singularity functions. Engineering Mechanics,19(6), 381–392. https://dx.doi.org/10.12989/sem.2011.37.1.061
7. Fryba, L. (1972). Vibrations of Solids and Structures under Moving Loads. Groningen: Noordhoff.
8. Hammed, F. A., Usman, M. A., Onitilo, S. A, Alade, F. A. and Omoteso, K. A. (2020). Forced response vibration of simply supported beams with an elastic Pasternak foundation under a distributed moving load, FUDMA Journal of Sciences (FJS),4 (2), 1 – 7.
9. Iancu-Bogdan, T. (2009). Beams on elastic foundation: The simplified continuum approach. Bulletin of the Polytechnic Institute of Jassy LV (LIX), 4, 37-45
10. Inglis, C. E. (1934). A mathematical treatise on vibrations of railway bridge. Cambridge: Cambridge University Press.
11. Jimoh, A. & Ajoge, E. O. (2018). Effect of rotatory inertia and load natural frequency on the response of uniform Rayleigh beam resting on Pasternak foundation subjected to a harmonic magnitude moving load. Applied Mathematical Sciences, 12(16), 783 – 795.
12. Jimoh, A. & Ajoge, E. O. (2020). Dynamic behavior of uniform Bernoulli-Euler beam resting on bi-parametric foundations and subjected to distributed moving load with damping effect. Journal of Basic and Applied Research International, 26(3), 33-39.
13. Jiya, M. & Shaba, A. I. (2018). Analysis of a Uniform Bernoulli – Euler beam on Winkler foundation subjected to harmonic moving load. Appl. Sci. Environ. Manage., 22 (3), 368 – 372.
14. Kargarmovin, M. H. & Younesian, D. (2004). Dynamics of Timoshenko beams on Pasternak foundation under moving load. Mechanics Research Communications, 31, 713-723.
15. Kenney, J. T. (1954). Steady-state vibrations of beam on elastic foundation for moving load. Journal of Applied Mechanics, 76, 359–364.
16. Krylov, A. N. (1905). Mathematical collection of papers (61). Petersburg: Academy of Sciences.
17. Lowan, A. N. (1935). On transverse oscillations of beam under the moving variable loads. Mag. Ser. 7, 19 (127), 708 – 715.
18. Lu, C. F., Lim, C. W. & Yao, W. A. (2009). A new analytic symplectic elasticity approach for beams resting on Pasternak elastic foundations. Journal of Mechanics of Materials and Structures,4(10), 1741-1754. https://dx.doi.org/10.2140/jomms.2009.4.1741
19. Mohan, A. & Kuriakose, B. (2024). Analysis of beam on elastic foundation. International Journal of Creative Research Thoughts (IJCRT),12(1), 335-345.
20. Morfidis, K., Avramidis, I. E. (2002). Formulation of a generalized beam element on a two-parameter elastic foundation with semi-rigid connections and rigid offsets. Computers and Structures, 80, 1919–1934.
21. Mustafa, A. & Metin, A. (2022). A Ritz Formulation for Vibration Analysis of axially Functionally Graded Timoshenko-Ehrenfest Beams. Journal of Computational Applied Mechanics, 53(1), 102-115.
22. Ogunbamike, O. K. (2021). A new approach on the response of non-uniform prestressed Timoshenko beams on elastic foundation subjected to harmonic loads. African Journal of Mathematics and Statistics Studies, 4(2), 66-87.
23. Ogunyebi, S. N. (2020). Dynamic influences of constant and variable elastic foundations on elastic beam under exponentially varying magnitude moving load. International Journal of Scientific and Research Publications, 10(6), 997-1003. https://dx.doi.org/10.29322/ijsrp.10.06.2020.p102120
24. Oni S. T. & Ogunyebi, S. N. (2018). Dynamic response of uniform Rayleigh beams on variable bi-parametric elastic foundation under partially distributed loads. Asian J. Applied Sci., 11, 199-212. https://dx.doi.org/10.3923/ajaps.2018.199.212
25. Oni, S. T. & Awodola, T.O. (2003). Vibrations under a moving load of a non-uniform Rayleigh beam on variable elastic foundation. Journal of the Nigerian Association of Mathematics Physics, 7, 191-206.
26. Onyia, M. E. & Kwaghvihi, S. J. (2020). Dynamic analysis of beams on elastic foundation using fourier sine transform method. International Journal of Civil Engineering and Technology (IJCIET), 11(4),1-13.
27. Steel, C. R. (1967). The finite beam with moving load. Journal of Applied Mechanics, 34, 11-118.
28. Stokes, G. (1849). Discussion of a differential equation relating to the breaking of railway bridges. Translations of the Cambridge philosophical society, 8(5), 707 – 735.
29. Timoshenko, S. (1921). On the correction for shear of the differential equation for transverse vibration of prismatic bars. Mag. S. 6. 41, 744-776.
30. Usman, M. A, Makinde, T. A. and Daniel, D. O. (2020). Modal Analysis of Vibration of Euler-Bernoulli Beam Subjected to Concentrated Moving Load. Iraqi Journal of Science, 61(9), 2324-2334.https://dx.doi.org/10.24996/ijs.2020.61.9.19
31. Uzzal, R. U. A., Bhat, R. B. & Ahmed, W. (2012). Dynamic response of a beam subjected to moving load and moving mass supported by Pasternak foundation. Shock and Vibration, 19, 205–220. https://dx.doi.org/10.3233/SAV-2011-0624
32. Westergaard, H. M. (1948). New formulas for stresses in concrete pavements of airfields. Transactions of the American Society of Civil Engineers, 113, 425–444.
33. Willis, et al (1851). Preliminary essay to appendix B.: Experiments for determining the effects produced by causing weights to travel over bars with different velocities. In: Barlow, P. (1851). Treatise on the strength of timber, cast iron and malleable iron. London.
34. Winkler, E. (1867). Die Lehre Von Der Elastizitat Und Festigkeit. Dominicus: Prague.
35. Wstawska, I., Magnucki, K. & Kędzia, P. (2022). Analytical investigation of a beam on elastic foundation with nonsymmetrical properties. Engineering Transactions, 70(1), 77–93. https://dx.doi.org/10.24423/EngTrans.1254.20220314

This study presents an analytical investigation of the dynamic behavior of a simply supported uniform Bernoulli-Euler beam resting on both constant and variable Pasternak foundations subjected to concentrated moving load with damping. The governing equations of motion were formulated by incorporating the effects of foundation stiffness, shear interaction, damping, and load inertia. Analytical solutions were obtained for both moving-force and moving-mass problems. For the constant foundation case, the Generalized Finite Integral Transform technique (GFIT) combined with Laplace transformation and convolution theory is employed, while Galerkin’s method was utilized for the variable foundation case, in place of GFIT. Numerical simulations were performed to compare the responses of the beam under the different loading and foundation conditions. The results showed that moving-mass responses exhibit larger amplitudes than the corresponding moving-force responses and that beams resting on variable Pasternak foundations experience greater deflections than those on constant foundations. The study further demonstrates that the moving-force model cannot be considered a reliable approximation for the moving-mass problem and that constant Pasternak foundations provide a stiffer and safer structural response. The findings contribute to a better understanding of beam-foundation interaction and provide useful information for the design and analysis of engineering structures subjected to moving loads.

🔒 This is a subscription article

Full text is available to subscribers and institutional members. Please choose an option below to access it.

1. Abbas, W., Bakr, O.K., Nassar, M. M, Abdeen, A. M. M. & Shabrawy, M. (2021). Analysis of tapered Timoshenko and Euler–Bernoulli beams on an elastic foundation with moving loads. Hindawi Journal of Mathematics, 1-13.
2. Adeoye, A. S. & Awodola, T. O. (2018). Dynamic response to moving distributed masses of pre-stressed uniform Rayleigh beam resting on variable elastic Pasternak foundation. Edelweiss Appli Sci Tech,2, 1-9.
3. Ayre, R. S., Jacobsen, L. S. & Hsu, C. S. (1951). Transverse vibration of one and of two-span beams under the action of a moving mass load, Proceedings of the first U.S National Congress of Applied Mechanics, 81-90.
4. Bechtold, J. E. & Riley, D. R. (1991). Application of beams on elastic foundation and B-spline solution methodologies to parametric analysis of intramedullary implant systems. Journal Biomechanics, 24(6), 441-448. https://dx.doi.org/10.1016/0021-9290(91)90032-I
5. Celep, Z., Güler, K & Demir, F. (2011). Response of a completely free beam on a tensionless Pasternak foundation subjected to dynamic load. Structural Engineering and Mechanics, 37 (1) 61-77. DOI: 10.12989/sem.2011.37.1.061
6. Dobromir, D. (2012). Analytical solution of beam on elastic foundation by singularity functions. Engineering Mechanics,19(6), 381–392. https://dx.doi.org/10.12989/sem.2011.37.1.061
7. Fryba, L. (1972). Vibrations of Solids and Structures under Moving Loads. Groningen: Noordhoff.
8. Hammed, F. A., Usman, M. A., Onitilo, S. A, Alade, F. A. and Omoteso, K. A. (2020). Forced response vibration of simply supported beams with an elastic Pasternak foundation under a distributed moving load, FUDMA Journal of Sciences (FJS),4 (2), 1 – 7.
9. Iancu-Bogdan, T. (2009). Beams on elastic foundation: The simplified continuum approach. Bulletin of the Polytechnic Institute of Jassy LV (LIX), 4, 37-45
10. Inglis, C. E. (1934). A mathematical treatise on vibrations of railway bridge. Cambridge: Cambridge University Press.
11. Jimoh, A. & Ajoge, E. O. (2018). Effect of rotatory inertia and load natural frequency on the response of uniform Rayleigh beam resting on Pasternak foundation subjected to a harmonic magnitude moving load. Applied Mathematical Sciences, 12(16), 783 – 795.
12. Jimoh, A. & Ajoge, E. O. (2020). Dynamic behavior of uniform Bernoulli-Euler beam resting on bi-parametric foundations and subjected to distributed moving load with damping effect. Journal of Basic and Applied Research International, 26(3), 33-39.
13. Jiya, M. & Shaba, A. I. (2018). Analysis of a Uniform Bernoulli – Euler beam on Winkler foundation subjected to harmonic moving load. Appl. Sci. Environ. Manage., 22 (3), 368 – 372.
14. Kargarmovin, M. H. & Younesian, D. (2004). Dynamics of Timoshenko beams on Pasternak foundation under moving load. Mechanics Research Communications, 31, 713-723.
15. Kenney, J. T. (1954). Steady-state vibrations of beam on elastic foundation for moving load. Journal of Applied Mechanics, 76, 359–364.
16. Krylov, A. N. (1905). Mathematical collection of papers (61). Petersburg: Academy of Sciences.
17. Lowan, A. N. (1935). On transverse oscillations of beam under the moving variable loads. Mag. Ser. 7, 19 (127), 708 – 715.
18. Lu, C. F., Lim, C. W. & Yao, W. A. (2009). A new analytic symplectic elasticity approach for beams resting on Pasternak elastic foundations. Journal of Mechanics of Materials and Structures,4(10), 1741-1754. https://dx.doi.org/10.2140/jomms.2009.4.1741
19. Mohan, A. & Kuriakose, B. (2024). Analysis of beam on elastic foundation. International Journal of Creative Research Thoughts (IJCRT),12(1), 335-345.
20. Morfidis, K., Avramidis, I. E. (2002). Formulation of a generalized beam element on a two-parameter elastic foundation with semi-rigid connections and rigid offsets. Computers and Structures, 80, 1919–1934.
21. Mustafa, A. & Metin, A. (2022). A Ritz Formulation for Vibration Analysis of axially Functionally Graded Timoshenko-Ehrenfest Beams. Journal of Computational Applied Mechanics, 53(1), 102-115.
22. Ogunbamike, O. K. (2021). A new approach on the response of non-uniform prestressed Timoshenko beams on elastic foundation subjected to harmonic loads. African Journal of Mathematics and Statistics Studies, 4(2), 66-87.
23. Ogunyebi, S. N. (2020). Dynamic influences of constant and variable elastic foundations on elastic beam under exponentially varying magnitude moving load. International Journal of Scientific and Research Publications, 10(6), 997-1003. https://dx.doi.org/10.29322/ijsrp.10.06.2020.p102120
24. Oni S. T. & Ogunyebi, S. N. (2018). Dynamic response of uniform Rayleigh beams on variable bi-parametric elastic foundation under partially distributed loads. Asian J. Applied Sci., 11, 199-212. https://dx.doi.org/10.3923/ajaps.2018.199.212
25. Oni, S. T. & Awodola, T.O. (2003). Vibrations under a moving load of a non-uniform Rayleigh beam on variable elastic foundation. Journal of the Nigerian Association of Mathematics Physics, 7, 191-206.
26. Onyia, M. E. & Kwaghvihi, S. J. (2020). Dynamic analysis of beams on elastic foundation using fourier sine transform method. International Journal of Civil Engineering and Technology (IJCIET), 11(4),1-13.
27. Steel, C. R. (1967). The finite beam with moving load. Journal of Applied Mechanics, 34, 11-118.
28. Stokes, G. (1849). Discussion of a differential equation relating to the breaking of railway bridges. Translations of the Cambridge philosophical society, 8(5), 707 – 735.
29. Timoshenko, S. (1921). On the correction for shear of the differential equation for transverse vibration of prismatic bars. Mag. S. 6. 41, 744-776.
30. Usman, M. A, Makinde, T. A. and Daniel, D. O. (2020). Modal Analysis of Vibration of Euler-Bernoulli Beam Subjected to Concentrated Moving Load. Iraqi Journal of Science, 61(9), 2324-2334.https://dx.doi.org/10.24996/ijs.2020.61.9.19
31. Uzzal, R. U. A., Bhat, R. B. & Ahmed, W. (2012). Dynamic response of a beam subjected to moving load and moving mass supported by Pasternak foundation. Shock and Vibration, 19, 205–220. https://dx.doi.org/10.3233/SAV-2011-0624
32. Westergaard, H. M. (1948). New formulas for stresses in concrete pavements of airfields. Transactions of the American Society of Civil Engineers, 113, 425–444.
33. Willis, et al (1851). Preliminary essay to appendix B.: Experiments for determining the effects produced by causing weights to travel over bars with different velocities. In: Barlow, P. (1851). Treatise on the strength of timber, cast iron and malleable iron. London.
34. Winkler, E. (1867). Die Lehre Von Der Elastizitat Und Festigkeit. Dominicus: Prague.
35. Wstawska, I., Magnucki, K. & Kędzia, P. (2022). Analytical investigation of a beam on elastic foundation with nonsymmetrical properties. Engineering Transactions, 70(1), 77–93. https://dx.doi.org/10.24423/EngTrans.1254.20220314