Acta Mechanica

, Volume 229, Issue 4, pp 1721–1739 | Cite as

Approximate solutions for the problem of a load moving on the surface of a half-plane

  • Niki D. Beskou
  • Jiang Qian
  • Dimitri E. Beskos
Original Paper


The classical problem of determining the dynamic response of an elastic half-plane to a load moving with constant speed on its surface is revisited. The problem is first solved analytically in an exact manner by a simple and efficient method that employs complex Fourier series involving the horizontal coordinate and the time to reduce the partial differential equations of motion into ordinary ones, which can be easily solved to provide the system response. Then, the problem is solved again by the same method under various simplifying assumptions that effectively reduce the system of two partial differential equations of the problem into a single equation. These assumptions are zero horizontal displacement, zero horizontal normal stress and zero horizontal normal stress plus zero derivative of the horizontal displacement with respect to the vertical coordinate. The resulting three approximate solutions are much easier to derive and simpler than the exact solution but do not satisfy the zero shear stress on the surface boundary condition and the equation of motion along the horizontal direction. Nevertheless, comparison of these approximate solutions against the exact solution by means of numerical parametric studies demonstrates that only one of them is practically acceptable in the range of sub-Rayleigh load speeds, which are of interest in road pavement dynamics.


Moving load Elastic half-plane Exact solution Approximate solutions Approximation errors 


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  1. 1.
    Beskou, N.D., Theodorakopoulos, D.D.: Dynamic effects of moving loads on road pavements: a review. Soil Dyn. Earthq. Eng. 31, 547–567 (2011)CrossRefGoogle Scholar
  2. 2.
    Sneddon, I.N.: Stress produced by a pulse of pressure moving along the surface of a semi-infinite solid. Rendiconti Circolo Matematico di Palermo 2, 57–62 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cole, J., Huth, J.: Stresses produced in a half plane by moving loads. J. Appl. Mech. ASME 25, 433–436 (1958)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Georgiadis, H.G., Barber, J.R.: Steady-state transonic motion of a line load over an elastic half-space: the corrected Cole–Huth solution. J. Appl. Mech. ASME 60, 772–774 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ang, D.D.: Transient motion of a line load on the surface of an elastic half-space. Q. Appl. Math. 18, 251–256 (1960)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Eason, G.: The stresses produced in semi-infinite solid by a moving surface force. Int. J. Eng. Sci. 2, 581–609 (1965)CrossRefzbMATHGoogle Scholar
  7. 7.
    Payton, R.G.: Transient motion of an elastic half-space due to a moving surface line load. Int. J. Eng. Sci. 5, 49–79 (1967)CrossRefGoogle Scholar
  8. 8.
    Gakenheimer, D.C., Miklowitz, J.: Transient excitation of an elastic half space by a point load travelling on the surface. J. Appl. Mech. ASME 36, 505–515 (1969)CrossRefzbMATHGoogle Scholar
  9. 9.
    De Barros, F.C.P., Luco, J.E.: Stresses and displacements in a layered half-space for a moving line load. Appl. Math. Comput. 67, 103–134 (1995)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Barber, J.R.: Surface displacements due to a steadily moving point force. J. Appl. Mech. ASME 63, 245–251 (1996)CrossRefzbMATHGoogle Scholar
  11. 11.
    Lefeuve-Mesguez, G., Le Houedec, D., Peplow, A.T.: Ground vibration in the vicinity of a high-speed moving harmonic strip load. J. Sound Vib. 231, 1289–1309 (2000)CrossRefGoogle Scholar
  12. 12.
    Georgiadis, H.G., Lykotrafitis, G.: A method based on the Radon transform for three-dimensional elastodynamic problems of moving loads. J. Elast. 65, 87–129 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Liao, W.I., Teng, T.J., Yeh, C.S.: A method for the response of an elastic half- space to moving sub-Rayleigh point loads. J. Sound Vib. 284, 173–188 (2005)CrossRefGoogle Scholar
  14. 14.
    De Hoop, A.T.: The moving load problem in soil dynamics-the vertical displacement approximation. Wave Motion 36, 335–346 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Matuo H., Ohara S.: Lateral earth pressure and stability of quay walls during earthquakes. In: Proceedings of 2nd World Conference on Earthquake Engineering, Tokyo, Japan (1960)Google Scholar
  16. 16.
    Arias, A., Sanchez-Sesma, F.J., Ovando-Shelley, E.: A simplified elastic model for seismic analysis of earth-retaining structures with limited displacements. Proc. Int. Conf. Recent Adv. Geotech. Earthq. Eng. Soil Dyn. St Louis, MO 1, 235–240 (1981)Google Scholar
  17. 17.
    Veletsos, A.S., Younan, A.H.: Dynamic soil pressures on rigid vertical walls. Earthq. Eng. Struct. Dyn. 23, 275–301 (1994)CrossRefGoogle Scholar
  18. 18.
    Siddharthan, R., Zafir, Z., Norris, G.M.: Moving load response of layered soil. I: formulation. J. Eng. Mech. ASCE 119(10), 2052–2071 (1993)CrossRefGoogle Scholar
  19. 19.
    Siddharthan, R., Zafir, Z., Norris, G.M.: Moving load response of layered soil. II: verification and application. J. Eng. Mech. ASCE 119(10), 2072–2089 (1993)CrossRefGoogle Scholar
  20. 20.
    Theodorakopoulos, D.D.: Dynamic analysis of a poroelastic half-plane soil medium under moving loads. Soil Dyn. Earthq. 23, 521–533 (2003)CrossRefGoogle Scholar
  21. 21.
    Beskos, N.D., Chen, Y., Qian, J.: Dynamic response of an elastic plate on a cross-anisotropic elastic half-plane to a load moving on its surface. Transp. Geotech. 14, 98–106 (2018)Google Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2017

Authors and Affiliations

  • Niki D. Beskou
    • 1
  • Jiang Qian
    • 2
  • Dimitri E. Beskos
    • 1
    • 2
  1. 1.Department of Civil EngineeringUniversity of PatrasPatrasGreece
  2. 2.Institute of Structural Engineering and Disaster Reduction, College of Civil EngineeringTongji UniversityShanghaiPeople’s Republic of China

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