# Closed-form solution of beam on Pasternak foundation under inclined dynamic load

## Abstract

The dynamic response of an infinite Euler—Bernoulli beam resting on Pasternak foundation under inclined harmonic line loads is developed in this study in a closed-form solution. The conventional Pasternak foundation is modeled by two parameters wherein the second parameter can account for the actual shearing effect of soils in the vertical direction. Thus, it is more realistic than the Winkler model, which only represents compressive soil resistance. However, the Pasternak model does not consider the tangential interaction between the bottom of the beam and the foundation; hence, the beam under inclined loads cannot be considered in the model. In this study, a series of horizontal springs is diverted to the face between the bottom of the beam and the foundation to address the limitation of the Pasternak model, which tends to disregard the tangential interaction between the beam and the foundation. The horizontal spring reaction is assumed to be proportional to the relative tangential displacement. The governing equation can be deduced by theory of elasticity and Newton’s laws, combined with the linearly elastic constitutive relation and the geometric equation of the beam body under small deformation condition. Double Fourier transformation is used to simplify the geometric equation into an algebraic equation, thereby conveniently obtaining the analytical solution in the frequency domain for the dynamic response of the beam. Double Fourier inverse transform and residue theorem are also adopted to derive the closed-form solution. The proposed solution is verified by comparing the degraded solution with the known results and comparing the analytical results with numerical results using ANSYS. Numerical computations of distinct cases are provided to investigate the effects of the angle of incidence and shear stiffness on the dynamic response of the beam. Results are realistic and can be used as reference for future engineering designs.

## Keywords

Beam Harmonic line load Pasternak foundation Tangential interaction between the beam and the foundation Fourier transform## Preview

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## References

- 1.H.T. Yu, C. Cai, X.F. Guan, Y. Yuan, Analytical solution for long lined tunnels subjected to travelling loads, Tunn. Undergr. Sp. Tech. 58 (2016) 209–215.CrossRefGoogle Scholar
- 2.L. Andersen, S.R.K. Nielsen, P.H. Kirkegaard, Finite element modeling of infinite Euler beams on Kelvin foundations exposed to moving loads in convected coordinates, J. Sound. Vib. 241 (4) (2001) 587–604.CrossRefGoogle Scholar
- 3.T.P. Chang, Y.N. Liu, Dynamic finite element analysis of a nonlinear beam subjected to a moving load, Int. J. Solids Struct. 33 (12) (1996) 1673–1688.CrossRefGoogle Scholar
- 4.H.T. Yu, Y. Yuan, A. Bobet, Multiscale method for long tunnels subjected to seismic loading, Int. J. Numer. Anal. Methods 37 (4) (2013) 374–398.CrossRefGoogle Scholar
- 5.G.S. Payette, J.N. Reddy, Nonlinear quasi-static finite element formulations for viscoelastic Euler-Bernoulli and Timoshenko beams, Int. J. Numer. Methods Biomed. 26 (12) (2010) 1736–1755.MathSciNetMATHGoogle Scholar
- 6.Y.X. Wu, Y.F. Gao, Analytical solutions for simply supported viscously damped double-beam system under moving harmonic loads, J. Eng. Mech.–ASCE 141 (7) (2015) 04015004.CrossRefGoogle Scholar
- 7.H.T. Yu, Y. Yuan, Z.Z. Qiao, Y. Gu, Z.H. Yang, X.D. Li, Seismic analysis of a long tunnel based on multi-scale method, Eng. Struct. 49 (2013) 572–587.CrossRefGoogle Scholar
- 8.J.T. Kenney, Steady-state vibrations of beam on elastic foundation for moving load, J. Appl. Mech. 21 (4) (1954) 359–364.MATHGoogle Scholar
- 9.M.A. Biot, Bending of an infinite beam on an elastic foundation, J. Appl. Mech. 4 (1) (1937) A1–A7.Google Scholar
- 10.P.M. Mathews, Vibration of a beam on elastic foundation, J. Appl. Math. Mech. 38 (3–4) (1958) 105–115.MathSciNetMATHGoogle Scholar
- 11.P.M. Mathews, Vibration of a beam on elastic foundation II, J. Appl. Math. Mech. 39 (1–2) (1959) 13–19.MathSciNetMATHGoogle Scholar
- 12.J.D. Achenbach, C. Sun, Dynamic response of beam on viscoelastic subgrade, J. Eng. Mech. Div. 91 (5) (1965) 61–76.Google Scholar
- 13.L. Sun, A closed-form solution of a Bernoulli-Euler beam on a viscoelastic foundation under harmonic line loads, J. Sound Vib. 242 (4) (2001) 619–627.CrossRefGoogle Scholar
- 14.L. Sun, A closed-form solution of beam on viscoelastic subgrade subjected to moving loads, Comput. Struct. 80 (1) (2002) 1–8.MathSciNetCrossRefGoogle Scholar
- 15.L. Sun, An explicit representation of steady state response of a beam on an elastic foundation to moving harmonic line loads, Int. J. Numer. Anal. Methods 27 (1) (2003) 69–84.CrossRefGoogle Scholar
- 16.S.M. Kim, J.M. Roesset, Dynamic response of a beam on a frequency-independent damped elastic foundation to moving load, Can. J. Civil Eng. 30 (2) (2003) 460–467.CrossRefGoogle Scholar
- 17.M. Shamalta, A.V. Metrikin, Analytical study of the dynamic response of an embedded railway track to a moving load, Arch. Appl. Mech. 73 (2003) 131–146.CrossRefGoogle Scholar
- 18.H.T. Yu, Y. Yuan, Analytical solution for an infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to arbitrary dynamic loads, J. Eng. Mech.–ASCE 140 (3) (2014) 542–551.CrossRefGoogle Scholar
- 19.Y. Yuan, H.T. Yu, C. Li, X. Yan, J.Y. Yuan, Multi-point shaking table test for long tunnels subjected to non-uniform seismic loadings – Part I: theory and validation, Soil Dyn. Earthquake Eng. (2017), doi:10.1016/j.soildyn.2016.08.017.CrossRefGoogle Scholar
- 20.H.T. Yu, Y. Yuan, G.P. Xu, Q.K. Su, X. Yan, C. Li, Multi-point shaking table test for long tunnels subjected to non-uniform seismic loadings – Part II: application to the HZM immersed tunnel, Soil Dyn. Earthquake Eng. (2017), doi:10.1016/j.soildyn.2016.08.018.CrossRefGoogle Scholar
- 21.D. Kerr A, Elastic and viscoelastic foundation models, Int. J. Appl. Mech. 31 (3) (1964) 23–32.MATHGoogle Scholar
- 22.M.H. Karganovin, D. Younesian, Dynamics of Timoshenko beams on Pasternak foundation under moving load, Mech. Res. Commun. 31 (2004) 713–723.CrossRefGoogle Scholar
- 23.R.U.A. Uzzal, R.B. Bhat, W. Ahmed, Dynamic response of a beam subjected to moving load and moving mass supported by Pasternak foundation, Shock Vib. 19 (2012) 205–220.CrossRefGoogle Scholar
- 24.D. Basu, N.S.V.K. Rao, Analytical solutions for Euler-Bernoulli beam on visco-elastic foundation subjected to moving load, Int. J. Numer. Anal. Methods 37 (8) (2012) 945–960.CrossRefGoogle Scholar
- 25.H.T. Yu, C. Cai, Y. Yuan, M.C. Jia, Analytical solutions for Euler–Bernoulli beam on Pasternak foundation subjected to arbitrary dynamic loads, Int. J. Numer. Anal. Methods 41 (8) (2017) 1125–1137.CrossRefGoogle Scholar