Advertisement

Earthquake Engineering and Engineering Vibration

, Volume 18, Issue 1, pp 129–140 | Cite as

A closed-form solution for a double infinite Euler-Bernoulli beam on a viscoelastic foundation subjected to harmonic line load

  • Bing LiEmail author
  • Yongfeng Cheng
  • Zhaoqing Zhu
  • Fuyou Zhang
Article
  • 25 Downloads

Abstract

The dynamic response of a double infinite beam system connected by a viscoelastic foundation under the harmonic line load is studied. The double infinite beam system consists of two identical and parallel beams, and the two beams are infinite elastic homogeneous and isotropic. A viscoelastic layer connects the two beams continuously. To decouple the two coupled equations governing the response of the double infinite beam system, a variable substitution method is introduced. The frequency domain solutions of the decoupled equations are obtained by using Fourier transforms as well as Laplace transforms successively. The time domain solution in the generalized integral form are then obtained by employing the corresponding inverse transforms, i.e. Fourier transform and inverse Laplace transform. The solution is verified by numerical examples, and the effects of parameters on the response are also investigated.

Keywords

beam harmonic line load viscoelastic foundation integration transform convolution residue theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgement

This research was supported by the National Natural Science Foundation of China (Grant No. 51578145). The authors are grateful to the reviewers for their helpful comments.

References

  1. Abu-Hilal M (2006), “Dynamic Response of a Double Euler-Bernoulli Beam Due to a Moving Constant Load,” Journal of Sound and Vibration, 297: 477–491.CrossRefGoogle Scholar
  2. Arani AG, Shiravand A, Rahi M and Kolahchi R (2012), “Nonlocal Vibration of Coupled DLGS Systems Embedded on Visco-Pasternak Foundation,” Physica B: Condensed Matter, 407(21): 4123–4131CrossRefGoogle Scholar
  3. Basu D and Rao NSV (2013), “Analytical Solutions for Euler-Bernoulli Beam on Visco-Elastic Foundation Subjected to Moving Load,” International Journal for Numerical and Analytical Methods in Geomechanics, 37(8): 945–960.CrossRefGoogle Scholar
  4. Guan XF, Yu HT and Tian X (2016), “A Stochastic Second-order and Two-scale Thermo-mechanical Model for Strength Prediction of Concrete Materials,” International Journal for Numerical Methods in Engineering, 108(8): 885–901.CrossRefGoogle Scholar
  5. Gurgoze M and Erol H (2004), “On Laterally Vibrating Beams Carrying Tip Masses, Coupled by Several Double Spring-Mass Systems,” Journal of Sound and Vibration, 269: 431–438.CrossRefGoogle Scholar
  6. Hamada TR, Nakayama H and Hayashi K (1983), “Free and Forced Vibration of Elastically Connected Double-beam Systems,” Bulletin of the Japan Society of Mechanics Engineers, 1936–1942.Google Scholar
  7. Jang TS (2013), “A New Semi-analytical Approach to Large Deflections of Bernoulli-Euler-v. Karman Beams on a Linear Elastic Foundation: Nonlinear Analysis of Infinite Beams,” International Journal of Mechanical Sciences, 66: 22–32.CrossRefGoogle Scholar
  8. Jang TS (2017), “A New Dispersion-relation Preserving Method for Integrating the Classical Boussinesq Equation,” Communications in Nonlinear Science and Numerical Simulation, 43: 118–138.CrossRefGoogle Scholar
  9. Kenney JT (1954), “Steady-State Vibrations of Beam on Elastic Foundation for Moving Load,” Journal of Applied Mechanics, 21(4): 359–364.Google Scholar
  10. Kim SM and Roesset JM (2003), “Dynamic Response of a Beam on a Frequency-Independent Damped Elastic Foundation to Moving Load,” Canadian Journal of Civil Engineering, 30(2): 460–467.CrossRefGoogle Scholar
  11. Li MG, Yu HT, Wang JH, Xia XH and Chen JJ (2015), “A Multiscale Coupling Approach Between Discrete Element Method and Finite Difference Method for Dynamic Analysis,” International Journal for Numerical Methods in Engineering, 102(1): 1–21.CrossRefGoogle Scholar
  12. Mathews PM (1958), “Vibration of a Beam on Elastic Foundation,” Journal of Applied Mathematics and Mechanics, 38(3–4): 105–115.Google Scholar
  13. Mathews PM (1959), “Vibration of a Beam on Elastic Foundation,” Journal of Applied Mathematics and Mechanics, 39(1–2): 13–19.Google Scholar
  14. Oniszczuk Z (2000), “Free Transverse Vibrations of Elastically Connected Simply Supported Double-Beams Complex System,” Journal of Sound and Vibration, 232: 387–403.CrossRefGoogle Scholar
  15. Oniszczuk Z (2003), “Forced Transverse Vibrations of an Elastically Connected Complex Simply Supported Double-Beam System,” Journal of Sound and Vibration, 264: 273–286.CrossRefGoogle Scholar
  16. Palmeri A and Adhikari S (2011), “A Galerkin-Type State-Space Approach for Transverse Vibrations of Slender Double-beam Systems with Viscoelastic Inner Layer,” Journal of Sound Vibration, 330: 6372–6386.CrossRefGoogle Scholar
  17. Saito J and Terasawa T (1980), “Steady-State Vibrations of a Beam on a Pasternak Foundation for Moving Loads,” Journal of Applied Mechanics, 47(4): 879–883.CrossRefGoogle Scholar
  18. Sun L (2001), “A Closed-Form Solution of a Bernoulli-Euler Beam on a Viscoelastic Foundation under Harmonic Line Loads,” Journal of Sound and Vibration, 242(4): 619–627.CrossRefGoogle Scholar
  19. Sun L (2002), “A Closed-Form Solution of Beam on Viscoelastic Subgrade Subjected to Moving Loads,” Computer and Structure, 80(1): 1–8.CrossRefGoogle Scholar
  20. Sun L (2003), “An Explicit Representation of Steady State Response of a Beam on an Elastic Foundation to Moving Harmonic Line Loads,” International Journal for Numerical and Analytical Methods in Geomechanics, 27(1): 69–84.CrossRefGoogle Scholar
  21. Vu HV, Ordonez AM and Karnopp BH (2000), “Vibration of a Double-Beam System,” Journal of Sound and Vibration, 229: 807–822.CrossRefGoogle Scholar
  22. Wu YX and Gao YF (2015), “Analytical Solutions for Simply Supported Viscously Damped Double-Beam System under Moving Harmonic Loads,” Journal of Engineering Mechanics, ASCE, 141(7): 04015004.CrossRefGoogle Scholar
  23. Wu YX and Gao YF (2016), “Dynamic Response of a Simply Supported Viscously Damped Double-Beam System under the Moving Oscillator,” Journal of Sound and Vibration, 384: 194–209.CrossRefGoogle Scholar
  24. Yan X, Yu HT, Yuan Y and Yuan JY (2015), “Multi-Point Shaking Table Test of the Free Field under Non-uniform Earthquake Excitation,” Soils and Foundations, 55(5): 985–1000.CrossRefGoogle Scholar
  25. Yuan Y, Yu HT. Li C, Yan X and Yuan JY (2018), “Multi-Point Shaking Table Test for Long Tunnels Subjected to Non-Uniform Seismic Loadings-Part I: Theory and Validation,” Soil Dynamics and Earthquake Engineering, 108: 177–186.CrossRefGoogle Scholar
  26. Yu HT, Cai C, Guan XF and Yuan Y (2016), “Analytical Solution for Long Lined Tunnels Subjected to Travelling Loads,” Tunnelling and Underground Space Technology, 58: 209–215.CrossRefGoogle Scholar
  27. Yu HT, Cai C, Yuan Y and Jia MC (2017), “Analytical Solutions for Euler-Bernoulli Beam on Pasternak Foundation Subjected to Arbitrary Dynamic Loads,” International Journal for Numerical and Analytical Methods in Geomechanics, DOI: 10.1002/nag.2672.Google Scholar
  28. Yu H and Yuan Y (2014), “Analytical Solution for an Infinite Euler-Bernoulli Beam on a Viscoelastic Foundation Subjected to Arbitrary Dynamics Loads,” Journal of Engineering Mechanics, 140(3): 542–551.CrossRefGoogle Scholar
  29. Yu HT, Yuan Y and Bobet A (2013a), “Multiscale Method for Long Tunnels Subjected to Seismic Loading,” International Journal for Numerical and Analytical Methods in Geomechanics, 37(4): 374–398.CrossRefGoogle Scholar
  30. Yu HT, Yuan Y, Qiao ZZ, Gu Y, Yang ZH and Li XD (2013b), “Seismic Analysis of a Long Tunnel Based on Multi-Scale Method,” Engineering Structures, 49: 572–587.CrossRefGoogle Scholar
  31. Yu HT, Yuan Y, Xu GP, Su QK, Yan X and Li C (2018), “Multi-Point Shaking Table Test for Long Tunnels Subjected to Non-Uniform Seismic Loadings-Part II: Application to the HZM Immersed Tunnel,” Soil Dynamics and Earthquake Engineering, 108: 187–195.CrossRefGoogle Scholar

Copyright information

© Institute of Engineering Mechanics, China Earthquake Administration and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Bing Li
    • 1
    Email author
  • Yongfeng Cheng
    • 2
  • Zhaoqing Zhu
    • 2
  • Fuyou Zhang
    • 3
  1. 1.Key Laboratory of Concrete and Prestressed Concrete Structures of the Ministry of EducationSoutheast UniversityNanjingChina
  2. 2.China Electric Power Research InstituteBeijingChina
  3. 3.Institute of Engineering Safety and Disaster PreventionHohai UniversityNanjingChina

Personalised recommendations