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Three dimensional solution of thick rectangular simply supported plates under a moving load

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Abstract

Dynamic behavior of continuous systems such as beams and plates, under a moving load is an important engineering subject. In this paper, 3D elasticity equations are solved by use of the displacement potential functions and the exact solution of a simply supported thick rectangular plate under moving load is presented. For this purpose, the governing equations in terms of displacements, Navier’s equations, are converted to two linear partial differential equations of forth and second order using displacement potential functions. Then the governing equations in terms of the potential functions are solved using the separation of variables and Laplace integral transform, satisfying exact initial and boundary conditions. In order to validate the present approach, the obtained results of this study are compared with the results of the classical theory of plates for thin and existing solutions for moderately thick plates. Also, it is observed that the speed of a moving load has an important effect on the dynamic response of plate.

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References

  1. Ouyang H (2010) Moving-load dynamic problems: a tutorial (with a brief overview). Mech Syst Signal Process 25:2039–2060

    Article  ADS  Google Scholar 

  2. Vaseghi Amiri J, Nikkhoo A, Davoodi MR, Ebrahimzadeh Hassanabadi M (2013) Vibration analysis of a Mindlin elastic plate under a moving mass excitation by eigenfunction expansion method. Thin Walled Struct 62:53–64

    Article  Google Scholar 

  3. Willis R (1849) The effect produced by causing weights to travel over elastic bars, Report of the commissioners appointed to inquire into the application of iron to railway structures, Appendix B. Stationery Office, London

    Google Scholar 

  4. Stokes GG (1849) Discussion of a differential equation relating to the breaking of railway bridges. Trans Camb Philos Soc 8(5):707–735

    Google Scholar 

  5. Fryba L (1999) Vibration of solids and structures under moving loads. Thomas Telford, London

    Book  MATH  Google Scholar 

  6. Yavari A, Nouri M, Mofid M (2002) Discrete element analysis of dynamic response of Timoshenko beams under moving mass. Adv Eng Softw 33:143–153

    Article  MATH  Google Scholar 

  7. DeSalov V, Muscolino G, Palmeri A (2010) A substructure approach tailored to the dynamic analysis of multi-span continuous beams under moving loads. J Sound Vib 329:3101–3120

    Article  ADS  Google Scholar 

  8. Kiani K, Nikkhoo A, Mehri B (2009) Prediction capabilities of classical and shear deformable beam models excited by a moving mass. J Sound Vib 320(3):632–648

    Article  ADS  Google Scholar 

  9. Stanisic MM, Hardin JC (1969) On the response of beams to an arbitrary number of moving masses. J Frankl Inset 287:115

    Article  Google Scholar 

  10. Akin JE, Mofid M (1989) Numerical solution for response of beams with moving mass. J Struct Eng ASCE 115(1):120–131

    Article  Google Scholar 

  11. Mackertich S (1992) Response of a beam to a moving mass. J Acoust Soc Am 92(3):1766–1769

    Article  ADS  Google Scholar 

  12. Green MF, Cebon D (1994) Dynamic responses of highway bridges to heavy vehicle loads, theory and experimental validation. J Sound Vib 170(1):51–78

    Article  ADS  Google Scholar 

  13. Esmailzadeh E, Ghorashi M (1995) Vibration analysis of beams traversed by a moving mass. J Eng 8(4):214

    MATH  Google Scholar 

  14. Lee HP (1996) The dynamic response of a Timoshenko beam subjected to a moving mass. J Sound Vib 198(2):249–256

    Article  ADS  Google Scholar 

  15. Mofid M, Akin JE (1996) Discrete element response of beams with traveling mass. Adv Eng Softw 25:321–331

    Article  Google Scholar 

  16. Mofid M, Shadnam M (2000) On the response of beams with internal hinges, under moving mass. Adv Eng Softw 31:323–328

    Article  Google Scholar 

  17. Bilello C, Bergman LA, Kuchma D (2004) Experimental investigation of a small-scale bridge model under a moving mass. J Struct Eng 130:799–804

    Article  Google Scholar 

  18. Nikkhoo A, Rofooei FR, Shadnam MR (2007) Dynamic behavior and modal control of beams under moving mass. J Sound Vib 306(3–5):712–724

    Article  ADS  MathSciNet  Google Scholar 

  19. Dehestani M, Mofid M, Vafai A (2009) Investigation of critical influential speed for moving mass problems on beams. Appl Math Model 33:3885–3895

    Article  MATH  MathSciNet  Google Scholar 

  20. Eftekhar Azam S, Mofid M, Afghani Khoraskani R (2013) Dynamic response of Timoshenko beam under moving mass. Scientia Iranica A 20(1):50–56

    Google Scholar 

  21. Morley LSD (1962) Elastic plate with loads travelling at uniform velocity along the bounding surfaces. Q J Mech Appl Math XV(Pt.a):194–213

    MATH  Google Scholar 

  22. Reissmann H (1963) Response of elastic plate strip to line load. AIAA J 1:354

    Article  ADS  Google Scholar 

  23. Jahanshahi A, Monzel FJ (1965) Effects of rotatory inertia and transverse shear on the response of elastic plates to moving forces. Arch  Appl  Mech 34:401–411

    Google Scholar 

  24. Rasek TF, Schlack AL (1967) Dynamic response of plates due to moving loads. J Acoust Soc Am 42(3):625–636

    Article  ADS  Google Scholar 

  25. Shirakawa Osaka K (1981) Response of rectangular thick plates to moving single loads. Ingenieur-Archiv 50:165–175

    Article  MATH  Google Scholar 

  26. Taheri MR, Ting E (1989) Dynamic response of plate to moving loads: structural Impedance method. Comput Struct 33(60):1379–1393

    Article  MATH  Google Scholar 

  27. Gbadeyan JA, Oni ST (1995) Dynamic behavior of beams and rectangular plates under moving mass. J Sound Vib 182(5):677–695

    Article  ADS  MATH  Google Scholar 

  28. Gbadeyan JA, Dada MS (2006) Dynamic response of a Mindlin elastic rectangular plate under a distributed moving mass. Int J Mech Sci 48:323–340

    Article  MATH  Google Scholar 

  29. Wu JJ (2005) Dynamic analysis of rectangular plate under a moving line load using scale beams and scaling laws. Comput Struct 83:1646–1658

    Article  Google Scholar 

  30. Nikkhoo A, Rofooei FR (2012) Parametric study of the dynamic response of thin rectangular plates traversed by a moving mass. Acta Mech 223(1):15–27

    Article  MATH  MathSciNet  Google Scholar 

  31. Nikkhoo A, Ebrahimzadeh M, Eftekhar Azam S, Vaseghi Amiri J (2014) Vibration of thin rectangular plate subjected to series of moving inertial loads. Mech Res Commun 55:105–113

    Article  Google Scholar 

  32. Niaz M, Nikkhoo A (2015) Inspection of a rectangular plate dynamics under a moving mass with varying velocity utilizing BCOPs. Latin Am J Solids Struct 12:317–332

    Article  Google Scholar 

  33. Lekhnitskii SG (1981) Theory of elasticity of an anisotropic body, Mir Publication, Moscow

    MATH  Google Scholar 

  34. Hu HC (1953) On the three dimensional problems of the theory of elasticity of a transversely isotropic body. Acta Phys Sin 9:131–147

    MathSciNet  Google Scholar 

  35. Nowacki W (1954) The stress function in three-dimensional problems concerning an elastic body characterized by transverse isotropy. Bulletin de l’Academie Polonaise Sciences 4:21–25

    Google Scholar 

  36. Eskandari-Ghadi M (2005) A complete solution of the wave equations for transversely isotropic media. J Elast 81:1–19

    Article  MATH  MathSciNet  Google Scholar 

  37. Eskandari-Ghadi M, Ardeshir-Behrestaghi A (2010) Forced vertical vibration of rigid circular disc buried in an arbitrary depth of a transversely isotropic half-space. Soil Dyn Earthq Eng 30:547–560

    Article  Google Scholar 

  38. Eskandari-Ghadi M, Mirzapour A, Ardeshir-Behrestaghi A (2011) Rocking vibration of a rigid circular disc in a transversely isotropic full-space. Int J Numer Anal Met 35:1587–1603

    Article  Google Scholar 

  39. Eskandari-Ghadi M, Fallahi M, Ardeshir-Behrestaghi A (2009) Forced vertical vibration of rigid circular disc on a transversely isotropic half-space. J Eng Mech 136:913–922

    Article  Google Scholar 

  40. Eskandari-Ghadi M, Pak R, Ardeshir-Behrestaghi A (2008) Transversely isotropic elastodynamic solution of a finite layer on an infinite subgrade under surface loads. Soil Dyn Earthq Eng 28:986–1003

    Article  Google Scholar 

  41. Eskandari-Ghadi M, Ardeshir-Behrestaghi A, Navayi-Neya B (2013) Mathematical analysis for an axisymmetric disc-shaped crack in transversely isotropic half-space. Int J Mech Sci 68:171–179

    Article  Google Scholar 

  42. Rahimian M, Eskandari-Ghadi M, Pak RY, Khojasteh A (2007) Elastodynamic potential method for transversely isotropic solid. J Eng Mech 133:1134–1145

    Article  Google Scholar 

  43. Rahimian M, Ghorbani-Tanha AK, Eskandari-Ghadi M (2006) The Reissner–Sagoci problem for a transversely isotropic half-space. Int J Numer Anal Met 30:1063–1074

    Article  MATH  Google Scholar 

  44. Nematzadeh M, Eskandari-Ghadi M, Navayi-Neya B (2011) An analytical solution for transversely isotropic simply supported thick rectangular plates using displacement potential functions. J Strain Anal Eng 46:121–142

    Google Scholar 

  45. Moslemi A, Navayi Neya B, Vaseghi Amiri J (2016) 3-D elasticity buckling solution for simply supported thick rectangular plates using displacement potential functions. Appl Math Model 40:5717–5730

    Article  MathSciNet  Google Scholar 

  46. Sadd MH (2009) Elasticity theory, applications, and numeric. Academic Press, San Diego

    Google Scholar 

  47. Myint-U Tyn, Debnath Lokenath (2007) Linear partial differential equations for scientists and engineers. Birkhauser, Cambridge

    MATH  Google Scholar 

  48. Cohen A (2007) Numerical methods for Laplace transform inversion. Springer, New York

    MATH  Google Scholar 

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Authors and Affiliations

  1. Civil Engineering Department, Babol Noshirvani University of Technology, P.O. Box 484, Shariati Ave, Babol, Mazandaran, 47148-71167, Iran

    Parvaneh Nateghi Babagi, Bahram Navayi Neya & Mehdi Dehestani

Authors
  1. Parvaneh Nateghi Babagi
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  2. Bahram Navayi Neya
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  3. Mehdi Dehestani
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Correspondence to Bahram Navayi Neya.

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Babagi, P.N., Navayi Neya, B. & Dehestani, M. Three dimensional solution of thick rectangular simply supported plates under a moving load. Meccanica 52, 3675–3692 (2017). https://doi.org/10.1007/s11012-017-0653-x

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  • DOI: https://doi.org/10.1007/s11012-017-0653-x

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