Applied Mathematics and Mechanics

, Volume 40, Issue 4, pp 549–562 | Cite as

Quasi-static and dynamical analyses of a thermoviscoelastic Timoshenko beam using the differential quadrature method

  • Qiang Lyu
  • Jingjing Li
  • Nenghui ZhangEmail author


The quasi-static and dynamic responses of a thermoviscoelastic Timoshenko beam subject to thermal loads are analyzed. First, based on the small geometric deformation assumption and Boltzmann constitutive relation, the governing equations for the beam are presented. Second, an extended differential quadrature method (DQM) in the spatial domain and a differential method in the temporal domain are combined to transform the integro-partial-differential governing equations into the ordinary differential equations. Third, the accuracy of the present discrete method is verified by elastic/viscoelastic examples, and the effects of thermal load parameters, material and geometrical parameters on the quasi-static and dynamic responses of the beam are discussed. Numerical results show that the thermal function parameter has a great effect on quasi-static and dynamic responses of the beam. Compared with the thermal relaxation time, the initial vibrational responses of the beam are more sensitive to the mechanical relaxation time of the thermoviscoelastic material.

Key words

Timoshenko beam thermoviscoelasticity thermal load dynamic response differential quadrature method (DQM) 

Chinese Library Classification


2010 Mathematics Subject Classification

74A15 74H45 


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  1. [1]
    AMIRIAN, B., HOSSEINI-ARA, R., and MOOSAVI, H. Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model. Applied Mathematics and Mechanics (English Edition), 35, 875–886 (2014) Scholar
  2. [2]
    ZOCHER, M. A., GROVES, S. E., and ALLEN, D. H. A three-dimensional finite element formu-lation for thermoviscoelastic orthotropic media. International Journal for Numerical Methods in Engineering, 40, 2267–2288 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    ARAKI, W., ADACHI, T., and YAMAJI, A. Thermal stress analysis of thermoviscoelastic hollow cylinder with temperature-dependent thermal properties. Journal of Thermal Stresses, 28, 29–46 (2005)CrossRefGoogle Scholar
  4. [4]
    ZHANG, N. H. and XING, J. J. Vibration analysis of linear coupled thermoviscoelastic thin plates by a variational approach. International Journal of Solids and Structures, 45, 2583–2597 (2008)CrossRefzbMATHGoogle Scholar
  5. [5]
    EZZAT, M. A., EL-KARAMANY, A. S., and EL-BARY, A. A. On thermo-viscoelasticity with variable thermal conductivity and fractional-order heat transfer. International Journal of Ther-mophysics, 36, 1–14 (2015)CrossRefGoogle Scholar
  6. [6]
    CHEN, L. Q. and CHENG, C. J. Dynamical behavior of nonlinear viscoelastic beams. Applied Mathematics and Mechanics (English Edition), 21, 995–1001 (2000) Scholar
  7. [7]
    MANOACH, E. and RIBEIRO, P. Coupled, thermoelastic, large amplitude vibrations of Timo-shenko beams. International Journal of Mechanical Sciences, 46, 1589–1606 (2004)CrossRefzbMATHGoogle Scholar
  8. [8]
    PARAYIL, D. V., KULKARNI, S. S., and PAWASKAR, D. N. Analytical and numerical solutions for thick beams with thermoelastic damping. International Journal of Mechanical Sciences, 94-95, 10–19 (2015)CrossRefGoogle Scholar
  9. [9]
    DARBAN, H. and MASSABO, R. Thermo-elastic solutions for multilayered wide plates and beams with interfacial imperfections through the transfer matrix method. Meccanica, 53, 553–571 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    BERTI, A., RIVERA, J. E. M., and NASO, M. G. A contact problem for a thermoelastic Timo-shenko beam. Zeitschrift f¨ur Angewandte Mathematik und Physik, 66, 1969–1986 (2015)CrossRefzbMATHGoogle Scholar
  11. [11]
    YANG, X. D. and ZHANG, W. Nonlinear dynamics of axially moving beam with coupled longitudinal-transversal vibrations. Nonlinear Dynamics, 78, 2547–2556 (2014)CrossRefGoogle Scholar
  12. [12]
    CHEN, L. Q. and DING, H. Steady-state transverse response in coupled planar vibration of axially moving viscoelastic beams. Journal of Vibration and Acoustics, 132, 011009 (2010)CrossRefGoogle Scholar
  13. [13]
    IESAN, D. First-strain gradient theory of thermoviscoelasticity. Journal of Thermal Stresses, 38, 701–715 (2015)CrossRefGoogle Scholar
  14. [14]
    IESAN, D. On the nonlinear theory of thermoviscoelastic materials with voids. Journal of Elas-ticity, 128, 1–16 (2016)MathSciNetzbMATHGoogle Scholar
  15. [15]
    BERNARDI, C. and COPETTI, M. I. M. Discretization of a nonlinear dynamic thermoviscoelastic Timoshenko beam model. Zeitschrift f¨ur Angewandte Mathematik und Mechanik, 97, 532–549 (2017)MathSciNetCrossRefGoogle Scholar
  16. [16]
    FU, Y. M. and TAO, C. Nonlinear dynamic responses of viscoelastic fiber-metal-laminated beams under the thermal shock. Journal of Engineering Mathematics, 98, 113–128 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    CHEN, L. Q., DING, H., and LIM, C. W. Principal parametric resonance of axially accelerat-ing viscoelastic beams: multi-scale analysis and differential quadrature verification. Shock and Vibration, 19, 527–543 (2012)CrossRefGoogle Scholar
  18. [18]
    EFTEKHARI, S. A. A differential quadrature procedure for linear and nonlinear steady state vibrations of infinite beams traversed by a moving point load. Meccanica, 51, 1–18 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    SAVIZ, M. R. Electro-elasto-dynamic analysis of functionally graded cylindrical shell with piezo-electric rings using differential quadrature method. Acta Mechanica, 228, 1645–1670 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    ZHANG, R., LIANG, X., and SHEN, S. A Timoshenko dielectric beam model with flexoelectric effect. Meccanica, 51, 1181–1188 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    WANG, X. and BERT, C. W. A new approach in applying differential quadrature to static and free vibrational analyses of beams and plates. Journal of Sound and Vibration, 162, 566–572 (1993)CrossRefzbMATHGoogle Scholar
  22. [22]
    LI, J. J. and CHENG, C. J. Differential quadrature method for analyzing nonlinear dynamic characteristics of viscoelastic plates with shear effects. Nonlinear Dynamics, 61, 57–70 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    AMOOZGAR, M. R. and SHAHVERDI, H. Analysis of nonlinear fully intrinsic equations of geometrically exact beams using generalized differential quadrature method. Acta Mechanica, 227, 1265–1277 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    KIENDL, J., AURICCHIO, F., HUGHES, T. J. R., and REALI, A. Single-variable formulations and isogeometric discretizations for shear deformable beams. Computer Methods in Applied Me-chanics and Engineering, 284, 988–1004 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    TIMOSHENKO, S. and GERE, J. Mechanics of Materials, Van Nostrand Reinhold Company, New York, 315–426 (1972)Google Scholar
  26. [26]
    CHEN, T. M. The hybrid Laplace transform/finite element method applied to the quasi-static and dynamic analysis of viscoelastic Timoshenko beams. International Journal for Numerical Methods in Engineering, 38, 509–522 (1995)CrossRefzbMATHGoogle Scholar
  27. [27]
    YU, O. Y., JIANG, Y., and ZHOU, L. Analytical solution of bending of viscoelastic timber beam reinforced with fibre reinforcement polymer (FRP) sheet (in Chinese). Journal of Shanghai University (Natural Science), 22, 609–622 (2016)Google Scholar
  28. [28]
    LI, S. R. and ZHOU, Y. H. Geometrically nonlinear analysis of Timoshenko beams under ther-momechanical loadings. Journal of Thermal Stresses, 26, 691–700 (2003)CrossRefGoogle Scholar
  29. [29]
    ABOUDI, J., PINDERA, M. J., and ARNOLD, S. M. Linear thermoelastic higher-order theory for periodic multiphase materials. Journal of Applied Mechanics, 68, 697–707 (2001)CrossRefzbMATHGoogle Scholar
  30. [30]
    UYGUNOGLU, T. and TOPCU, I. B. Thermal expansion of self-consolidating normal and lightweight aggregate concrete at elevated temperature. Construction and Building Materials, 23, 3063–3069 (2009)CrossRefGoogle Scholar
  31. [31]
    CHIBA, R. Stochastic thermal stresses in an FGM annular disc of variable thickness with spatially random heat transfer coefficients. Meccanica, 44, 159–176 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    SUN, Y. X., FANG, D. N., and SOH, A. K. Thermoelastic damping in micro-beam resonators. International Journal of Solids and Structures, 43, 3213–3229 (2006)CrossRefzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  2. 2.Department of Mechanics, College of SciencesShanghai UniversityShanghaiChina

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