Journal of Vibration Engineering & Technologies

, Volume 7, Issue 6, pp 579–589 | Cite as

Differential Quadrature Element Method for Free Vibration of Strain Gradient Beams with Elastic Boundary Conditions

  • Jingnong Jiang
  • Lifeng WangEmail author
  • Xinwei Wang
Original Paper



Numerous literatures on the vibrational analysis of structures based on the strain gradient elasticity theory (SGET) are only restricted to classic boundary conditions. However, the boundary conditions of structures in the engineering are different from those classic cases in nature. The purpose of this work is to develop a strong form differential quadrature element method (DQEM) to study the vibration of the strain gradient beams (SGB) with elastic boundary conditions.


Vibration of gradient elastic materials with elastic boundary conditions is studied by an SGB model. This model is characterized by a sixth-order differential equation (SODE). A strong-form DQEM is developed to calculate six-order boundary-value problems. Excellent accuracy, simplicity, and high computational efficiency of the DQEM have been demonstrated by comparing with exact solution and available results.

Results and conclusions

Numerical results verify the good reliability and accuracy of the DQEM. Numerical results show that the nonlocal effect parameter, boundary spring Stiffness, and high-order boundary conditions have important influence on the vibrational behaviors of the SGBs.


Strain gradient beam Differential quadrature element method Vibration Elastic boundary conditions 



This work was supported in part by the National Natural Science Foundation of China under Grants 11522217 and 11632003, and in part by the Fundamental Research Funds for the Central Universities of China.


  1. 1.
    Bellman R, Casti J (1971) Differential quadrature and long term integration. J Math Anal Appl 34:235–238MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bert CW, Jang SK, Striz AG (1988) Two new approximate methods for analyzing free vibration of structural components. AIAA J 26:612–618CrossRefGoogle Scholar
  3. 3.
    Jang SK, Bert CW, Striz AG (1989) Application of differential quadrature to deflection and buckling of structural components. Int J Numer Methods Eng 28:561–577CrossRefGoogle Scholar
  4. 4.
    Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49:1–28CrossRefGoogle Scholar
  5. 5.
    Shu C (2000) Differential Quadrature and Its Application in Engineering. Springer, LondonCrossRefGoogle Scholar
  6. 6.
    Chen W, Shu C, He W, Zhong T (2000) The application of special matrix product to differential quadrature solution of geometrically nonlinear bending of orthotropic rectangular plates. Comput Struct 74(1):65–76MathSciNetCrossRefGoogle Scholar
  7. 7.
    Civalek Ö (2004) Application of differential quadrature (DQ) and harmonic differential quadrature (HDQ) for buckling analysis of thin isotropic plates and elastic columns. Eng Struct 26(2):171–186CrossRefGoogle Scholar
  8. 8.
    Zong Z, Zhang YY (2009) Advanced differential quadrature methods. CRC Press, BeijingCrossRefGoogle Scholar
  9. 9.
    Liew KM, Han JB, Xiao ZM, Du H (1996) Differential quadrature method for Mindlin plates on Winkler foundations. Int J Mech Sci 38(4):405–421CrossRefGoogle Scholar
  10. 10.
    Ansari R, Gholami R, Rouhi H (2012) Various gradient elasticity theories in predicting vibrational response of single-walled carbon nanotubes with arbitrary boundary conditions. J Vib Control 19(5):708–719MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ke LL, Liu C, Wang YS (2015) Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions. Physica E 66:93–106CrossRefGoogle Scholar
  12. 12.
    Striz AG, Chen WL, Bert CW (1994) Static analysis of structures by the quadrature element method (QEM). Int J Solids Struct 31:2807–2818CrossRefGoogle Scholar
  13. 13.
    Striz AG, Chen WL, Bert CW (1997) Free vibration of plates by the high accuracy quadrature element method. J Sound Vib 202(5):689–702CrossRefGoogle Scholar
  14. 14.
    Wang XW, Gu HZ (1997) Static analysis of frame structures by the differential quadrature element method. Int J Numer Meth Eng 40:759–772CrossRefGoogle Scholar
  15. 15.
    Liu FL, Liew KM (1999) Vibration Analysis of discontinuous Mindlin plates by differential quadrature element method. J Vib Acoust ASME 121:204–208CrossRefGoogle Scholar
  16. 16.
    Chen CN (2000) Differential quadrature element analysis using extended differential quadrature. Comput Math Appl 39:65–79MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhong HZ, Wang Y (2010) Weak form quadrature element analysis of Bickford beams. Eur J Mech A Solids 29:851–858CrossRefGoogle Scholar
  18. 18.
    Wang XW (2015) Differential quadrature and differential quadrature based element methods: theory and applications. Butterworth-Heinemann, OxfordzbMATHGoogle Scholar
  19. 19.
    Tornabene F, Fantuzzi N, Ubertini F, Viola E (2015) Strong formulation finite element method based on differential quadrature: a survey. Appl Mech Rev 67:020801CrossRefGoogle Scholar
  20. 20.
    Wang XW, Yuan Z, Jin CH (2017) Weak form quadrature element method and its applications in science and engineering: a state-of-the-art review. Appl Mech Rev 69:030801CrossRefGoogle Scholar
  21. 21.
    Aifantis EC (1999) Strain gradient interpretation of size effects. Int J Fract 95:299–314CrossRefGoogle Scholar
  22. 22.
    Lam DCC, Yang F, Chong ACM, Wang J, Tong P (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508CrossRefGoogle Scholar
  23. 23.
    Gao XL, Park SK (2007) Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem. Int J Solids Struct 44:7486–7499CrossRefGoogle Scholar
  24. 24.
    Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313MathSciNetCrossRefGoogle Scholar
  25. 25.
    Akgoz B, Civalek O (2012) Analysis of microtubules based on strain gradient elasticity and modified couple stress theories. Adv Vib Eng 11(4):385–400zbMATHGoogle Scholar
  26. 26.
    Papargyri-Beskou S, Polyzos D, Beskos DE (2003) Dynamic analysis of gradient elastic flexural beams. Struct Eng Mech 15:705–716CrossRefGoogle Scholar
  27. 27.
    Wang LF, Hu HY (2005) Flexural wave propagation in single-walled carbon nanotubes. Phys Rev B 71:195412CrossRefGoogle Scholar
  28. 28.
    Papargyri-Beskou S, Beskos DE (2008) Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates. Arch Appl Mech 78:625–635CrossRefGoogle Scholar
  29. 29.
    Papargyri-Beskou S, Giannakopoulos AE, Beskos DE (2010) Variational analysis of gradient elastic flexural plates under static loading. Int J Solids Struct 47:2755–2766CrossRefGoogle Scholar
  30. 30.
    Pegios IP, Papargyri-Beskou S, Beskos DE (2015) Finite element static and stability analysis of gradient elastic beam structures. Acta Mech 226:745–768MathSciNetCrossRefGoogle Scholar
  31. 31.
    Artan R, Batra RC (2012) Free vibrations of a strain gradient beam by the method of initial values. Acta Mech 223:2393–2409MathSciNetCrossRefGoogle Scholar
  32. 32.
    Xu W, Wang LF, Jiang JN (2016) Strain gradient finite element analysis on the vibration of double-layered graphene sheets. Int J Comput Methods 13:1650011MathSciNetCrossRefGoogle Scholar
  33. 33.
    Askes H, Aifantis EC (2011) Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results. Int J Solids Struct 48:1962–1990CrossRefGoogle Scholar
  34. 34.
    Jiang JN, Wang LF, Zhang YQ (2017) Vibration of single-walled carbon nanotubes with elastic boundary conditions. Int J Mech Sci 122:156–166CrossRefGoogle Scholar
  35. 35.
    Jiang JN, Wang LF (2018) Analytical solutions for the thermal vibration of strain, gradient beams with elastic boundary conditions. Acta Mech 229:2203–2219MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.State Key Laboratory of Mechanics and Control of Mechanical StructuresNanjing University of Aeronautics and AstronauticsNanjingChina

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