Skip to main content
SpringerLink
Log in

Homogenized and classical expressions for static bending solutions for functionally graded material Levinson beams

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The exact relationship between the bending solutions of functionally graded material (FGM) beams based on the Levinson beam theory and those of the corresponding homogenous beams based on the classical beam theory is presented for the material properties of the FGM beams changing continuously in the thickness direction. The deflection, the rotational angle, the bending moment, and the shear force of FGM Levinson beams (FGMLBs) are given analytically in terms of the deflection of the reference homogenous Euler-Bernoulli beams (HEBBs) with the same loading, geometry, and end supports. Consequently, the solution of the bending of non-homogenous Levinson beams can be simplified to the calculation of transition coefficients, which can be easily determined by variation of the gradient of material properties and the geometry of beams. This is because the classical beam theory solutions of homogenous beams can be easily determined or are available in the textbook of material strength under a variety of boundary conditions. As examples, for different end constraints, particular solutions are given for the FGMLBs under specified loadings to illustrate validity of this approach. These analytical solutions can be used as benchmarks to check numerical results in the investigation of static bending of FGM beams based on higher-order shear deformation theories.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Pradhan, S. C. and Murmu, T. Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method. Journal of Sound and Vibration, 321, 342–362 (2009)

    Article  Google Scholar 

  2. Alshorbagy, A. E., Eltaher, M. A., and Mahmoud, F. F. Free vibration characteristics of a functionally graded beam by finite element method. Applied Mathematical Modelling, 35, 412–425 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Simsek, M. and Kocatürk, T. Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load. Composite Structures, 90, 465–473 (2009)

    Article  Google Scholar 

  4. Yang, J. and Chen, Y. Free vibration and buckling analyses of functionally graded beams with edge cracks. Composite Structures. 83, 48–60 (2008)

    Article  Google Scholar 

  5. Li, S. R. and Liu, P. Analogous transformation of static and dynamic solutions between functionally graded material and uniform beams (in Chinese). Mechanics and Engineering, 32(5), 45–49 (2010)

    Google Scholar 

  6. Li, S. R., Su, H. D., and Cheng, C. J. Free vibration of functionally graded material beams with surface-bonded piezoelectric layers in thermal environment. Applied Mathematics and Mechanics (English Edition), 30(8), 969–982 (2009) DOI 10.1007/s10483-009-0803-7

    Article  MATH  Google Scholar 

  7. Yaghoobi, H. and Torabi, H. M. Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation. Applied Mathematical Modelling, 37, 8324–8340 (2013)

    Article  MathSciNet  Google Scholar 

  8. Levyakov, S. V. Elastica solution for thermal bending of a functionally graded beam. Acta Mechanica, 224, 1731–1740 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Sina, S. A., Navazi, H. M., and Haddadpour, H. M. H. An analytical method for free vibration analysis of functionally graded beams. Materials and Design, 30, 741–747 (2009)

    Article  Google Scholar 

  10. Li, X. F. A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams. Journal of Sound and Vibration, 318, 1210–1229 (2008)

    Article  Google Scholar 

  11. Huang, Y. and Li, X. F. Bending and vibration of cylindrical beams with arbitrary radial nonhomogeneity. International Journal of Mechanical Science, 52, 595–601 (2010)

    Article  Google Scholar 

  12. Kiani, Y. and Eslami, M. R. Thermomechanical buckling of temperature dependent FGM beams. Latin American Journal of Solids and Structures, 10, 223–246 (2013)

    Article  Google Scholar 

  13. Murin, J., Aminbaghai, M., Hrabovsky, J., Kutiš, V., and Kugler, S. Modal analysis of the FGM beams with effect of the shear correction function. Composites: Part B, 45, 1575–1582 (2013)

    Article  Google Scholar 

  14. Fu, Y. M., Chen, Y., and Zhang, P. Thermal buckling analysis of functionally graded beam with longitudinal crack. Meccanica, 48, 1227–1237 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Pradhan, K. K. and Chakraverty, S. Free vibration of Euler and Timoshenko functionally graded beams by Rayleigh-Ritz method. Composites Part B: Engineering, 51, 175–184 (2013)

    Article  Google Scholar 

  16. Ansari, R., Gholami, R., Shojaei, M. F., Mohammadi, V., and Sahmani, S. Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory. Composite Structures, 100, 385–397 (2013)

    Article  Google Scholar 

  17. Ma, L. S. and Lee, D. W. Exact solutions for nonlinear static responses of a shear deformable FGM beam under in-plane thermal loading. European Journal of Mechanics A, Solids, 31, 13–20 (2011)

    Article  MathSciNet  Google Scholar 

  18. Ma, L. S. and Lee, D. W. A further discussion of nonlinear mechanical behavior of FGM beam under in-plane thermal loading. Composite Structures, 93, 831–842 (2011)

    Article  Google Scholar 

  19. Esfahani, S. E., Kiani, Y., and Eslami, M. R. Non-linear thermal stability analysis of temperature dependent FGM beams supported on non-linear hardening elastic foundations. International Journal of Mechanical Sciences, 69, 10–20 (2013)

    Article  Google Scholar 

  20. Benatta, M. A., Tounsi, A., Mechab, I., and Bouiadjra, M. B. Mathematical solution for bending of short hybrid composite beams with variable fibers spacing. Applied Mathematics and Computation, 212, 337–348 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sallai, B. O., Tounsi, A., Mechab, I., Bachir, M. B., Meradjah, M. B., and Adda, E. A. A theoretical analysis of flexional bending of Al/Al2O3 S-FGM thick beams. Computational Materials Science, 44, 1344–1350 (2009)

    Google Scholar 

  22. Kadoli, R., Akhtar, K., and Ganesan, N. Static analysis of functionally graded beams using higher order shear deformation theory. Applied Mathematical Modeling, 32, 2509–2523 (2008)

    Article  MATH  Google Scholar 

  23. Aydogdu, M. and Taskin, V. Free vibration analysis of functionally graded beams with simply supported edges. Material Design, 28, 1651–1656 (2007)

    Article  Google Scholar 

  24. Şimşek, M. Fundamental frequency analysis of functionally graded beams by using different higherorder beam theories. Nuclear Engineering and Design, 240, 697–705 (2010)

    Article  Google Scholar 

  25. Mahi, A., Bedia, E. A. A., Tounsi, A., and Mechab, I. An analytical method for temperaturedependent free vibration analysis of functionally graded beams. Composite Structures, 92, 1877–1887 (2010)

    Article  Google Scholar 

  26. Zhang, D. G. Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory. Composite Structures, 100, 121–126 (2013)

    Article  Google Scholar 

  27. Shen, H. S. and Wang, Z. X. Nonlinear analysis of shear deformable FGM beams resting on elastic foundations in thermal environments. International Journal of Mechanical Sciences, 81, 195–206 (2014)

    Article  Google Scholar 

  28. Sankar, B. V. An elasticity solution for functionally graded beams. Composites Science and Technology, 61, 689–696 (2001)

    Article  Google Scholar 

  29. Zhong, Z. and Yu, T. Analytical solution of cantilever functionally graded beam. Composite Science and Technology, 67, 481–488 (2007)

    Article  Google Scholar 

  30. Ding, H. J., Huang, D. J., and Chen, W. Q. Elastic solution for plane anisotropic functionally graded beams. International Journal of Solids and Structures, 44, 176–196 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Abrate, S. Free vibration, buckling and static deflections of functionally graded plates. Composites Science and Technology, 66, 2383–2394 (2006)

    Article  Google Scholar 

  32. Abrate, S. Functionally graded plates behave like homogenous plates. Composites Part B: Engineering, 39, 151–158 (2008)

    Article  Google Scholar 

  33. Reddy, J. R., Wang, C. M., and Kitipornchai, S. Axisymmetric bending of functionally graded circular and annular plates. European Journal of Mechanics-A/Solids, 18, 185–199 (1999)

    Article  MATH  Google Scholar 

  34. Ma, L. S. and Wang, T. J. Relationships between axisymmetric bending and buckling solutions of FGM circular plates based on third-order plate theory and classical plate theory. International Journal of Solids and Structures, 41, 85–101 (2004)

    Article  MATH  Google Scholar 

  35. Cheng, Z. Q. and Kitipornchai, S. Exact bending solution of inhomogenous plates from homogenous thin-plate deflection. AIAA Journal, 38, 1289–1291 (2000)

    Article  Google Scholar 

  36. Levinson, M. A new rectangular beam theory. Journal of Sound and Vibration, 74, 81–87 (1981)

    Article  MATH  Google Scholar 

  37. Reddy, J. N., Wang, C. M., and Lee, K. H. Relationships between bending solutions of classical and shear deformation beam theories. International Journal of Solids and Structures, 26, 3373–3384 (1997)

    Article  Google Scholar 

  38. Reddy, J. N., Wang, C. M., Lee, K. H., and Ng, K. H. Bending solutions of Levinson beams and plates in terms of the classical theories. International Journal of Solids and Structures, 38, 4701–4720 (2001)

    Article  MATH  Google Scholar 

  39. Wang, C. M. Timoshenko beam-bending solutions in terms of Euler-Bernoulli solutions. Journal of Engineering Mechanics ASCE, 121, 763–765 (1995)

    Article  Google Scholar 

  40. Reddy, J. N., Wang, C. M., and Lee, K. H. Shear Deformable Beams and Plates-Relationship with Classical Solutions, Elsevier, Singapore (2000)

    Google Scholar 

  41. Li, S. R., Cao, D. F., and Wan, Z. Q. Bending solutions of FGM Timoshenko beams from those of the homogenous Euler-Bernoulli beams. Applied Mathematical Modelling, 37, 7077–7085 (2013)

    Article  MathSciNet  Google Scholar 

  42. Li, S. R. and Batra, R. C. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler-Bernoulli beams. Composites and Structures, 95, 5–9 (2013)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Civil Science and Engineering, Yangzhou University, Yangzhou, 225127, Jiangsu Province, China

    Shirong Li, Zeqing Wan & Xuan Wang

  2. School of Hydraulic, Energy and Power Engineering, Yangzhou University, Yangzhou, 225127, Jiangsu Province, China

    Shirong Li, Zeqing Wan & Xuan Wang

Authors
  1. Shirong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zeqing Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shirong Li.

Additional information

Project supported by the National Natural Science Foundation of China (No. 11272278)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S., Wan, Z. & Wang, X. Homogenized and classical expressions for static bending solutions for functionally graded material Levinson beams. Appl. Math. Mech.-Engl. Ed. 36, 895–910 (2015). https://doi.org/10.1007/s10483-015-1956-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-015-1956-9

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation