International Journal of Steel Structures

, Volume 19, Issue 2, pp 446–468 | Cite as

Stability Analysis of Frame Having FG Tapered Beam–Column

  • M. Rezaiee-PajandEmail author
  • Amir R. Masoodi


This study focuses on the analysis of the tapered beam–column having functionally graded section. By using two different methods, the substantial effects of the coupling–extensional bending of this structure are considered. Both second-order effects and flexibility of connections are considered in the analysis. The suggested exact stiffness matrix is very general and suitable for analyzing any plane frame, which has non-prismatic members and various types of connections. By using this formulation, the frame stability analysis of the FG beam with power function will be performed. Furthermore, the effects of semi-rigid connections and supports on the buckling load of 2D frames will be studied. Finally, comparing the new responses with the available solutions shows the accuracy, efficiency and capabilities of the proposed stiffness matrix. In addition, some new problems, including 2D frame, especially gabled frame, are solved using authors’ schemes.


Coupling–extensional effect Functionally graded materials Semi-rigid connections Exact stiffness matrix Buckling load Non-prismatic beam Second-order effects 


  1. Adamek, V., & Vales, F. (2015). Analytical solution for a heterogeneous Timoshenko beam subjected to an arbitrary dynamic transverse load. European Journal of Mechanics A/Solids, 49, 373–381.MathSciNetCrossRefzbMATHGoogle Scholar
  2. AI-Gahtani, J. (1996). Exact stiffness for tapered members. Journal of Structural Engineering, 122, 1234–1239.CrossRefGoogle Scholar
  3. AI-Gahtani, J., & Khan, M. S. (1998). Exact analysis of nonprismatic beams. Journal of Engineering Mechanics, 12, 1290–1293.CrossRefGoogle Scholar
  4. Aristizabal-Ochoa, J. D. (1993). Static, stability and vibration of non-prismatic beam and column. Journal of Sound and Vibration, 162, 441–455.CrossRefzbMATHGoogle Scholar
  5. Avraam, T. P., & Fasoulakis, Z. C. (2013). Nonlinear postbuckling analysis of frames with varying cross-section columns. Journal of Engineering Structures, 56, 1–7.CrossRefGoogle Scholar
  6. Banerjee, J. R. (1978). Compact computation of buckling loads for plane frames consisting of tapered members. Advances in Engineering Software, 9, 162–170.CrossRefzbMATHGoogle Scholar
  7. Banerjee, J. R., & Williams, F. W. (1986). Exact Bernoulli–Euler static stiffness matrix for a range of tapered beam–columns. International Journal for Numerical Methods in Engineering, 23, 1615–1628.CrossRefzbMATHGoogle Scholar
  8. Chen, D., Yang, J., & Kitipornchai, S. (2015). Elastic buckling and static bending of shear deformable functionally graded porous beam. Composite Structures, 133, 54–61.CrossRefGoogle Scholar
  9. Darbandi, S. M., Firouz-Abadi, R. D., & Haddadpour, H. (2010). Buckling of variable section columns under axial loading. Journal of Engineering Mechanics, 136, 472–476.CrossRefGoogle Scholar
  10. Faella, C., Piluso, V., & Rizzano, G. (2000). Structural steel semi-rigid connections: Theory, design, and software. Boca Raton: CRC Press.Google Scholar
  11. Fang, J.-S., & Zhou, D. (2016). Free vibration analysis of rotating axially functionally graded tapered Timoshenko beams. International Journal of Structural Stability and Dynamics, 16, 1550007.MathSciNetCrossRefzbMATHGoogle Scholar
  12. Farhatnia, F., Bagheri, M. A., & Ghobadi, A. (2012). Buckling analysis of FGM thick beam under different boundary conditions using GDQM. Advanced Materials Research, 433, 4920–4924.CrossRefGoogle Scholar
  13. Huang, Y., & Li, X. F. (2011). Buckling analysis of non-uniform and axially graded columns with varying flexural rigidity. Journal of Engineering Mechanics, 137, 73–81.CrossRefGoogle Scholar
  14. Ichikawa, K. (2001). Functionally graded materials in the 21st century. New York: Springer.CrossRefGoogle Scholar
  15. Ihaddoudène, A. N. T., Saidani, M., & Chemrouk, M. (2009). Mechanical model for the analysis of steel frames with semi-rigid joints. Journal of Constructional Steel Research, 65, 631–640.CrossRefGoogle Scholar
  16. Jin, C., & Wang, X. (2015). Accurate free vibration analysis of Euler functionally graded beams by the weak form quadrature element method. Composite Structures, 125, 41–50.CrossRefGoogle Scholar
  17. Kar, V. R., Mahapatra, T. R., & Panda, S. K. (2017). Effect of different temperature load on thermal postbuckling behaviour of functionally graded shallow curved shell panels. Composite Structures, 160, 1236–1247.CrossRefGoogle Scholar
  18. Kar, V. R., & Panda, S. K. (2015a). Large deformation bending analysis of functionally graded spherical shell using FEM. Structural Engineering and Mechanics, 53, 661–679.CrossRefGoogle Scholar
  19. Kar, V. R., & Panda, S. K. (2015b). Thermoelastic analysis of functionally graded doubly curved shell panels using nonlinear finite element method. Composite Structures, 129, 202–212.CrossRefGoogle Scholar
  20. Kar, V. R., & Panda, S. K. (2016a). Nonlinear free vibration of functionally graded doubly curved shear deformable panels using finite element method. Journal of Vibration and Control, 22, 1935–1949.MathSciNetCrossRefGoogle Scholar
  21. Kar, V. R., & Panda, S. K. (2016b). Nonlinear thermomechanical behavior of functionally graded material cylindrical/hyperbolic/elliptical shell panel with temperature-dependent and temperature-independent properties. Journal of Pressure Vessel Technology, 138, 061202–061202–061202–061213.CrossRefGoogle Scholar
  22. Kar, V. R., & Panda, S. K. (2016c). Post-buckling behaviour of shear deformable functionally graded curved shell panel under edge compression. International Journal of Mechanical Sciences, 115–116, 318–324.CrossRefGoogle Scholar
  23. Kar, V. R., & Panda, S. K. (2017). Large-amplitude vibration of functionally graded doubly-curved panels under heat conduction. AIAA Journal, 55, 4376–4386.CrossRefGoogle Scholar
  24. Kataoka, M., & de Cresce El Debs, A. (2015). Beam–column composite connections under cyclic loading: An experimental study. Materials and Structures, 48, 929–946.CrossRefGoogle Scholar
  25. Khandan, R., Noroozi, S., Sewell, P., et al. (2012). The development of laminated composite plate theories: A review. Journal of Materials Science, 47, 5901–5910.CrossRefGoogle Scholar
  26. Li, S. R., & Batra, R. C. (2013). Relations between buckling loads of functionally graded Timoshe nko and homogeneous Euler–Bernoulli beams. Journal of Composite Structures, 95, 5–9.CrossRefGoogle Scholar
  27. Li, Y. S., Feng, W. J., & Cai, Z. Y. (2014). Bending and free vibration of functionally graded piezoelectric beam based on modified strain gradient theory. Composite Structures, 115, 41–50.CrossRefGoogle Scholar
  28. Lim, T.-C. (2004). Elastic properties of a Poisson–shear material. Journal of Materials Science, 39, 4965–4969.CrossRefGoogle Scholar
  29. Luo, Y.-Z., Xu, X., & Wu, F. (2007). Accurate stiffness matrix for non-prismatic members. Journal of Structural Engineering, 133, 1168–1175.CrossRefGoogle Scholar
  30. Mahapatra, T. R., Panda, S. K., & Kar, V. R. (2016). Geometrically nonlinear flexural analysis of hygro-thermo-elastic laminated composite doubly curved shell panel. International Journal of Mechanics and Materials in Design, 12, 153–171.CrossRefGoogle Scholar
  31. Motavalli, M., Farshad, M., & Flüeler, P. (1993). Buckling of polymer pipes under internal pressure. Materials and Structures, 26, 348–354.CrossRefGoogle Scholar
  32. Navadeh, N., Hewson, R., & Fallah, A. (2018). Dynamics of transversally vibrating non-prismatic Timoshenko cantilever beams. Engineering Structures, 166, 511–525.CrossRefGoogle Scholar
  33. Nguyen, N. T., Kim, N. I., Cho, I., et al. (2014). Static analysis of transversely or axially functionally graded tapered beams. Materials Research Innovations, 18, S2-260–S262-264.CrossRefGoogle Scholar
  34. Niknam, H., Fallah, A., & Aghdam, M. M. (2014). Nonlinear bending of functionally graded tapered beams subjected to thermal and mechanical loading. International Journal of Non-Linear Mechanics, 65, 141–147.CrossRefGoogle Scholar
  35. Nikolić, A., & Šalinić, S. (2017). Buckling analysis of non-prismatic columns: A rigid multibody approach. Engineering Structures, 143, 511–521.CrossRefGoogle Scholar
  36. Rezaiee-Pajand, M., Bambaeechee, M., & Sarafrazi, M. (2011). Static and dynamic nonlinear analysis of semi-rigid steel frames with new beam–column element. International Journal of Engineering (IJE), 24, 203–221.Google Scholar
  37. Rezaiee-Pajand, M., & Masoodi, A. R. (2016). Exact natural frequencies and buckling load of functionally graded material tapered beam–columns considering semi-rigid connections. Journal of Vibration and Control, 24, 1787–1808.MathSciNetCrossRefzbMATHGoogle Scholar
  38. Rezaiee-Pajand, M., Masoodi, A. R., & Mokhtari, M. (2018). Static analysis of functionally graded non-prismatic sandwich beams. Advances in Computational Design, 3, 165–190.Google Scholar
  39. Salmon, C. G., & Johnson, J. E. (1992). Steel structures: Design and behavior. New York: Happer and Rrow.Google Scholar
  40. Sankar, B. V. (2001). An elasticity solution for functionally graded beams. Composites Science and Technology, 61, 689–696.CrossRefGoogle Scholar
  41. Schoutens, J., & Higa, D. (1986). Linear stress relations for a metal matrix composite sandwich beam with any core material. Journal of Materials Science, 21, 1943–1946.CrossRefGoogle Scholar
  42. Shooshtari, A., Heyrani Moghaddam, S., & Masoodi, A. R. (2015). Pushover analysis of gabled frames with semi-rigid connections. Steel and Composite Structures, 18, 1557–1568.CrossRefGoogle Scholar
  43. Shooshtari, A., & Khajavi, R. (2010). An efficient procedure to find shape functions and stiffness matrices of non-prismatic Euler–Bernoulli and Timoshenko beam elements. European Journal of Mechanics A/Solids, 29, 826–836.CrossRefGoogle Scholar
  44. Steiger, R., & Fontana, M. (2005). Bending moment and axial force interacting on solid timber beams. Materials and Structures, 38, 507–513.CrossRefGoogle Scholar
  45. Su, H., Banerjee, J. R., & Cheung, C. W. (2013). Dynamic stiffness formulation and free vibration analysis of functionally graded beams. Composite Structures, 106, 854–862.CrossRefGoogle Scholar
  46. Sun, G., & Sun, C. T. (1995). Bending of shape-memory alloy-reinforced composite beam. Journal of Materials Science, 30, 5750–5754.CrossRefGoogle Scholar
  47. Tomandl, G., Mangler, M., Stoyan, D., et al. (2006). Characterisation methods for functionally graded materials. Journal of Materials Science, 41, 4143–4151.CrossRefGoogle Scholar
  48. Valipour, H. R., & Bradford, M. A. (2012). A new shape function for tapered three-dimensional beams with flexible connections. Journal of Constructional Steel Research, 70, 43–50.CrossRefGoogle Scholar
  49. Yan, J.-B. (2015). Finite element analysis on steel–concrete–steel sandwich beams. Materials and Structures, 48, 1645–1667.CrossRefGoogle Scholar
  50. Ying, J., Lu, C. F., & Chen, W. Q. (2008). Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations. Journal of Composite Structures, 84, 209–219.CrossRefGoogle Scholar
  51. Yoshihara, H. (2010). Analysis of the elastic buckling of a plywood column. Materials and Structures, 43, 1075–1083.CrossRefGoogle Scholar
  52. Zeinali, Y. H., Jamali, S. M., & Musician, S. (2013). General form of the stiffness matrix of a tapered beam–column. International Journal of Mining, Metallurgy & Mechanical Engineering (IJMMME), 1, 149–153.Google Scholar
  53. Zenkour, A. M. (2007). Elastic deformation of the rotating functionally graded annular disk with rigid casing. Journal of Materials Science, 42, 9717–9724.CrossRefGoogle Scholar
  54. Zhou, Z., & Chan, S. (1995). Self-equilibrating element for second-order analysis of semirigid jointed frames. Journal of Engineering Mechanics, 121, 896–902.CrossRefGoogle Scholar

Copyright information

© Korean Society of Steel Construction 2018

Authors and Affiliations

  1. 1.Department of Civil EngineeringFerdowsi University of MashhadMashhadIran

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