DOI QR코드

DOI QR Code

Analytical approximate solutions for large post-buckling response of a hygrothermal beam

  • Yu, Yongping (College of Construction Engineering, Jilin University) ;
  • Sun, Youhong (College of Construction Engineering, Jilin University)
  • 투고 : 2011.05.06
  • 심사 : 2012.06.01
  • 발행 : 2012.07.25

초록

This paper deals with large deformation post-buckling of a linear-elastic and hygrothermal beam with axially nonmovable pinned-pinned ends and subjected to a significant increase in swelling by an alternative method. Analytical approximate solutions for the geometrically nonlinear problem are presented. The solution for the limiting case of a string is also obtained. By coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials, the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-nonlinear equation, and supplementary condition with cosinoidal nonlinearity can also be simplified to be a polynomial integral equation. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. Two approximate formulae for load along axis, potential strain for free hygrothermal expansion and periodic solution are established for small as well as large angle of rotation at the end of the beam. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.

키워드

참고문헌

  1. Anandrao, K.S., Gupta, R.K., Ramchandran, P. and Rao, G.V. (2010), "Thermal post-buckling analysis of uniform slender functionally graded material beams", Struct. Eng. Mech., 36(5), 545-560. https://doi.org/10.12989/sem.2010.36.5.545
  2. Abramowitz, M. and Stegun, I.A. (1965), Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York.
  3. Aoki, Y., Yamada, K. and Ishikawa, T. (2008), "Effect of hygrothermal condition on compression after impact strength of CFRP laminates", Compos. Sci. Technol., 68, 1376-1383. https://doi.org/10.1016/j.compscitech.2007.11.015
  4. Ascher, U.M., Mattheij, R.M.M. and Russell, R.D. (1988), Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall, Englewood Cliffs.
  5. Belendez, A., Alvarez, M.L., Fernandez, E. and Pascual, I. (2009), "Linearization of conservative nonlinear oscillators", Eur. J. Phys., 30, 259-270. https://doi.org/10.1088/0143-0807/30/2/004
  6. Boley, B.A. and Weiner, J.H. (1997), Theory of Thermal Stresses, Dover Publications, New York.
  7. Cisternas, J. and Holmes, P. (2002), "Buckling of extensible thermoelastic rods", Math. Comput. Model., 36, 233-243. https://doi.org/10.1016/S0895-7177(02)00122-X
  8. Coffin, D.W. and Bloom, F. (1999), "Elastica solution for the hygrothermal buckling of a beam", Int. J. Nonlin. Mech., 34, 935-947. https://doi.org/10.1016/S0020-7462(98)00067-5
  9. Dafedar, J.B. and Desai1, Y.M. (2002), "Thermomechanical buckling of laminated composite plates using mixed, higher-order analytical formulation", J. Appl. Mech., 69, 790-799. https://doi.org/10.1115/1.1490372
  10. Denman, H.H. (1969), "An approximate equivalent linearization technique for nonlinear oscillations", J. Appl. Mech.-T. ASME, 36, 358-360. https://doi.org/10.1115/1.3564651
  11. El Naschie, M.S. (1976), "Thermal initial post-buckling of the extensional elastica", Int. J. Mech. Sci., 18, 321-324. https://doi.org/10.1016/0020-7403(76)90034-5
  12. Jekot, T. (1996), "Non-linear problems of thermal buckling of a beam", J. Therm. Stresses, 19(4), 359-367. https://doi.org/10.1080/01495739608946180
  13. Jonckheere, R.E. (1971), "Determination of the period of nonlinear oscillations by means of Chebyshev polynomials", ZAMM-Z. Angew. Math. Mech., 51, 389-393. https://doi.org/10.1002/zamm.19710510508
  14. Kundu, C.K. and Han, J.H. (2009), "Nonlinear buckling analysis of hygrothermoelastic composite shell panels using finite element method", Compos. Part B, 40, 313-C328. https://doi.org/10.1016/j.compositesb.2008.12.001
  15. Kocaturk, T. and Akbas, S.D. (2011), "Post-buckling analysis of Timoshenko beams with various boundary conditions under non-uniform thermal loading", Struct. Eng. Mech., 40(3), 347-371. https://doi.org/10.12989/sem.2011.40.3.347
  16. Kocaturk, T. and Akbas, S.D. (2012), "Post-buckling analysis of Timoshenko beams made of functionally graded material under thermal loading", Struct. Eng. Mech., 41(6), 347-371.
  17. Lal, A., Singh, B.N. and Kale, S. (2011), "Stochastic post buckling analysis of laminated composite cylindrical shell panel subjected to hygrothermomechanical loading", Compos. Struct., 93, 1187-1200. https://doi.org/10.1016/j.compstruct.2010.11.005
  18. Li, P.S., Sun, W.P. and Wu, B.S. (2008), "Analytical approximate solutions to large amplitude oscillation of a simple pendulum", J. Vib. Shock, 27, 72-74. (In Chinese)
  19. Li, G.Q., Wang, P.J. and Hou, H.T. (2009), "Post-buckling behaviours of axially restrained steel columns in fire", Steel Compos. Struct., 9(2), 89-101. https://doi.org/10.12989/scs.2009.9.2.089
  20. Li, S.R. and Cheng, C.J. (2000), "Analysis of thermal post-buckling of heated elastic rods", Appl. Math. Mech. (English Ed.), 21, 133-140. https://doi.org/10.1007/BF02458513
  21. Liu, L., Kardomateas, G.A., Birman, V., Holmes, J.W. and Simitses, G.J. (2006), "Thermal buckling of a heatexposed, axially restrained composite column", Compos. Part A, 37, 972-980. https://doi.org/10.1016/j.compositesa.2005.04.006
  22. Nowinski, J.L. (1978), Theory of Thermoelasticity with Applications, Sijthoff and Noordhoff, Alphen a/d Rijn.
  23. Vaz, M.A. and Solano, R.F. (2003), "Post-buckling analysis of slender elastic rods subjected to uniform thermal loads", J. Therm. Stresses, 26, 847-860. https://doi.org/10.1080/01495730306293
  24. Wang, X. and Dong, K. (2007), "Local buckling for triangular and lemniscate delaminations near the surface of laminated cylindrical shells under hygrothermal effects", Compos. Struct., 79, 67-75. https://doi.org/10.1016/j.compstruct.2005.11.029
  25. Wu, B.S., Lim, C.W. and Sun, W.P. (2006a), "Improved harmonic balance approach to periodic solutions of nonlinear jerk equations", Phys. Lett. A, 354, 95-100. https://doi.org/10.1016/j.physleta.2006.01.020
  26. Wu, B.S., Sun, W.P. and Lim, C.W. (2006b), "An analytical approximate technique for a class of strong nonlinear oscillators", Int. J. Nonlin. Mech., 41, 766-774. https://doi.org/10.1016/j.ijnonlinmec.2006.01.006
  27. Wu, B.S., Yu, Y.P. and Li, Z.G. (2007), "Analytical approximations to large post-buckling deformation of elastic rings under uniform hydrostatic pressure", Int. J. Mech. Sci., 49, 661-668. https://doi.org/10.1016/j.ijmecsci.2006.11.003
  28. Ziegler, F. and Rammerstorfer, F.G. (1989), Thermoelastic Stabiliy, Ed. Hetnarski, R.B., Temperature Stresses III Ch. 2, Elsevier, Amsterdam.

피인용 문헌

  1. Brief and accurate analytical approximations to nonlinear static response of curled cantilever micro beams vol.56, pp.3, 2015, https://doi.org/10.12989/sem.2015.56.3.461
  2. Probabilistic analysis of micro-film buckling with parametric uncertainty vol.50, pp.5, 2014, https://doi.org/10.12989/sem.2014.50.5.697
  3. Post-buckling analysis of Timoshenko beams with temperature-dependent physical properties under uniform thermal loading vol.44, pp.1, 2012, https://doi.org/10.12989/sem.2012.44.1.109
  4. Theoretical modelling of post - buckling contact interaction of a drill string with inclined bore-hole surface vol.49, pp.4, 2014, https://doi.org/10.12989/sem.2014.49.4.427