Structural Engineering and Mechanics

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Marguerre shell type secant matrices for the postbuckling analysis of thin, shallow composite shells

  • Arul Jayachandran, S. (Structural Engineering Research Centre, CSIR Campus) ;
  • Kalyanaraman, V. (Department of Civil Engineering, Indian Institute of Technology) ;
  • Narayanan, R. (Structural Engineering Research Centre, CSIR Campus)
  • Received : 2003.08.18
  • Accepted : 2004.02.04
  • Published : 2004.07.25
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Abstract

The postbuckling behaviour of thin shells has fascinated researchers because the theoretical prediction and their experimental verification are often different. In reality, shell panels possess small imperfections and these can cause large reduction in static buckling strength. This is more relevant in thin laminated composite shells. To study the postbuckling behaviour of thin, imperfect laminated composite shells using finite elements, explicit incremental or secant matrices have been presented in this paper. These incremental matrices which are derived using Marguerre's shallow shell theory can be used in combination with any thin plate/shell finite element (Classical Laminated Plate Theory - CLPT) and can be easily extended to the First Order Shear deformation Theory (FOST). The advantage of the present formulation is that it involves no numerical approximation in forming total potential energy of the shell during large deformations as opposed to earlier approximate formulations published in the literature. The initial imperfection in shells could be modeled by simply adjusting the ordinate of the shell forms. The present formulation is very easy to implement in any existing finite element codes. The secant matrices presented in this paper are shown to be very accurate in tracing the postbuckling behaviour of thin isotropic and laminated composite shells with general initial imperfections.

Keywords

References

  1. Allman, D.J. (1995), "On the assumed displacement fields of a shallow curved shell finite element", Int. J. Num. Meth. Eng., 11, 159-166. https://doi.org/10.1002/cnm.1640110209
  2. Arul Jayachandran, S., Gopalakrishnan, S. and Narayanan, R. (2003), "Improved secant matrices for the postbuckling analysis of thin composite plates", Int. J. Structural Stability and Dynamics, 3(3), 355-375. https://doi.org/10.1142/S0219455403000938
  3. Atluri, S.N. (1997), Structural Integrity and Durability, Tech. Science Press, Forsyth, G.A.
  4. Bogner, F.K., Fox, R.L. and Schmidt, L.A. (1966), "The generation of inter - element compatible matrices by the use of interpolation formulas", Proc. of the Conf. on Matrix Methods in Structural Mechanics, TR 66-80, Air Force Flight Dynamics Lab., Wright - Patterson Airforce Base, Ohio.
  5. Boisse, P., Daniel, J.L. and Gelin, J.C. (1994), "A $C^{0}$ three node shell element for nonlinear structural analysis", Int. J. Num. Meth. Eng., 37, 2339-2364. https://doi.org/10.1002/nme.1620371402
  6. Chan, S.L. (1988), "Geometric and Material nonlinear analysis of beam - columns and frames using the minimum residual displacement method", Int. J. Num. Meth. Eng., 26, 2657-2669. https://doi.org/10.1002/nme.1620261206
  7. Chaudri, R.A. and Hsia, R.L. (1999), "Effect of thickness on the large elastic deformation behaviour of laminated shells", Comp. Struct., 44(2/3), 117-128. https://doi.org/10.1016/S0263-8223(98)00115-9
  8. Chia, F.Y. (1987), "Nonlinear free vibration and postbuckling of symmetrically laminated imperfect shallow cylindrical panels with mixed boundary conditions resting on elastic foundation", Int. J. Eng. Sci., 25, 427- 441. https://doi.org/10.1016/0020-7225(87)90069-3
  9. Crisfield, M.A. (1991), Nonlinear Finite Element Analysis of Solids and Structures - Vol. 1, Essentials, John Wiley and Sons.
  10. Datoo, M.H. (1991), Mechanics of Fibrous Composites, Elsevier App. Sci. Co.
  11. Erasmo Carrera and Horst Parish (1998), "An evaluation of geometrically nonlinear effects of thin and moderately thick multilayered composite shells", Comp. Struct., 40(1), 11-24.
  12. Ferreira, A.J.M. and Barbosa, J.T. (2000), "Buckling behaviour of composite shells", Comp. Struct., 50(1), 93-98. https://doi.org/10.1016/S0263-8223(00)00090-8
  13. Fu, Y.M. and Chia, C.Y. (1989), "Multimode nonlinear vibration and post buckling of anti-symmetric imperfect angle ply cylindrical thick panels", Int. J. Nonlin. Mech., 24, 365-381. https://doi.org/10.1016/0020-7462(89)90025-5
  14. Ganapathi, M. and Varadhan, T.K. (1995), "Nonlinear free flexural vibrations of laminated circular cylindrical shells", Comp. Struct., 30, 33-49. https://doi.org/10.1016/0263-8223(94)00025-5
  15. Horrigmoe Geis (1977), "Finite element instability analysis of free form shells", Report No. 77/2, Div. of Structural Mechanics, The Norwegian Institute of Technology, The University of Trondheim, Norway.
  16. Horrigmoe Geis and Bergan, P.G. (1978), "Nonlinear analysis of free-form shells by flat finite elements", Comp. Meth. Appl. Mech. Eng., 16, 11-35. https://doi.org/10.1016/0045-7825(78)90030-0
  17. Jetteur, P. and Fray, F. (1986), "A four node Marguerre element for non-linear shell analysis", Eng. Comp., 3, 276-282. https://doi.org/10.1108/eb023667
  18. Kheyrkhahan, M. and Peek, R. (1999), "Postbuckling analysis and imperfection sensitivity of general shells by the finite element method", Int. J. Solids Struct., 36(18), 2641-2681. https://doi.org/10.1016/S0020-7683(98)00129-2
  19. Kim, K.D. (1996), "Buckling behaviour of composite panels using the finite element method", Comp. Struct., 36, 3-43.
  20. Kim, K.D., Park, T. and Voyiadjis, G.J. (1998), "Postbuckling analysis of composite panels with imperfection damage", Comp. Mech., 22, 375-387. https://doi.org/10.1007/s004660050369
  21. Koiter, W.T. (1945), "The stability of elastic equilibrium", Doctoral Thesis, (1945) in Dutch. English Translation by E. Riks, Techn. Report AFFDL-TR-70-25, Wright Patterson Airforce Base, 1970.
  22. Kroplin, B.H. (1982), Postbuckling Instability Analysis of Shells Using Mixed Method in Buckling of Shells, Ed. E. Ramm, Univ. of Stuttgart, 175-1992.
  23. Levy, S. (1942), "Square plate with clamped edges under normal pressure producing large deflections", NACA TR 740.
  24. Librescu, L. and Chang, M.Y. (1992), "Imperfection sensitivity and post buckling behaviour of shear deformable composite doubly curved shallow panels", Int. J. Solids Struct., 29(9), 1065-1083. https://doi.org/10.1016/0020-7683(92)90136-H
  25. Madenci, E. and Barut, A. (1995), "A free formulation based flat shell element for nonlinear analysis of thin composite structures", Int. J. Num. Meth. Eng., 37, 3825-3842. https://doi.org/10.1002/nme.1620372206
  26. Madasamy, C.M. (1995), "Spline finite strip method for linear and nonlinear analysis of isotropic, orthotropic and laminated fibre composite thin plated members", Ph.D thesis, Dept. of Civil Engg., IIT, Madras, India.
  27. Mallet, R.H. and Marcal, P.V. (1968), "Finite element analysis of nonlinear structures", J. Struct. Div., ASCE, 94(ST9), 2081-2105.
  28. Marguerre, K. (1938), "Zur theric der gekrummfess platte grosser Formanderuny", Proc. the Fifth International Congress of Applied Mechanics, Cambridge, Massachusetts, 93-101.
  29. Noor, A.K. and Burton, W.S. (1990), "Assessment of computational models for multi-layered composite shells", Applied Mechanics Review, 43, 67-97. https://doi.org/10.1115/1.3119162
  30. Noor, A.K. and Anderson, C.M. (1981), "Mixed models and reduced/selective integration displacement models for nonlinear shell analysis", In Nonlinear Finite Element Analysis of Plates and Shells, Ed.Hughes, T.J.R., Pifko, A. and Jay, A. ASME, 119-146.
  31. Oliver, J. and Onate, E. (1984), "A total Lagrangian formulation for geometrically nonlinear analysis of structures using finite elements Part I. Two Dimensional problems: shell and plate structures", Int. J. Num. Meth. Eng., 20, 2253-2281. https://doi.org/10.1002/nme.1620201208
  32. Onate, E., Zaarate, F. and Flores, F. (1994), "A simple triangular element for thick and thin plate and shell analysis", Int. J. Num. Meth. Eng., 37, 2569-2582. https://doi.org/10.1002/nme.1620371505
  33. Onate, E. (1995), "On the derivation and possibilities of the secant stiffness matrix for nonlinear finite element analysis", Comp. Mech., 15, 572-593. https://doi.org/10.1007/BF00350269
  34. Pai, P.F. and Palazotto, A.N. (1995), "Nonlinear displacement based finite element analysis of composite shells - A new total Lagrangian formulation", Int. J. Solids Struct., 32, 3047-3073. https://doi.org/10.1016/0020-7683(94)00273-Y
  35. Palazotto, A.N., Chien, L.S. and Taylor, W.W. (1992), "Stability characteristics of laminated cylindrical panels under transverse loading", AIAA J., 30(6), 1649-1653. https://doi.org/10.2514/3.11113
  36. Parish, H. (1995), "A continuum based shell theory for nonlinear applications", Int. J. Num. Meth. Eng., 38, 1855-1883. https://doi.org/10.1002/nme.1620381105
  37. Rajasekaran Sundaramoorthy and David W. Murray (1973), "Incremental finite element matrices", J. Struct. Div., ASCE, 99(ST12), 2423-2438.
  38. Ramm, E. (1982), "Strategies for tracing the nonlinear response near limit points", In Nonlinear Finite Element Analysis in Structural Mechanics, Ed. Wunderlich, W., Springer Verlaag, Berlin.
  39. Reddy, J.N. (1981), "A finite element analysis of large deflection bending of laminated anisotrophic shells", In Nonlinear Finite Element Analysis of Plates and Shells, Ed.Hughes, T.J.R., Pifko, A. and Jay, A., ASME, 249- 259.
  40. Riks, E. (1984), "Bifurcation and stability - a numerical approach", In Innovative Methods for Nonlinear Problems, Ed. Liu et al., W.K., Pineridge Press, Swansea, 313-344.
  41. Riks, E., Brogan, F.A. and Rankin, C.C. (1990), "Numerical aspects of shell instability analysis", In Computational Methods for Nonlinear Response of Shells, Ed. Kratzig, W.N. and Onate, E., Springer Verlaag, 125-151.
  42. Sabir, A.B. and Lock (1973), "The application of finite elements to the large deflection geometrically nonlinear behaviour of cylindrical shells", In Variational Methods in Engg., Ed. Brebbia, C.A. and tottenham, H., Southampton University Press, 7176-7175.
  43. Sabir, A.B. and Djoudi, M.S. (1995), "Shallow shell finite element for the large deflection geometrically nonlinear analysis of shells and plates", Thin Walled Struct., 21, 253-267. https://doi.org/10.1016/0263-8231(94)00005-K
  44. Sheinman, I. and Frostig, Y. (1990), "Postbuckling analysis of stiffened laminated curved panels", J. Eng. Mech., ASCE, 116(10), 2223-2236. https://doi.org/10.1061/(ASCE)0733-9399(1990)116:10(2223)
  45. Sheinman, I. and Simitses (1983), "Buckling and post buckling of imperfect cylindrical shells under axial compression", Comp. Struct., 17, 471-481.
  46. Stolarski, H., Belytschko, T., Carpenter, N. and Kennedy, J.M. (1984), "A simple triangular curved shell element for collapse analysis", Eng. Comp., 1, 210-218. https://doi.org/10.1108/eb023574
  47. Saigal, S., Rakesh K. Kapania, and Yang, T. (1986), "Geometrically nonlinear finite element analysis of imperfect laminated shells", J. Comp. Mat., 20, 197-213. https://doi.org/10.1177/002199838602000206
  48. Surana, K.S. (1983), "Geometrically nonlinear formulation for the curved shell elements", IJNME, 19, 581-615. https://doi.org/10.1002/nme.1620190409
  49. Teng, J.G. and Song, C.Y. (2001), "Numerical models for nonlinear analysis of elastic shells with eigenmode affine imperfections", Int. J. Solids Struct., 38(18), 3263-3280. https://doi.org/10.1016/S0020-7683(00)00222-5
  50. Yang, H.T.Y. (2000), "A survey of recent shell finite elements", Int. J. Num. Meth. Eng., 47(1/3), 101-127. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3<101::AID-NME763>3.0.CO;2-C
  51. Yang, H.T.Y. (1972), "Elastic snap - through analysis of curved plates using discrete elements", AIAA J., 10(4), 371-372. https://doi.org/10.2514/3.50104
  52. Yang, H.T.Y., Kapania, R.K. and Sunil Saigal (1989), "Accurate rigid body mode representation for a nonlinear curved thin-shell element", AIAA J., 27(2), 211-218. https://doi.org/10.2514/3.10083
  53. Zhang, Y. and Mathews, F.L. (1985), "Large deflection behaviour of simply supported laminated panels under inplane loading", J. Appl. Mech., 52, 553-585. https://doi.org/10.1115/1.3169100

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