Advertisement

Single Layer Modelling and Effective Stiffness Estimations of Laminated Plates

  • Holm Altenbach
  • Johannes Meenen
Part of the International Centre for Mechanical Sciences book series (CISM, volume 448)

Abstract

Two-dimensional laminated plate theories can be derived by reducing the three-dimensional equations of continuum mechanics or by postulating the existence of a two-dimensional Cosserat surface (the so-called direct approach). Theories derived by both approaches can be further classified into equivalent single-layer theories and layer-wise theories which possess individual unknowns for each layer. Layer-wise theories generally lead to better approximations than equivalent single-layer theories, but the large number of independent unknowns and governing equations increases the amount of computational power which is necessary to find a numerical solution, and the possibilities to obtain simple analytical solutions are limited.

Introducing assumptions into the principle of virtual displacements, classical and refined plate theories such as the classical laminated plate theory based on extended Kirchhoff’s assumptions, the first order shear deformation theories considering the transverse shear deformations of the plate, the refined plate theories working with additional degrees of freedom and the power series approach can be derived. After a brief overview of different approaches the modelling with respect to the scale size is discussed. Then, refined plate theories derived from the principle of virtual displacements are presented in detail. In contrast to this approach, by exploiting a direct formulation based on the theory of deformable surfaces and the introduction of five independent degrees of freedom for each material point of this surface, a theory similar to the first order shear deformation plate theory is developed. Finally, the extension of classical and refined plate theories to non-mechanical loadings is explained for the example of the laminated plate under thermal loadings.

Although many different plate theories result in identical stiffness properties for bending, tension/compression, in-plane shear and torsion, differences are present in the formulation of the transverse stiffness. From the analysis of three-dimensional problems, it is known that the accuracy of results obtained from single-layer theories is strongly dependent on the correct determination of the effective stiffness. It is shown that the classical stiffness tensors are identical for several different formulations of laminated plate theories. For the calculation of these properties closed solutions can be obtained. For the transverse shear stiffness closed solutions can be obtained too, but the comparison of numerical solutions with experimental data or calculations based on the three-dimensional theory sometimes does not lead to a satisfying agreement. For the first order shear deformation theory, a method for the determination of the transverse shear stiffness tensors is proposed, which leads to a Sturm-Liouville problem. For special cases the stiffness expressions are estimated and compared with results obtained from other authors. The approach is applicable to laminated plates as well as to sandwich plates.

The paper is limited to the calculation of laminated and sandwich plates. The effective properties of the individual layers are assumed to be known with respect to the global coordinate system. The estimation of these effective properties from the properties of the different constituents of each laminae is briefly presented and the transformation of the effective properties from a local coordinate system to a global coordinate system is discussed.

Keywords

Transverse Shear Plate Theory Laminate Plate Sandwich Plate Shear Deformation Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Aleksandrov, A.Y., Bryukker, L.E., Kurshin, L.M., and Prusakov, A.P. (1960). Analysis of three-layered panels (in Russ.). Moskva: Oborongiz.Google Scholar
  2. Altenbach, H. (1987). Definition of elastic moduli for plates made from thickness-uneven anisotropic material. Mech. Solids 22 (1): 135–141.Google Scholar
  3. Altenbach, H., Altenbach, J., and Zolochevsky, A. (1995). Erweiterte Deformationsmodelle und Versagenskriterien der Werkstoffinechanik. Stuttgart: Deutscher Verlag für Grundstoffindustrie.Google Scholar
  4. Altenbach, H., Altenbach, J., and Rikards, R. (1996). Einführung in die Mechanik der Laminat-und Sandwichtragwerke. Stuttgart: Deutscher Verlag für Grundstoffindustrie.Google Scholar
  5. Altenbach, H., and Zhilin, P.A. (1988). A general theory of elastic simple shells (in Russ.). Uspekhi Mekhaniki 11 (4): 107–148.MathSciNetGoogle Scholar
  6. Ambarcumyan, S.A. (1958). On the theory of bending of anisotropic plates and shallow shells (in Russ.). Izv. AN SSSR. Otd. tekhn. nauk (5): 69–77.Google Scholar
  7. Bollé, L. (1947a). Contribution au problème linéaire de flexin d’une plaque élastique. Bull. Techn. Suisse Romande 73 (21): 281–285.Google Scholar
  8. Bollé, L. (1947b). Contribution au problème linéaire de flexin d’une plaque élastique. Bull. Techn. Suisse Romande 73 (22): 293–298.Google Scholar
  9. Bolotin, V.V., and Novichkov, Yu.N. (1980). Mechanics of multi-layered structures (in Russ.). Moscow: Mashinostroenie.Google Scholar
  10. Buffer, H. (1961). Der Spannungszustand in einer geschichteten Scheibe. ZAMM41:158–180.Google Scholar
  11. Burton, W.S., and Noor, A.K. (1995). Assessment of computational models for sandwich panels and shells. Comput. Meth. Appl. Mech. Engng. 124: 125–151.CrossRefGoogle Scholar
  12. Cauchy, A.-L. (1828). Sur l’équilibre et le mouvement d’une plaque solide. Exercices de Mathématiques (Paris) 328–355.Google Scholar
  13. Chatterjee, S.N., and Kulkarni, S.V. (1979). Shear correction factors for laminated plates. AIAA Journal 17 (5): 498–499.CrossRefGoogle Scholar
  14. Chawla, K.K. (1987). Composite materials. Berlin: Springer.CrossRefGoogle Scholar
  15. Chow, T.S. (1971). On the propagation of flexural waves in an orthotropic laminated plate and its response to an impulsive load. J. Composite Materials 5: 306–319.CrossRefGoogle Scholar
  16. Christensen, R.M. (1979). Mechanics of composite materials. New York: Wiley. Dundrovâ, V., Kovafik, V., and Slapâk, P. (1970). Biegetheorie für Sandwichplatten. Praha: Academica.Google Scholar
  17. Ehrenstein, G. (1992). Faserverbund-Kunststoffe. München: Hanser.Google Scholar
  18. Goldenweizer, A.L. (1962). Formulation of approximative theory of shells with the help of the asymptotic integration of the equations of the theory of elasticity (in Russ.). Prikl. Mat. i Mekh. 26 (4): 668–686.Google Scholar
  19. Gould, P. (1988). Analysis of shells and plates. New York: Springer.CrossRefMATHGoogle Scholar
  20. Green, A., Naghdi, P.M., and Waniwright, W. (1965). A general theory of Cosserat surface. Arch. Rat. Mech. Anal. 20: 287–308.Google Scholar
  21. Grigolyuk, E.I., and Kogan, A.F. (1972). Present state of the theory of multi-layered shells (in Russ.). Prikl. Mekh. 8 (6): 3–17.Google Scholar
  22. Grigolyuk, E.I., and Seleznev, I.T. (1973). Nonclassical theories of vibration of beams, plates and shells (in Russ.). Itogi nauki i tekhniki. Mekhanika tverdogo deformiruemogo tela, Vol. 5. Moskva: VINITI.Google Scholar
  23. Halpin, J.C., and Tsai, S.W. (1969). Effects of Environmental Factors on Composite Materials. Report AFML-TR 67–423. Dayton, Ohio: U.S.A. Air Force Materials Laboratory.Google Scholar
  24. Halpin, J.C. (1992). Primer on Composite Materials Analysis. Lancaster: Technomic. Hencky, H. (1947). Über die Berücksichtigung der Schubverzerrung in ebenen Platten. Ingenieur-Archiv 16: 72–76.Google Scholar
  25. Hill, R. (1964). Theory of mechanical properties of fibre-strengthened materials: 1. Elastic behaviour. J. Mech. Phys. Solids 12 (4): 199–218.CrossRefMathSciNetGoogle Scholar
  26. Hult, J., and Rammerstorfer, F., eds., (1994). Engineering mechanics of fibre reinforced polymers and composite structures, CISM Courses and Lectures No 348. Wien, New York: Springer.Google Scholar
  27. Jemielita, G. (1990). On kinematical assumptions of refined theories of plates: a survey. Trans. ASME. J. Appl. Mech. 57: 1088–1091.CrossRefGoogle Scholar
  28. Jones, R.M. (1975). Mechanics of composite materials. Washington: McGraw-Hill. Kaplunov, J.D., Kossovich, E.V., and Nolde, E.V. (1998). Dynamics of thin walled elastic bodies. San Diego: Academic Press.Google Scholar
  29. Kienzler, R. (1982). Erweiterung der klassischen Schalentheorie; der Einfluß von Dickenverzerrung und Querschnittsverwölbungen. Ingenieur-Archiv 52: 311–322.CrossRefMATHGoogle Scholar
  30. Kienzler, R. (2002). On consistent plate theories. Arch. Appl. Mech. 72: 229–247.CrossRefMATHGoogle Scholar
  31. Kirchhoff, G. (1850). Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. J. Reine Angew. Math. 40: 51–88.CrossRefMATHGoogle Scholar
  32. Knight, Jr., N.F., and Qi, Y. (1997). Restatement of first-order shear-deformation theory for laminated plates. Int. J. Solids Structures 34 (4): 481–492.CrossRefMATHGoogle Scholar
  33. Koiter, W.T. (1970). On the foundations of the linear theory of thin elastic shells. Proc. Kon. Akad. Wet., Ser. B 73: 169–182.MATHMathSciNetGoogle Scholar
  34. Krenk, S. (1981). Theories for elastic plates via orthogonal polynomials. Trans. ASME. J. Appl. Mech. 48 (4): 900–904.CrossRefMATHGoogle Scholar
  35. Kromm, A. (1953). Verallgemeinerte Theorie der Plattenstatik. Ingenieur-Archiv 21: 266–286.CrossRefMATHMathSciNetGoogle Scholar
  36. Lai, W.M., Rubin, D., and Krempl, E. (1993). Introduction to continuum mechanics. Oxford: Pergamon Press.Google Scholar
  37. Lekhnitskii, S.G. (1968). Anisotropic plates. New York: Gordon and Breach.Google Scholar
  38. Levinson, M. (1980). An accurate, simple theory of the statics and dynamics of elastic plates. Mech. Res. Comm. 7 (6): 343–350.CrossRefMATHGoogle Scholar
  39. Lewinski, T. (1987). On refined plate models based on kinematical assumptions. Ingenieur-Archiv 57: 133–146.CrossRefMATHGoogle Scholar
  40. Librescu, L., and Reddy, J.N. (1989). A few remarks concerning several refined theories of anisotropic composite laminated plates. Int. J. Engng. Sci. 27 (5): 515–527.CrossRefMATHGoogle Scholar
  41. Lo, K.H., Christensen, R.M., and Wu, E.M. (1977). A high-order theory of plate deformation. Part I: Homogeneous plates. Trans. ASME. J. Appl. Mech. 44 (4): 663–668.CrossRefMATHGoogle Scholar
  42. Love, A.E.H. (1944). A treatise on the mathematical theory of elasticity. New York: Dover.MATHGoogle Scholar
  43. Lurie, A. (1990). Nonlinear theory of elasticity. Amsterdam: North-Holland.MATHGoogle Scholar
  44. Mau, S.T. (1973). A refined laminated plate theory. Trans. ASME. J. Appl. Mech. 40: 606–607.CrossRefGoogle Scholar
  45. Meenen, J., and Altenbach, H. (2001). A consistent deduction of von Kârm6,n-type plate theories from three-dimensional nonlinear continuum mechanics. Acta. Mech. 147: 117.Google Scholar
  46. Mindlin, R.D. (1951). Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Trans. ASME. J. Appl. Mech. 18 (2): 31–38.MATHGoogle Scholar
  47. Mushtari, K.h.M. (1959). On the theory of moderately thick plates in bending (in Russ.). Izv. AN SSSR. Otd. tekhn. nauk (2): 107–113.Google Scholar
  48. Naghdi, P.M. (1972). The theory of plates and shells. In Flügge, S., ed., Handbuch der Physik, Vol. VIa/2. Heidelberg: Springer. 425–640.Google Scholar
  49. Noor, A.K., and Burton, W.S. (1989a). Assessment of shear deformation theories for multilayered composite plates. Appl. Mech. Rev. 42 (1): 1–13.CrossRefGoogle Scholar
  50. Noor, A.K., and Burton, W.S. (1989b). Stress and free vibration analyses of multilayered composite plates. Composite Structures 11: 183–204.CrossRefGoogle Scholar
  51. Noor, A.K., Burton, W.S., and Bert, C. (1996). Computational models for sandwich panels and shells. Appl. Mech. Rev. 49 (3): 155–199.CrossRefGoogle Scholar
  52. Nye, J.F. (1992). Physical properties of crystals. Oxford: Oxford Science Publications. Pagano, N.J. (1969). Exact solutions for composite laminates in cylindrical bending. J. Composite Materials 3: 398–411.Google Scholar
  53. Pagano, N.J., and Soni, S.R. (1989). Models for studying free edge effects. In Pagano, N., ed., Interlaminar response of composite materials, Vol. 5, Composite Materials Series. Amsterdam: Elsevier.Google Scholar
  54. Pane, V. (1964). Verschärfte Theorie der elastischen Platten. Ingenieur-Archiv 33 (6): 351–371.CrossRefGoogle Scholar
  55. Plantema, F.J. (1966). Sandwich construction. New York: John Wiley Sons.Google Scholar
  56. Preußer, G. (1984). Eine systematische Herleitung verbesserter Plattentheorien. Ingenieur-Archiv 54: 51–61.CrossRefMATHGoogle Scholar
  57. Qi, Y., and Knight, Jr., N.F. (1996). A refined first-order shear-deformation theory and its justification by plane-strain bending problem of laminated plates. Int. J. Solids Structures 33 (1): 49–64.CrossRefMATHGoogle Scholar
  58. Reddy, J.N. (1984). A simple higher-order theory for laminated composite plates. Trans. ASME. J. Appl. Mech. 51: 745–752.CrossRefMATHGoogle Scholar
  59. Reddy, J.N. (1987). A generalization of two-dimensional theories of laminated composite plates. Comm. Appl. Num. Meth. 3: 173–180.CrossRefMATHGoogle Scholar
  60. Reddy, J.N. (1993). An evaluation of equivalent-single-layer and layerwise theories of composite laminates. Composite Structures 25: 21–35.CrossRefGoogle Scholar
  61. Reissner, E. (1944). On the theory of bending of elastic plates. J. Math. Phys. 23: 184–191.MATHMathSciNetGoogle Scholar
  62. Reissner, E. (1945). The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12 (11): A69 - A77.MATHMathSciNetGoogle Scholar
  63. Reissner, E. (1947). On the bending of elastic plates. Quart. Appl. Math. 5: 55–68.MATHMathSciNetGoogle Scholar
  64. Reissner, E. (1985). Reflections on the theory of elastic plates. Appl. Mech. Rev. 38 (11): 1453–1464.CrossRefGoogle Scholar
  65. Rikards, R., Chate, A., and Kenzer, M. (1990). Variants of averaging the shear stiffness of multi-layered structures in the case of estimation the eigenfrequencies applying the Timoshenko model (in Russ.). Voprosy Dinamiki i Prochnosti 52: 176–193.Google Scholar
  66. Rohwer, K. (1989). Transverse shear stiffness of composite and sandwich finite elements. Proc. Int. Conf.: Spacecraft Structures and Mechanical Testing, Nordwijk, 363–368, ESA SP-289.Google Scholar
  67. Sciuva, M. di (1984). A refined transverse shear deformation theory for multilayered anisotropic plates’. Atti della Accademia delle science die Torino. 1. Classe di scienze fische, mathematiche e naturali 118: 279–295.MATHGoogle Scholar
  68. Stamm, K., and Witte, H. (1974). Sandwichkonstruktionen. Wien, New York: Springer.Google Scholar
  69. Tessler, A. (1993). An improved plate theory of {1,2}-order for thick composite laminates. Int. J. Solids Struct. 30 (7): 981–1000.CrossRefMATHGoogle Scholar
  70. Timoshenko, S.P. (1921). On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. 41, Ser. 6 (245): 744–746.Google Scholar
  71. Todhunter, I., and Pearson, K. (1893). A History of the Theory of Elasticity and of the Strength of Materials, Vol. II. Saint-Venant to Lord Kelvin, Part I I. Cambridge: University Press.Google Scholar
  72. Touratier, M. (1988). A refined theory of thick composite plate. Mech. Res. Comm. 15 (4): 229–236.CrossRefMATHGoogle Scholar
  73. Touratier, M. (1991). An efficient standard plate theory. Int. J. Engng. Sci. 29 (8): 901–916.CrossRefMATHGoogle Scholar
  74. Tzou, H.S. (1993). Piezoelectric shells. Dordrecht: Kluwer.CrossRefGoogle Scholar
  75. Vekua, I.N. (1982). On some general methods of formulation different variants of the theory of shallow shells (in Russ.). Moskva: Nauka.Google Scholar
  76. Vinson, J. (2001). Sandwich structures. Appl. Mech. Rev. 54: 201–214.CrossRefGoogle Scholar
  77. Vlachoutsis, S. (1992). Shear correction factors for plates and shells. Int. J. Numer. Meth. Engng. 33: 1537–1552.CrossRefMATHGoogle Scholar
  78. Vlasov, B.F. (1957). On the equations in the plate bending theory (in Russ.). Izv. AN SSSR. Otd. tekhn. nauk (12): 57–60.Google Scholar
  79. Wlassow, W. (1958). Allgemeine Schalentheorie und ihre Anwendung in der Technik. Berlin: Akademie-Verlag.MATHGoogle Scholar
  80. Wunderlich, W. (1973). Vergleich verschiedener Approximationen der Theorie dünner Schalen (mit numerischen Beispielen). Institut für Konstruktiven Ingenieurbau der Ruhr-Universität Bochum, Techn.-Wiss. Mitt. No. 73–1.Google Scholar
  81. Zhilin, P.A. (1976). Mechanics of deformable directed surfaces. Int. J. Solids Struct. 12: 635–648.CrossRefMathSciNetGoogle Scholar
  82. Zhilin, P.A. (1992). On the theories of Poissons’s and Kirchhoff’s plates from the point of view of the modern plate theory. Izv. Ross. Ak. Nauk. Mekh. tv. tela (3): 48–64.Google Scholar

Copyright information

© Springer-Verlag Wien 2003

Authors and Affiliations

  • Holm Altenbach
    • 1
  • Johannes Meenen
    • 2
  1. 1.Technische Mechanik, Fachbereich IngenieurwissenschaftenMartin-Luther-UniversitätHalleGermany
  2. 2.Werkstofftechnik DWF/FBASF AGLudwigshafenGermany

Personalised recommendations