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Archives of Computational Methods in Engineering

, Volume 10, Issue 3, pp 215–296 | Cite as

Theories and Finite Elements for Multilayered Plates and Shells:A Unified compact formulation with numerical assessment and benchmarking

  • Erasmo Carrera
Article

Summary

This work is a sequel of a previous author’s article: “Theories and Finite Elements for Multilayered. Anisotropic, Composite Plates and Shell”, Archive of Computational Methods in Engineering Vol 9, no 2, 2002; in which a literature overview of available modelings for layered flat and curved structures was given. The two following topics, which were not addressed in the previous work, are detailed in this review: 1. derivation of governing equations and finite element matrices for some of the most relevant plate/shell theories; 2. to present an extensive numerical evaluations of available results, along with assessment and benchmarking.

The article content has been divided into four parts.

An introduction to this review content is given in Part I.

A unified description of several modelings based on displacements and transverse stress assumptions ins given in Part II. The order of the expansion in the thickness directions has been taken as a free parameter. Two-dimensional modelings which include Zig-Zag effects, Interlaminar Continuity as well as Layer-Wise (LW), and Equivalent Single Layer (ESL) description have been addressed.

Part III quotes governing equations and FE matrices which have been written in a unified manner by making an extensive use of arrays notations. Governing differential equations of double curved shells and finite element matrices of multilayered plates are considered. Principle of Virtual Displacement (PVD) and Reissner’s Mixed Variational Theorem (RMVT), have been employed as statements to drive variationally consistent conditions, e.g.C z 0 -Requirements, on the assumed displacements and stransverse stress fields. The number of the nodes in the element has been taken as a free parameter. As a results both differential governing equations and finite element matrices have been written in terms of a few 3×3 fundamental nuclei which have 9 only terms each.

A vast and detailed numerical investigation has been given in Part IV. Performances of available theories and finite elements have been compared by building about 40 tables and 16 figures. More than fifty available theories and finite elements have been compared to those developed in the framework of the unified notation discussed in Parts II and III. Closed form solutions and and finite element results related to bending and vibration of plates and shells have been addressed. Zig-zag effects and interlaminar continuity have been evaluated for a number of problems. Different possibilities to get transverse normal stresses have been compared. LW results have been systematically compared to ESL ones. Detailed evaluations of transverse normal stress effects are given. Exhaustive assessment has been conducted in the Tables 28–39 which compare more than 40 models to evaluate local and global response of layered structures. A final Meyer-Piening problem is used to asses two-dimensional modelings vs local effects description.

Keywords

Cylindrical Shell Laminate Plate Sandwich Plate Transverse Stress Cylindrical Panel 
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.

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References

  1. 1.
    Carrera, E. (2002). Theories and Finite Elements for Multilayered, Anisotropic, Composite Plates and Shell.Archive of Computational Methods in Engineering, Vol.9, no2, 87–140.MATHMathSciNetGoogle Scholar
  2. 2.
    Carrera, E. (1995). A class of two-dimensional theories for anisotropic multilayered plates analysis. Accademia delle Scienze di Torino, Memorie Scienze Fisiche, 19–20 (1995–1996), 1–39.Google Scholar
  3. 3.
    Carrera, E. (1997).C z0 Requirements—Models for the two dimensional analysis of multilayered structure.Composite Structures, Vol.37, 373–384, andAstronautics, Vol.32, 2135–2136.CrossRefGoogle Scholar
  4. 4.
    Carrera, E. (1998). Mixed Layer-Wise Models for Multilayered Plates Analysis.Composite Structures, Vol.43, 57–70.CrossRefGoogle Scholar
  5. 5.
    Carrera, E. (1998). Evaluation of Layer-Wise Mixed Theories for Laminated Plates Analysis.American Institute of Aeronautics and Astronautics Journal, Vol.26, 830–839.Google Scholar
  6. 6.
    Carrera, E. (1998). Layer-Wise Mixed Models for Accurate Vibration Analysis of Multilayered Plates.Journal of Applied Mechanics, Vol.65, 820–828.Google Scholar
  7. 7.
    Carrera, E. (1999). Multilayered Shell Theories that Account for a Layer-Wise Mixed Description. Part I. Governing Equations.American Institute of Aeronautics and Astronautics Journal, Vol.37, No.9, 1107–1116.Google Scholar
  8. 8.
    Carrera, E. (1999). Multilayered Shell Theories that Account for a Layer-Wise Mixed Description. Part II. Numerical Evaluations.American Institute of Aeronautics and Astronautics Journal, Vol.37, No.9, 1117–1124.Google Scholar
  9. 9.
    Carrera, E. (1999). A Reissner’s Mixed Variational Theorem Applied to Vibration Analysis of Multilayered Shells.Journal of Applied Mechanics, Vol.66, No.1, 69–78.Google Scholar
  10. 10.
    Carrera, E. (1999), A Study of Transverse Normal Stress Effects on Vibration of Multilayered Plates and Shells.Journal of Sound and Vibration, Vol.225, 803–829.CrossRefGoogle Scholar
  11. 11.
    Carrera, E. (1999). Transverse Normal Stress Effects in Multilayered Plates.Journal of Applied Mechanics, Vol.66, 1004–1012.Google Scholar
  12. 12.
    Carrera, E. (2000). Single-Layer vs Multi-Layers Plate Modelings on the Basis of Reissner’s Mixed Theorem.American Institute of Aeronautics and Astronautics Journal, Vol.38, 342–343.Google Scholar
  13. 13.
    Carrera, E. (2000). A Priori vs a Posteriori Evaluation of Transverse Stresses in Multilayered Orthotropic Plates.Composite Structures, Vol.48, 245–260.CrossRefGoogle Scholar
  14. 14.
    Carrera, E. (2000). An assessment of mixeed and classical theories for thermal stress analysis of orthotropic plates.Journal of Thermal Stress, Vol.23, 797–831.CrossRefGoogle Scholar
  15. 15.
    Carrera, E. (2000). An assessment of mixed and classical theories on global and local response of multilayered orthotropic plates.Composite Structures, Vol.40, 183–198.CrossRefGoogle Scholar
  16. 16.
    Carrera, E. (2001). Developments ideas and evaluations based upon Reissner’s Mixed Variational Theorem in the Modeling of Multilayered Plates and Shells.Applied Mechanics Review, Vol.54, 301–329.CrossRefGoogle Scholar
  17. 17.
    Washizu, K. (1968).Variational Methods in Elasticity and Plasticity. Pergamon Press, N.Y.MATHGoogle Scholar
  18. 18.
    Reissner, E. (1984). On a certain mixed variational theory and a proposed application.International Journal for Numerical Methods in Engineering, Vol.20, 1366–1368.MATHCrossRefGoogle Scholar
  19. 19.
    Reissner, E. (1985). Reflections on the theory of elastic plates.Applied Mechanics Review, Vol.38, 1453–1464.Google Scholar
  20. 20.
    Reissner, E. (1986a). On a mixed variational theorem and on a shear deformable plate theory.International Journal for Numerical Methods in Engineering, Vol.23, 193–198.MATHCrossRefGoogle Scholar
  21. 21.
    Reissner, E. (1986b). On a certain mixed variational theorem and on laminated elastic shell theory.Proceedings of the Euromech-Colloquium,219, 17–27.Google Scholar
  22. 22.
    Dvorkin, E.N. (1995). Nonlinear analysis of shells using the MITC formulation.Archive of Computational Methods in Engineering, Vol.2, No.2, 1–50.MathSciNetGoogle Scholar
  23. 23.
    Yang H.T., Saigal, S., Masud, A. and Kapania, R.K. (2000). A survey of recent shell finite elements.International Journal of Numerical Methods in Engineering, Vol.47, 101–127.MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Nurakami, H. (1986). Laminated composite plate theory with improved in-plane responses.Journal of Applied Mechanics, Vol.53, 661–666.Google Scholar
  25. 25.
    Reddy, J.N. (1997).Mechanics of Laminated Composite Plates, Theory and Analysis. CRC Press.Google Scholar
  26. 26.
    Librescu, L. (1975).Elasto-statics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures. Noordhoff Int., Leyden, Netherland.Google Scholar
  27. 27.
    Kraus, H. (1967).Thin Elastic Shells. John Wiley, N.Y.MATHGoogle Scholar
  28. 28.
    Carrera, E. (1991). The effects of shear deformation and curvature on buckling and vibrations of cross-ply laminated compasites shells.Journal of Sound and Vibration, Vol.150, 405–433.CrossRefGoogle Scholar
  29. 29.
    Pagano, N.J. (1969). Exact solutions for Composite Laminates in Cylindrical Bending.Journal of Composite materials, Vol.3, 398–411.CrossRefGoogle Scholar
  30. 30.
    Toledano, A. and Murakami, H. (1987). A composite plate theory for arbitrary laminate configurations.Journal of Applied Mechanics, Vol.54, 181–189.MATHCrossRefGoogle Scholar
  31. 31.
    Cho, M. and Parmerter, R.R. (1993). Efficient higher order composite plate theory for general lamination configuration.American Institute of Aeronautics and Astronautics Journal, Vol.31, 1299–1305.MATHGoogle Scholar
  32. 32.
    Lo, K.H., Christensen, R.M. and Wu, E.M. (1977). A Higher-Order Theory of Plate Deformation. Part 2: Laminated Plates.Journal of Applied Mechanics, Vol.44, 669–676.MATHGoogle Scholar
  33. 33.
    Kant, T. and Kommeneni, J.R. (1989). Large Amplitude Free Vibration Analysis of Cross-Ply Composite and Sandwich Laminates with a Refined Theory andC 0 Finite Elements.Computer & Structures, Vol.50, 123–134.CrossRefGoogle Scholar
  34. 34.
    Kheider, A.A. and Librescu, L. (1988). Analysis of Symmetric Cross-ply Laminated Elastic Plates Using a Higher order Theory: Part II-Buckling and Free Vibration.Composite Structures, Vol.9, 259–277.CrossRefGoogle Scholar
  35. 35.
    Reddy, J.N. and Phan, N.D. (1985). Stability and Vibration of Isotropic, Orthotropic, and Laminated Plates According to a Higher order Shear Deformation Theory.Journal of Sound and Vibration, Vol.98, 157–170.MATHCrossRefGoogle Scholar
  36. 36.
    Cho, K.N., Bert, C.W. and Striz, A.G. (1991). Free Vibrations of Laminated Rectangular Plates Analyzed by Higher order Individual-Laver Theory.Journal of Sound and Vibration, Vol.145, 429–442.CrossRefGoogle Scholar
  37. 37.
    Noor, A.K. and Burton W.S. (1989a). Stress and Free Vibration Analyses of Multilayered Composite Plates.Composite Structures, Vol.11, 183–204.CrossRefGoogle Scholar
  38. 38.
    Ren, J.G. (1986). A new theory of laminated plates,Composite Science and Technology, Vol.26, 225–239.CrossRefGoogle Scholar
  39. 39.
    Ren, J.G. (1986). Bending theory of laminated plates.Composite Science and Technology, Vol.27, 225–248.CrossRefGoogle Scholar
  40. 40.
    Dischiuvo, M. (1993). A general quadrilater multilayered plate element with continuous interlaminar stresses.Composite Structures, Vol.47, 91–105.CrossRefGoogle Scholar
  41. 41.
    Idlbi, A., Karama, M. and Touratier, M. (1997). Comparison of various laminated plate theories.Composite Structures, Vol.37, 173–184.CrossRefGoogle Scholar
  42. 42.
    Koiten, W.T. (1960). A Consistent First Approximations in the General Theory of Thin Elastic Shells. Proceedings ofFirst Symposium on the Theory of Thin Elastic Shells, Aug. 1959, North-Holland, Amsterdam, 12–23.Google Scholar
  43. 43.
    Ren, J.G. (1987). Exact Solutions for Laminated Cylindrical Shells in Cylindrical Bending.Composite Science and Technology, Vol.29, 169–187.CrossRefGoogle Scholar
  44. 44.
    Varadan T.K. and Bhaskar, K. (1991). Bending of Laminated Orthotropic Cylindrical Shells-An Elasticity Approach.Composite Structures, Vol.17, 141–156.CrossRefGoogle Scholar
  45. 45.
    Bhaskar, K. and Varadan, T.K. (1993). Exact elasticity solution for Laminated Anisotropic Cylindrical Shells.Journal of Applied Mechanics, Vol.60, 41–47.MATHGoogle Scholar
  46. 46.
    Jing, H. and Tzeng, K.G. (1993b). Refined Shear Deformation Theory of Laminated Shells.American Institute of Aeronautics and Astronautics Journal, Vol.31, 765–773.MATHGoogle Scholar
  47. 47.
    Noor, A.K. and Rarig, P.L. (1974). Three-Dimensional Solutions of Laminated Cylinders.Computer Methods in Applied Mechanics and Engineering, Vol.3, 319–334.CrossRefMATHGoogle Scholar
  48. 48.
    Nosier, A., Kapania, R.K., Reddy, J.N. (1993). Free Vibration Analysis of Laminated Plates Using a Layer-Wise Theory.American Institute of Aeronautics and Astronautics Journal. Vol.31, 2335–2346.MATHGoogle Scholar
  49. 49.
    Ye, J.Q. and Soldatos, K.P. (1994). Three-dimensional vibration of laminated cylinders and cylindrical panels with symmetric or antisymmetric cross-ply lay-up.Composite Engineering, Vol.4, 429–444.CrossRefGoogle Scholar
  50. 50.
    Timarci, T. and Soldatos, K.P. (1995). Comparative dynamic studies for symmetric cross-ply circular cylindrical shells on the basis a unified shear deformable shell theories.Journal of Sound and Vibration, Vol.187, 609–624.CrossRefGoogle Scholar
  51. 51.
    Pagano, N.J. and Hatfield, S.J. (1972). Elastic Behavior of Multilayered Bidirectional Composites.American Institute of Aeronautics and Astronautics Journal, Vol.10, 931–933.Google Scholar
  52. 52.
    Pandya, B.N. and Kant, T. (1985). Flexural analysis of, laminated composites using refined higher-order C0 Plate bending elements.Computer Methods in Applied Mechanics and Engineering, Vol.66, 173–198.CrossRefGoogle Scholar
  53. 53.
    Di, S. and Ramm, E. (1993). Hybrid stress formulation for higher-order theory of laminated shell analysis.Computer Methods in Applied Mechanics and Engineering. Vol.109, 359–376.MATHCrossRefGoogle Scholar
  54. 54.
    Liew, K.M., Han, B. and Xiao, M. (1996). Differential quadrature method for thick symmetric cross-ply laminates with first-order shear flexibility.International Journal of Solids and Structures, Vol.33, 2647–2658.MATHCrossRefGoogle Scholar
  55. 55.
    Liou, W.J. and Sun, C.T. (1985). A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates.Computer & Structures, Vol.25, 241–249.CrossRefGoogle Scholar
  56. 56.
    Pagano, N.J. (1970). Exact solutions for rectaugular bidirection composites and sandwich plates.Journal of Composite Materials, Vol.4, 20–34.Google Scholar
  57. 57.
    Meyer-Piening, H.-R. (2000). Experiences with ‘Exact’ linear sandwich beam and plate analyses regarding bending, instability and frequency investigations. Proceedings of theFifth International Conference On Sandwich Constructions Zurich, Switzerland, September 5–7, Vol.I, 37–48.Google Scholar
  58. 58.
    Meyer-Piening, H.R. and Stefanelli, R. (2000). Stresses, deflections, buckling and frequencies of a cylindrical curved rectangular sandwich panel based on the elasticity solutions. Proceedings of theFifth International Conference On Sandwich Constructions, Zurich, Switzerland, September 5–7, Vol.II 705–716.Google Scholar
  59. 59.
    Noor, A.K. (1973). Free vibrations of multilayerd composite plates.American Institute of Aeronautics and Astronautics Journal, Vol.11, 1038–1039.Google Scholar
  60. 60.
    Srinivas, S., Joga Rao, C.V. and Rao, A.K. (1970). Flexural Vibration of rectangular plates. American Society of Mechanical Engineering.Journal of Applied Mechanics, Vol.23, 430–436.Google Scholar
  61. 61.
    Dennis, S.T. and Palazotto, A.N. (1991). Laminated Shell in Cylindrical Bending, Two-Dimensional Approach vs Exact.American Institute of Aeronautics and Astronautics Journal, Vol.29, 647–650.Google Scholar
  62. 62.
    Reddy, J.N. and Liu, C.F. (1985). A Higher-Order Shear Deformation Theory of Laminated Elastic Shells.International Journal of Engineering Sciences, Vol.23, 319–330.MATHCrossRefGoogle Scholar
  63. 63.
    Pandya, B.N. and Kant, T. (1988). Higher-order shear deformable for flexural of sandwich plates. Finite element evaluations.International Journal of Solids and Structures. Vol.24, 1267–1286.MATHCrossRefGoogle Scholar
  64. 64.
    Reddy, J.N. (1987). A generalization of Two-Dimensional Theories of Laminated Composite Plates.Communication in Applied Numerical Methods, Vol.3, 173–180.MATHCrossRefGoogle Scholar
  65. 65.
    Reddy, J.N. and Chao, W.C. (1981). A comparison of closed-form and finite-element solutions of thick laminated anisotropic rectangular plates.Nuclear Engrg. Design, 153–167.Google Scholar
  66. 66.
    Bhaskar, K. and Varadan, T.K. (1991). A Higher-order Theory for bending analysis of laminated shells of revolution.Computer & Structures, Vol.40, 815–819.MATHCrossRefGoogle Scholar
  67. 67.
    Ren, J.G. and Owen, D.R.J. (1989). Vibration and buckling of laminated plates.International Journal of Solids and Structures, Vol.25, 95–106.MATHCrossRefGoogle Scholar
  68. 68.
    Dischiuva, M. (1993). A general quadrilater, multilayered plate element with continuous interlaminar stresses.Composite Structures, Vol.47, 91–105.CrossRefGoogle Scholar
  69. 69.
    Dischiuvo, M. and Carrera, E. (1992). Elasto-dynamic Behavior of relatively thick, symmetrically laminated, anisotropic circular cylindrical shells,Journal of Applied Mechanics, Vol.59, 222–223.Google Scholar
  70. 70.
    Polit, O. and Touratier, M. (2000). High-order triangular sandwich plate finite element for linear and non-linear analyses.Computer Methods in Applied Mechanics and Engineering, Vol.185, 305–324.MATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Toledano, A. and Murakami, H. (1987). A high-order laminated plate theory with improved in-plane responses.International Journal of Solids and Structures, Vol.23, 111–131.MATHCrossRefGoogle Scholar
  72. 72.
    Auricchio, F. and Sacco, E. (2001). Partial-mixed formulation and refined models for the analysis of composites laminated within FSDT.Composite Structures, Vol.46, 103–113.CrossRefGoogle Scholar

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© CIMNE 2003

Authors and Affiliations

  1. 1.Aerospace DepartmentPoliteonico di TorinoTorinoItaly

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