Skip to main content
SpringerLink
Log in

A refined assumed strain finite element model for statics and dynamics of laminated plates

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

This work discusses some alternate models of a mixed assumed strain finite element which has been developed for laminated plates. After a brief theoretical review about this kind of plates and their possible finite element formulation, specifically devised for predicting the mechanical behavior of such structures, we discuss four possible assumptions for strains generating four kinds of mixed assumed strain finite elements. Several numerical tests performed on the aforementioned finite elements are thoroughly discussed in order to sketch some guidelines which can be useful when dealing with laminated plate problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Reddy, J.N.: Mechanics of Laminated Composite Plates and Shells—Theory and Analysis, 2nd edn. CRC Press, Boca Raton (2004)

    MATH  Google Scholar 

  2. Qatu, M.S.: Vibration of Laminated Shells and Plates, 1st edn. Academic Press, Oxford (2004)

    MATH  Google Scholar 

  3. Yang, P.C., Norris, C.H., Stavsky, Y.: Elastic wave propagation in heterogeneous plates. Int. J. Solids Struct. 2, 665–684 (1966)

    Article  Google Scholar 

  4. Whitney, J.M., Pagano, N.J.: Shear deformation in heterogeneous anisotropic plates. J. Appl. Mech. ASME 37, 1031–1036 (1970)

    Article  ADS  MATH  Google Scholar 

  5. Rolfes, R., Rohwer, K.: Improved transverse shear stresses in composite finite elements based on first order shear deformation theory. Int. J. Numer. Methods Eng. 40, 51–60 (1997)

    Article  Google Scholar 

  6. Rolfes, R., Rohwer, K., Ballerstaedt, M.: Efficient linear transverse normal stress analysis of layered composite plates. Comput. Struct. 68, 643–652 (1998)

    Article  MATH  Google Scholar 

  7. Yu, W., Hodges, D.H., Volovoi, V.V.: Asymptotically accurate 3-D recovery from Reissner-like composite plate finite elements. Comput. Struct. 81, 439–454 (2003)

    Article  Google Scholar 

  8. Naumenko, K., Eremeyev, V.A.: A layer-wise theory for laminated glass and photovoltaic panels. Compos. Struct. 112, 283–291 (2014)

    Article  Google Scholar 

  9. Naumenko, K., Eremeyev, V.A.: A layer-wise theory of shallow shells with thin soft core for laminated glass and photovoltaic applications. Compos. Struct. 178, 434–446 (2017)

    Article  Google Scholar 

  10. Altenbach, H., Eremeyev, V.A., Naumenko, K.: On the use of the first order shear deformation plate theory for the analysis of three-layer plates with thin soft core layer. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 95(10), 1004–1011 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Chepiga, V.E.: Refined theory of multilayered shells. Sov. Appl. Mech. 12, 1127–1130 (1976). English translation from Prikladnaia Mekhanika, vol. 12, pp. 45–49

  12. Lo, K.H., Christensen, R.M., Wu, E.M.: A higher-order theory of plate deformation: part 2, laminated plates. J. Appl. Mech. ASME 44, 669–676 (1977)

    Article  ADS  MATH  Google Scholar 

  13. Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. ASME 51, 745–752 (1984)

    Article  ADS  MATH  Google Scholar 

  14. Pandya, B.N., Kant, T.: Flexure analysis of laminated composites using refined higher-order \(C^0\) plate bending elements. Comput. Methods Appl. Mech. Eng. 66, 173–198 (1988)

    Article  ADS  MATH  Google Scholar 

  15. Yoda, T., Atluri, S.N.: Post-buckling analysis of stiffened laminated composite panels, using a higher-order shear deformation theory. Comput. Mech. 9, 390–404 (1992)

    Article  MATH  Google Scholar 

  16. Yong, Y.-K., Cho, Y.: Higher-order, partial hybrid stress, finite element formulation for laminated plate and shell analysis. Comput. Struct. 57, 817–827 (1995)

    Article  MATH  Google Scholar 

  17. Gaudenzi, P., Mannini, A., Carbonaro, R.: Multi-layer higher order finite elements for the analysis of free-edge stresses in composite laminates. Int. J. Numer. Methods Eng. 41, 851–873 (1998)

    Article  MATH  Google Scholar 

  18. Poniatovskii, V.V.: Theory for plates of medium thickness. PMM 26, 478–486 (1962). English translation from Prikladnaia Matematika i Mekhanika, vol. 26, pp. 335–341

  19. Cicala, P.: Consistent approximations in shell theory. J. Eng. Mech. Div. ASCE 88, 45–74 (1962)

    Google Scholar 

  20. Reddy, J.N.: A generalization of two-dimensional theories of laminated composite plates. Commun. Appl. Numer. Methods 3, 173–180 (1987)

    Article  MATH  Google Scholar 

  21. Di Sciuva, M.: An improved shear deformation theory for moderately thick multi-layered anisotropic shells and plates. J. Appl. Mech. ASME 54, 589–596 (1987)

    Article  MATH  Google Scholar 

  22. Robbins, D.H., Reddy, J.N.: Modeling of thick composites using a layerwise laminate theory. Int. J. Numer. Methods Eng. 36, 655–677 (1993)

    Article  MATH  Google Scholar 

  23. Bisegna, P., Sacco, E.: A layer-wise laminate theory rationally deduced from the three-dimensional elasticity. J. Appl. Mech. ASME 64, 538–545 (1997)

    Article  ADS  MATH  Google Scholar 

  24. Pagano, N.J.: Exact solutions for rectangular bidirectional composites and sandwich plates. J. Compos. Mater. 4, 20–34 (1970)

    Article  ADS  Google Scholar 

  25. Pagano, N.J., Hatfield, S.J.: Elastic behavior of multilayered bidirectional composites. AIAA J. 10, 931–933 (1972)

    Article  ADS  Google Scholar 

  26. Liou, W., Sun, C.T.: A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates. Comput. Struct. 25, 241–249 (1987)

    Article  MATH  Google Scholar 

  27. Cen, S., Long, Y., Yao, Z.: A new hybrid-enhanced displacement-based element for the analysis of laminated composite plates. Comput. Struct. 80, 819–833 (2002)

    Article  Google Scholar 

  28. Whitney, J.M.: Shear correction factors for orthotropic laminates under static load. J. Appl. Mech. ASME 40, 302–304 (1973)

    Article  ADS  Google Scholar 

  29. Vlachoutsis, S.: Shear correction factors for plates and shells. Int. J. Numer. Methods Eng. 33, 1537–1552 (1992)

    Article  MATH  Google Scholar 

  30. Savoia, M., Laudiero, F., Tralli, A.: A two-dimensional theory for the analysis of laminated plates. Comput. Mech. 14, 38–51 (1994)

    Article  MATH  Google Scholar 

  31. Noor, A.K., Burton, W.S.: Assessment of computational models for multilayered anisotropic plates. Compos. Struct. 14, 223–265 (1990)

    Article  Google Scholar 

  32. Auricchio, F., Sacco, E.: Refined first-order shear deformation theory models for composite laminates. J. Appl. Mech. ASME 70, 381–390 (2003)

    Article  ADS  MATH  Google Scholar 

  33. Qi, Y., Knight, N.F.: A refined first-order shear-deformation theory and its justification by plane strain bending problem of laminated plates. Comput. Struct. 33, 49–64 (1996)

    MATH  Google Scholar 

  34. Mau, S.T., Tong, P., Pian, T.H.H.: Finite element solutions for laminated plates. J. Compos. Mater. 6, 304–311 (1972)

    Article  ADS  Google Scholar 

  35. Spilker, R.L., Orringer, O., Witmer, E.A.: Use of the hybrid-stress finite-element model for the static and dynamic analysis of composite plates and shell. Technical Report ASRL TR 181–2, MIT (1976)

  36. Spilker, R.L., Munir, N.I.: A hybrid-stress quadratic serendipity displacement Mindlin plate bending element. Comput. Struct. 12, 11–21 (1980)

    Article  MATH  Google Scholar 

  37. Spilker, R.L.: Hybrid-stress eight-node element for thin and thick multilayered laminated plates. Int. J. Numer. Methods Eng. 18, 801–828 (1982)

    Article  MATH  Google Scholar 

  38. Cazzani, A., Rizzi, N.L., Stochino, F., Turco, E.: Modal analysis of laminates by a mixed assumed-strain finite element model. Math. Mech. Solids 23(1), 99–119 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Garusi, E., Cazzani, A., Tralli, A.: An unsymmetric stress formulation for Reissner–Mindlin plates: a simple and locking-free hybrid rectangular element. Int. J. Comput. Eng. Sci. 5, 589–618 (2004)

    Article  Google Scholar 

  40. Cazzani, A., Garusi, E., Tralli, A., Atluri, S.N.: A four-node hybrid assumed-strain finite element for laminated composite plates. Comput. Mater. Continua 2, 23–38 (2005)

    MATH  Google Scholar 

  41. Bathe, K.-J.: Finite Element Procedures. Prentice-Hall, Upper Saddle River (1996)

    MATH  Google Scholar 

  42. Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice Hall, Englewood Cliffs (1969)

    Google Scholar 

  43. Cazzani, A., Atluri, S.N.: Four-noded mixed finite elements, using unsymmetric stresses, for linear analysis of membranes. Comput. Mech. 11, 229–251 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  44. Cook, R.D., Malkus, D.S., Plesha, M.E.: Concept and Applications of Finite Element Analysis. Wiley, New York (1989)

    MATH  Google Scholar 

  45. Clough, R.W., Penzien, J.: Dynamics of Structures. McGraw-Hill, New York (1975)

    MATH  Google Scholar 

  46. Savoia, M., Laudiero, F., Tralli, A.: A refined theory for laminated beams: part I—a new high order approach. Meccanica 28(1), 39–51 (1993)

    Article  ADS  MATH  Google Scholar 

  47. Savoia, M., Tralli, A., Laudiero, F.: A refined theory for laminated beams: part II—an iterative variational approach. Meccanica 28(3), 217–225 (1993)

    Article  MATH  Google Scholar 

  48. Pilkey, W.D.: Formulas for Stress, Strain, and Structural Matrices, 2nd edn. Wiley, New York (2005)

    MATH  Google Scholar 

  49. Qatu, M.S., Leissa, A.W.: Vibration of Continuous Systems, 1st edn. McGraw-Hill, New York (2011)

    Google Scholar 

  50. Bilotta, A., Formica, G., Turco, E.: Performance of a high-continuity finite element in three-dimensional elasticity. Int. J. Numer. Methods Biomed. Eng. 26, 1155–1175 (2010)

    Article  MATH  Google Scholar 

  51. Greco, L., Cuomo, M.: B-Spline interpolation of Kirchhoff-Love space rods. Comput. Methods Appl. Mech. Eng. 256, 251–269 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. Greco, L., Cuomo, M.: An implicit \(G^1\) multi patch B-spline interpolation for Kirchhoff-Love space rod. Comput. Methods Appl. Mech. Eng. 269, 173–197 (2014)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  53. Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis of plane curved beams. Math. Mech. Solids 21(5), 562–577 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  54. Cazzani, A., Malagù, M., Turco, E.: Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches. Contin. Mech. Thermodyn. 28(1), 139–156 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. Cazzani, A., Malagù, M., Turco, E., Stochino, F.: Constitutive models for strongly curved beams in the frame of isogeometric analysis. Math. Mech. Solids 21(2), 182–209 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  56. Greco, L., Cuomo, M.: An isogeometric implicit \(\text{ G }^1\) mixed finite element for Kirchhoff space rods. Comput. Methods Appl. Mech. Eng. 298, 325–349 (2016)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  57. Cazzani, A., Stochino, F., Turco, E.: An analytical assessment of finite elements and isogeometric analysis of the whole spectrum of Timoshenko beams. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 96(10), 1220–1244 (2016)

    Article  ADS  MATH  Google Scholar 

  58. Altenbach, H., Eremeyev, V.A.: On the linear theory of micropolar plates. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 89(4), 242–256 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. Altenbach, H., Eremeyev, V.A.: On the shell theory on the nanoscale with surface stresses. Int. J. Eng. Sci. 49(12), 1294–1301 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  60. Šilhavỳ, M.: A direct approach to nonlinear shells with application to surface-substrate interactions. Math. Mech. Complex Syst. 1(2), 211–232 (2013)

    Article  MATH  Google Scholar 

  61. Turco, E.: Tools for the numerical solution of inverse problems in structural mechanics: review and research perspectives. Eur. J. Environ. Civ. Eng. 21(5), 509–554 (2017)

    Article  MathSciNet  Google Scholar 

  62. dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math. Mech. Solids 20(8), 887–928 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  63. dell’Isola, F., Della Corte, A., Esposito, R., Russo, L.: Some cases of unrecognized transmission of scientific knowledge: from antiquity to Gabrio Piola’s peridynamics and generalized continuum theories. In: Altenbach, H., Forest, S. (eds.) Generalized Continua as Models for Classical and Advanced Materials, pp. 77–128. Springer, Cham (2016)

    Chapter  Google Scholar 

  64. Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A Math. Phys. Eng. Sci. 474(20170878), 1–19 (2018)

    MathSciNet  MATH  Google Scholar 

  65. Rizzi, N., Varano, V., Gabriele, S.: Initial postbuckling behavior of thin-walled frames under mode interaction. Thin Walled Struct. 68, 124–134 (2013)

    Article  Google Scholar 

  66. Gabriele, S., Rizzi, N., Varano, V.: A 1D higher gradient model derived from Koiter’s shell theory. Math. Mech. Solids 21(6), 737–746 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  67. Cazzani, A., Wagner, N., Ruge, P., Stochino, F.: Continuous transition between traveling mass and traveling oscillator using mixed variables. Int. J. Non-Linear Mech. 66, 82–95 (2015)

    Google Scholar 

  68. Acito, M., Stochino, F., Tattoni, S.: Structural response and reliability analysis of RC beam subjected to explosive loading. Appl. Mech. Mater. 82, 434–439 (2011)

    Article  ADS  Google Scholar 

  69. Stochino, F.: RC beams under blast load: reliability and sensitivity analysis. Eng. Fail. Anal. 66, 544–565 (2016)

    Article  Google Scholar 

  70. dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060804 (2015)

    Article  Google Scholar 

  71. Giorgio, I., Grygoruk, R., dell’Isola, F., Steigmann, D.J.: Pattern formation in the three-dimensional deformations of fibered sheets. Mech. Res. Commun. 69, 164–171 (2015)

    Article  Google Scholar 

  72. dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenisation, experimental and numerical examples of equilibrium. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. 472(2185), 1–23 (2016)

    Article  Google Scholar 

  73. Scerrato, D., Zhurba Eremeeva, I.A., Lekszycki, T., Rizzi, N.L.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 96(11), 1268–1279 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  74. Scerrato, D., Giorgio, I., Rizzi, N.L.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 67(53), 1–19 (2016)

    MathSciNet  MATH  Google Scholar 

  75. Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A Math. Phys. Eng. Sci. 473(20170636), 1–21 (2017)

    MathSciNet  MATH  Google Scholar 

  76. Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Contin. Mech. Thermodyn. 28(1), 215–234 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. Giorgio, I., Galantucci, L., Della Corte, A., Del Vescovo, D.: Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: current and upcoming applications. Int. J. Appl. Electromagn. Mech. 47(4), 1051–1084 (2015)

    Article  Google Scholar 

  78. Giorgio, I., Culla, A., Del Vescovo, D.: Multimode vibration control using several piezoelectric transducers shunted with a multiterminal network. Arch. Appl. Mech. 79, 859–879 (2009)

    Article  ADS  MATH  Google Scholar 

  79. Enakoutsa, K., Della Corte, A., Giorgio, I.: A model for elastic flexoelectric materials including strain gradient effects. Math. Mech. Solids 21(2), 242–254 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  80. Maurini, C., dell’Isola, F., Pouget, J.: On models of layered piezoelectric beams for passive vibration control. J. Phys. IV 115, 307–316 (2004)

    MATH  Google Scholar 

  81. Aymerich, F., Serra, M.: An ant colony optimization algorithm for stacking sequence design of composite laminates. Comput. Model. Eng. Sci. 13(1), 49–65 (2006)

    Google Scholar 

  82. Aymerich, F., Serra, M.: Optimization of laminate stacking sequence for maximum buckling load using the ant colony optimization (ACO) metaheuristic. Compos. Part A Appl. Sci. Manuf. 39(2), 262–272 (2008)

    Article  Google Scholar 

  83. Banichuk, N.V., Ivanova, S.Y., Ragnedda, F., Serra, M.: Multiobjective approach for optimal design of layered plates against penetration of strikers. Mech. Based Des. Struct. Mach. 41(2), 189–201 (2013)

    Article  Google Scholar 

  84. Altenbach, H., Eremeyev, V.A.: Direct approach-based analysis of plates composed of functionally graded materials. Arch. Appl. Mech. 78(10), 775–794 (2008)

    Article  ADS  MATH  Google Scholar 

  85. Altenbach, H., Eremeyev, V.A.: Eigen-vibrations of plates made of functionally graded material. Comput. Mater. Continua 9(2), 153–178 (2009)

    MATH  Google Scholar 

  86. Altenbach, H., Eremeyev, V.A.: Analysis of the viscoelastic behavior of plates made of functionally graded materials. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM) 88(5), 332–341 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  87. Altenbach, H., Eremeyev, V.A.: On the bending of viscoelastic plates made of polymer foams. Acta Mech. 204(3–4), 137 (2009)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

The initial part of this research (up to December 2015) has been partially funded by MIUR, the Italian Ministry of Education, University and Research (Grant Number PRIN 2010-2011; Project 2010MBJK5B, “Dynamic, Stability and Control of Flexible Structures”); such support is gratefully acknowledged. Subsequently the final part of the research has received a financial support by the research grant “Healthy Cities and Smart Territories” (2016/17) funded by Fondazione di Sardegna and by R.A.S., the Autonomous Region of Sardinia; this support is gratefully acknowledged, too. The authors are also indebted to professor Mohamad S. Qatu, from Eastern Michigan University, for fruitful discussions.

Author information

Authors and Affiliations

  1. Department of Civil and Environmental Engineering and Architecture (DICAAR), University of Cagliari, Cagliari, Italy

    Antonio Cazzani, Mauro Serra & Flavio Stochino

  2. Department of Architecture, Design and Urban Planning (DADU), University of Sassari, Alghero, Italy

    Emilio Turco

Authors
  1. Antonio Cazzani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mauro Serra
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Flavio Stochino
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Emilio Turco
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilio Turco.

Additional information

Communicated by Francesco dell’Isola.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cazzani, A., Serra, M., Stochino, F. et al. A refined assumed strain finite element model for statics and dynamics of laminated plates. Continuum Mech. Thermodyn. 32, 665–692 (2020). https://doi.org/10.1007/s00161-018-0707-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-018-0707-x

Keywords

Search

Navigation