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A two-dimensional theory for the analysis of laminated plates

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Abstract

A new displacement-based two-dimensional theory for the analysis of multilayered plates is presented. The theory is based on the only kinematic constraint of transverse inextensibility, whereas no restrictions are imposed on the representation of the in-plane displacement components. A governing system of integral-differential equations is obtained which can be given a closed-form solution for a number of problems where no boundary layer are present. It is also shown that most of the 2-D plate models can be directly derived from the presented theory. The possibility of developing asymptotic solutions in the boundary layers is discussed with reference to the problem of a plate in cylindrical bending. Finally some numerical solutions are compared with those given by the plate model by Lo et al. (1977) and with F.E.M. solutions.

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References

  • Alliney, S.; Carnicer, R. S. 1992: A hybrid finite element model for multilayered plates. Comput. Mech. 10: 319–333

    Google Scholar 

  • Andreev, A. N.; Nemirovskii, Y. V. 1977: On the theory of multilayered elastic anisotropic shells. Izv. Akad. Nauk. SSSR, Mekh. Tverd. Tela 12: 87–96 [Transl. Mech. Solids 12: 73–81]

    Google Scholar 

  • Arridge, R. G. C.; Barham, P. J.; Farrell, C. J.; Keller, A. 1976: The importance of end effects in the measurement of moduli of highly anisotropic materials. J. Mat. Sci. 11: 788–790

    Google Scholar 

  • Barrett, K. E.; Ellis, S. 1988: An exact theory of elastic plates. Int. J. Solids Struct. 24: 859–880

    Google Scholar 

  • Bhimaraddi, A.; Stevens, L. K. 1984: A higher order theory for free vibration of orthotropic, homogeneous, and laminated rectangular plates. J. Appl. Mech. ASME 51: 195–198

    Google Scholar 

  • Bickford, W. B. 1982: A consistent higher order beam theory. Dev. Theor. Appl. Mech. 11: 137–150

    Google Scholar 

  • Chatterjee, S. N.; Ramnath, V. 1988: Modeling laminated composite structures as assemblage of sublaminates. Int. J. Solids Struct. 24: 439–458

    Google Scholar 

  • Chepiga, V. E. 1977: On constructing a theory of multilayered anisotropic shells with prescribed arbitrary accuracy of order h N. Izv. Akad. Nauk. SSSR, Meckh. Tverd. Tela 12: 113–120 [Transl. Mech. Solids 12: 98–103]

    Google Scholar 

  • Choi, I.; Horgan, C. O. 1977: Saint-Venant's principle and end effects in anisotropic elasticity. J. Appl. Mech. ASME 44: 424–430

    Google Scholar 

  • Choi, I.; Horgan, C. O. 1978: Saint-Venant end effects for plane deformation of sandwich strips. Int. J. Solids Struct. 14: 187–195

    Google Scholar 

  • Ciarlet, P. G.; Destuynder, P. 1979: A justification of the two-dimensional linear plate model. J. Mechanique 18: 315–344

    Google Scholar 

  • Cicala, P. 1962: Consistent approximations in shell theory. J. Engng. Mech. Div., Proc. Am. Soc. Civ. Engng. 88: 45–74

    Google Scholar 

  • Davet, J. L.; Destuynder, Ph. 1985: Singularités logaritmiques dans les effects de bord d'une plaque en matériaux composites. J. Mec. Theor. Appl. 4: 56–71

    Google Scholar 

  • Davet, J. L.; Destuynder, Ph. 1986: Free-edge stress concentration in composite laminates: A boundary layer approach. Comp. Meth. Appl. Mech. Eng. 59: 129–140

    Google Scholar 

  • Ericksen, J. L.; Truesdell, C. 1957: Exact theory of stress and strain in rods and shells. Arch. Rat. Mech. Anal. 1: 295–323

    Google Scholar 

  • Folkes, M. J.; Arridge, R. G. C. 1975: The measurement of shear modulus in highly anisotropic materials: the valiity of St. Venant principle. J. Phys. D 8: 1053–1064

    Google Scholar 

  • Gol'denveizer, A. L. 1968: Methods for justifying and refining the theory of shells. Prik. Mat. Mekh. 32: 684–695 [Transl. PMM 32: 704–718]

    Google Scholar 

  • Gol'denveizer, A. L.; Kolos, A. V. 1965: On the derivation of two-dimensional equations in the theory of thin elastic plates. Prik. Mat. Mekh. 29: 141–155 [Transl. PMM 29: 151–166]

    Google Scholar 

  • Green, A. E.; Naghdi, P. M. 1982: A theory of laminated composite plates. IMA J. Appl. Math. 29: 1–23

    Google Scholar 

  • Green, A. E.; Naghdi, P. M.; Wainwrigth, W. L. 1965: A general theory of a Cosserat surface. Arch. Rat. Mech. Anal. 20: 287–308

    Google Scholar 

  • Gregory, R. D.; Wan, F. Y. M. 1985: On plate theories and Saint Venant's principle. Int. J. Solids Struct. 10: 1005–1024

    Google Scholar 

  • Grigolyuk, E. I.; Kogan, F. A. 1972: State of the art of the theory of multilayered shells. Priklad. Mekh. 8: 3–17 [Transl. Sov. Appl. Mech. 8: 583–595

    Google Scholar 

  • Horgan, C. O. 1972: On Saint-Venant's principle in plane anisotropic elasticity. J. Elasticity 2: 169–180

    Google Scholar 

  • Horgan, C. O. 1982: Saint-Venant end effects in composites. J. Comp. Mat. 16: 411–422

    Google Scholar 

  • Koiter, W. T. 1966: On the nonlinear theory of thin elastic shells. Proc. Kon. Ned. ak. Wet. B 69: 1–54

    Google Scholar 

  • Koiter, W. T. 1969: Foundation and basic equations of shell theory: a survey of recent progress. In: Niordson, F. I. (ed.): IUTAM Symposium on the theory of thin shells. Springer Verlag, Berlin, 93–105

    Google Scholar 

  • Koiter, W. T.: Simmonds, J. G. 1972: Foundation of shell theory. In: Becker, E.; Mikhailov, G. K. (eds.): Theoretical and applied mechanics. Mosca, Springer Verlag, Berlin, pp. 150–176

    Google Scholar 

  • Ladeveze, P.1976: Justification de la théorie linéaire des coques élastiques. J. Mecanique 15: 813–856

    Google Scholar 

  • Lembo, M.; Podio Guidugli, P. 1991: Plate theory as an exact consequence of three-dimensional linear elasticity. Eur. J. Mech., A/Solids 10:485–516

    Google Scholar 

  • Lewis, E. L. V. 1979: An experimental study of end effects in the extensional deformation of polymers. J. Mat. Sci. 14: 2343–2352

    Google Scholar 

  • Librescu, L. 1967: On the theory of anisotropic elastic shells and plates. Int. J.Solids Struct. 3:53–68

    Google Scholar 

  • Librescu, L., 1975a: Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures. Noordhoff I. P., Leyden

    Google Scholar 

  • Librescu, L. 1975b: Improved linear theory of elastic anisotropic multilayered shells. Mekh. Polimerov 11: 1038–1050 [Transl. Polymer Mech. 11: 885–896]

    Google Scholar 

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

    Google Scholar 

  • Murakami, H. 1986: Laminated composite plate theory with improved in-plate response. J. Appl. Mech. ASME 53: 661–666

    Google Scholar 

  • Naghdi, P. M. 1972: The theory of shells and plates. In: Flugge, S. (ed.):Handbuch der Physik vol VI-2, Berlin, Springer-Verlag, 425–640

    Google Scholar 

  • Nayfeh, A. 1973: Perturbation methods. Wiley, J. & Sons, New York

    Google Scholar 

  • Noor, A. K.; Burton, W. S. 1989: Assessment of shear deformation theories for multilayered composite plates. Appl. Mech. Rev. 42: 1–12

    Google Scholar 

  • Pagano, N. J. 1970: Exact solutions for rectangular bidirectional composites and sandwich plates. J. Comp. Mater. 4: 20–34

    Google Scholar 

  • Podio Guidugli, P. 1989: An exact derivation of the thin plate equation. J. Elast. 22: 121–133

    Google Scholar 

  • Poniatovskii, V. V. 1962: Theory for plates of medium thickness. Prik. Mat. Mekh. 26: 335–341 [Transl. PMM 26: 478–486]

    Google Scholar 

  • Reddy, J. N. 1984: A simple higher-order theory for laminates composite plates. J. Appl. Mech. ASME 51: 745–752

    Google Scholar 

  • Reddy, J. N. 1986: Applied functional analysis and variational methods in engineering. McGraw-Hill, New York

    Google Scholar 

  • Reddy, J. N. 1987: A generalization of two-dimensional theories of laminated composite plates. Comm. Appl. Num. Meth. 3: 173–180

    Google Scholar 

  • Reddy, J. N. 1990: A general non-linear third-order theory of plates with moderate thickness. Int. J. Non-Lin. Mech. 25: 677–686

    Google Scholar 

  • Rehfield, L. W.; Murthy, P. L. N. 1982: Toward a new engineering theory of bending: fundamentals. AIAA J. 20: 693–699

    Google Scholar 

  • Reissner, E.; Stavsky, Y. 1961: Bending and stretching of certain types of heterogeneous aelotropic elastic plates. J. Appl. Mech. ASME 28:402–408

    Google Scholar 

  • Savoia, M. 1992: On the analysis of multilayered composite structures. (In Italian), PhD. Dissertation, Bologna, Italy

  • Savoia, M. 1993: A new 2-D approach for an accurate analysis of multilayered plates in bending. To appear

  • Savoia, M.; Reddy, J. N. 1992: A variational approach to three-dimensional elasticity solutions of laminated composite plates. J. Appl. Mech. ASME 59:S166-S175

    Google Scholar 

  • Savoia, M.; Tralli, A. 1992: On the accuracy of one-dimensional models for multilayered composite beams. Proceed. of the XI AIMETA Conference, pp. 487–492

  • Savoia, M.; Laudiero, F.; Tralli, A. 1993a: A refined theory for laminated beams—Part I: A new high order approach. Meccanica 28:38–51

    Google Scholar 

  • Savoia, M.; Tralli, A.; Laudiero, F. 1993b: A refined theory for laminated beams—Part II: An iterative variational approach. Meccanica 28:217–225

    Google Scholar 

  • Schmidt, R. 1977: A refined nonlinear theory of plates with transverse shear deformation. J. Indust. Math. Soc. 27:23–38

    Google Scholar 

  • Srinivas, S. 1973: A refined analysis of composite laminates. J. Sound Vibr. 30:495–507

    Google Scholar 

  • Srinivas, S.; Rao, A. K. 1973: Flexure of thick rectangular plates. J. Appl. Mech. ASME 40:208–299

    Google Scholar 

  • Truesdell, C.; Noll, W. 1965: The non-linear field theories of mechanics. In: Flugge, S. (ed.): Encylopedia of Physics, III/3, Springer-Verlag, Berlin

    Google Scholar 

  • Vashakmadze, T. S. 1987: On the problem of constructing the mathematical theory of plates and shells. In: Besseling, J. F.; Eckhaus, W. (eds.): Trends in applications of mathematics in mechanics. Springer-Verlag

  • Vianello, M. 1990: On the active part of the stress for elastic materials with internal constraints. J. Elast. 24: 289–294

    Google Scholar 

  • Wang, S. S.; Choi, I. 1982: Boundary layer effect in composite laminates. Part I: Free edge stress singularities. J. Appl. Mech. ASME 49:541–548

    Google Scholar 

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Authors and Affiliations

  1. Instituto di Tecnica delle Costruzioni, Facoltà di Ingegneria, Università di Bologna, Viale Risorgimento 2, Bologna, Italy

    M. Savoia

  2. Dipartimento di Costruzioni, Università di Firenze, Piazza Brunelleschi 6, Firenze, Italy

    F. Laudiero & A. Tralli

Authors
  1. M. Savoia
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  2. F. Laudiero
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  3. A. Tralli
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Communicated by B. Schrefler, 3 December 1993

The authors gratefully acknowledge the financial support of the (Italian) Ministry of University and Scientific and Technological Research (MURST) and of the National Council of Research (CNR, contr. 92.03045.07)

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Savoia, M., Laudiero, F. & Tralli, A. A two-dimensional theory for the analysis of laminated plates. Computational Mechanics 14, 38–51 (1994). https://doi.org/10.1007/BF00350156

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