, Volume 28, Issue 3, pp 217–225 | Cite as

A refined theory for laminated beams: Part II—An iterative variational approach

  • Marco Savoia
  • Antonio Tralli
  • Ferdinando Laudiero


This paper presents a displacement-based one-dimensional model for the analysis of laminated composite beams, based on the assumption of cross sections rigid in their own planes. The proposed model is mainly focused on the boundary layer analysis. The representation of the axial displacements is given as products between line functions and warping modes of the cross section. Both the sets of unknown functions are determined by means of a variational formulation in order to obtain the ‘best choice’ for the thickness coordinate functions. The minimization of the total potential energy functional is reduced to a sequence of linear problems by means of a gradient technique. Various examples referring to simply supported and cantilever beams, subjected to distributed or concentrated loads, are solved. The results for stress distributions are found to be in excellent agreement with exact plane strain and finite element plane stress solutions even at very low distances from the end sections.

Key words

Composite structures Laminated beams Boundary layer analysis 


In questo lavoro viene presentato un modello monodimens onale agli spostamenti per l'analisi di travi in laminato multistrato fondato sull'ipotesi di sezioni indeformabili nel proprio piano. Il modello è finalizzato principalmente allo studio degli effetti di bordo. Gli spostamenti assiali vengono espressi attraverso prodotti fra funzioni di linea e modi di ingobbamento della sezione. Entrambi gli insiemi di funzioni incognite sono determinati attraverso una formulazione variazionale allo scopo di ottimizzare le forme di ingobbamento della sezione. La minimizzazione del funzionale dell'energia potenziale totale viene ridotta ad una sequenza di problemi lineari mediante una tecnica tipo gradiente operando alternativamente le varizioni rispetto ai due insiemi di funzioni incognite. Negli esempi risolti, gli stati tensionali ottenuti risultano in eccellente accordo con soluzioni esatte (in stato piano di deformazione) e con soluzioni agli elementi finiti (in stato piano di tensione) anche in prossimità delle sezioni estreme libere o vincolate.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Pagano, N.J., ‘Exact solutions for composite laminates in cylindrical bending’,J. Compos. Mater.,3 (1969) 398–411.Google Scholar
  2. 2.
    Srinivas, S. and Rao, A.K., ‘A three-dimensional solution for plates and laminates’,J. Franklin Inst.,291 (1971) 469–481.CrossRefGoogle Scholar
  3. 3.
    Reddy, J.N., ‘On new developments in the refined theories of laminated composite plates’,Meccanica,25 (1990) 230–238.MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Librescu, L.,Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-type Structures, Noordhoff I.P., Leyden, 1975.MATHGoogle Scholar
  5. 5.
    Lo, K.H., Christensen, R.M. and Wu, E.M., ‘A higher-order theory of plate deformation: Part 2, Laminated plates’,J. Appl. Mech. ASME,44 (1977) 663–676.MATHGoogle Scholar
  6. 6.
    Khoma, I.Y. and Medvedeva, Z.A., ‘Construction of a general solution for a system of equilibrium equations of plates made of reinforced materials’,Prik. Mekh.,18 (1982) 67–74 [Engl. transl.Sov. Appl. Mech.,18, 450–456].MATHGoogle Scholar
  7. 7.
    Librescu, L. and Reddy, J.N., ‘A few remarks concerning several refined theories of anisotropic composite laminated plates’,Internat. J. Engng. Sci.,27 (1989) 515–527.MATHCrossRefGoogle Scholar
  8. 8.
    Srinivas, S., ‘A refined analysis of composite laminates’,J. Sound Vibration,30 (1973) 495–507.MATHADSGoogle Scholar
  9. 9.
    Epstein, M. and Glockner, P.G., ‘Nonlinear analysis of multilayered shells’,Internat. J. Solids Struct.,13 (1977) 1081–1089.MATHCrossRefGoogle Scholar
  10. 10.
    Green, A.E. and Naghdi, P.M., ‘A theory of laminated composite plates’,IMA J. Appl. Math.,29 (1982) 1–23.MATHGoogle Scholar
  11. 11.
    Reddy, J.N., ‘A generalization of two-dimensional theories of laminated composite plates’,Comm. Appl. Numer. Methods,3 (1987) 173–180.MATHCrossRefGoogle Scholar
  12. 12.
    Reddy, J.N., ‘On the generalization of displacement-based laminate theories’,Appl. Mech. Rev.,42 (1989) S213-S222.CrossRefGoogle Scholar
  13. 13.
    Savoia, M., Tralli, A. and Laudiero, F., ‘A model for the analysis of laminated composite beams’,X AIMETA Congress (1990), Pisa, pp. 285–290.Google Scholar
  14. 14.
    Villaggio, P.,Qualitative Methods in Elasticity, Noordhoff I.P., Leyden, 1977.MATHGoogle Scholar
  15. 15.
    Gregory, R.D. and Gladwell, I., ‘The cantilever beam under tension, bending or flexure at infinity’,J. Elasticity,12 (1982) 317–343.MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Choi, I. and Horgan, C.O., ‘Saint-Venant ends effects for plane deformation of sandwich strips’,Internat. J. Solids Struct.,14 (1978) 187–195.MATHCrossRefGoogle Scholar
  17. 17.
    Friedrichs, K.O. and Dressler, R.F., ‘A boundary-layer theory for elastic plates’,Comm. Pure Appl. Math.,14 (1961) 1–33.MATHMathSciNetGoogle Scholar
  18. 18.
    Gol'denveizer, A.L., ‘Derivation of an approximate theory of bending of a plate by the method of asymptotic integration of the equations of the theory of elasticity’,Prik. Mat. Mekh.,26 (1962) 668–686 [Engl. transl.PMM,26, 1000–1025].MATHMathSciNetGoogle Scholar
  19. 19.
    Savoia, M., Laudiero, F. and Tralli, A., ‘A refined theory for laminated beams: Part I-A new high order approach’,Meceanica,28 (1993) 39–51.MATHCrossRefADSGoogle Scholar
  20. 20.
    Oden, J.T. and Reddy, J.N.,Variational Methods in Theoretical Mechanics, Springer-Verlag, Berlin, 1976.MATHGoogle Scholar
  21. 21.
    Curry, H.B., ‘The method of steepest descent for nonlinear minimization problems’,Quart. Appl. Math.,2 (1955) 258–261.MathSciNetGoogle Scholar
  22. 22.
    Cheney, E.W.,Introduction to Approximation Theory, McGraw-Hill, New York, 1966.MATHGoogle Scholar
  23. 23.
    Hildebrand, F.B.,Advanced Calculus for Applications, Prentice-Hall, Englewood Cliffs, N.J., 1962.MATHGoogle Scholar
  24. 24.
    Weinstein, S.E., ‘Approximation of functions of several variables: product Chebyshev approximation’,J. Approx. Theory,2 (1969) 433–447.MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Lanczos, C.,Applied Analysis, Pitman, London, 1964.MATHGoogle Scholar
  26. 26.
    Barret, J.W. and Morton, K.W., ‘Approximate symmetrization and Petrov-Galerkin methods for diffusion-convection problems’,Comput. Meth. Appl. Mech. Engng,45 (1984) 97–122.CrossRefADSGoogle Scholar
  27. 27.
    Laudiero, F. and Savoia, M., ‘Shear strain effects in flexure and torsion of thin-walled beams with open or closed cross-section’,Thin-Walled Struct.,10 (1990) 87–119.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1993

Authors and Affiliations

  • Marco Savoia
    • 1
  • Antonio Tralli
    • 2
  • Ferdinando Laudiero
    • 2
  1. 1.1 stituto di Tecnica delle CostruzioniUniversità di BolognaBolognaItaly
  2. 2.Di partimento di CostruzioniUniversità di FirenzeFirenzeItaly

Personalised recommendations