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Calculating 3D stresses in layered composite plates and shells

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Advanced failure criteria for fiber composites account for all six components of the stress tensor. Plate and shell analysis, however, is sensibly performed by assuming the plane state of stress, which results in global displacements, cross-sectional membrane forces, and bending moments of suitable accuracy. Based on these results, equilibrium conditions can be applied to locally determine the stress components in the transverse direction. Therewith, the transverse shear stresses require first derivatives and transverse normal stresses even second derivatives of the membrane stresses. Higher-order finite elements would be necessary if these stress components are to be determined on the element level. To ease this deficiency, a procedure is proposed based on neglecting the in-plane derivatives of the membrane forces and twisting moments as well as the mixed derivatives of the bending moments. This allows us to reduce the order of differentiation by one. Applicability of this procedure is demonstrated by calculating the transverse shear and normal stresses for layered composite structures of different geometric dimensions and various stacking orders under mechanical as well as thermal loads. Comparison with results from 3D analyses shows excellent accuracy and efficiency of the proposed procedure.

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  1. 1.

    A. Puck, Festigkeitsanalyse von Faser-Matrix-Laminaten, Modelle für die Praxis, Carl Hanser Verlag, München—Wien (1996).

  2. 2.

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

  3. 3.

    R. L. Spilker, S. C. Chou, and O. Orringer, “Alternate hybrid-stress elements for analysis of multilayer composite plates,” J. Compos. Mater.,11, 51–70 (1977).

  4. 4.

    R. L. Spilker, “Hybrid-stress eight-node elements for thin and thick multilayer laminated plates,” Int. J. Numer. Meth. Eng.,18, 801–828 (1982).

  5. 5.

    D. R. J. Owen and Z. H. Li, “A refined analysis of laminated plates by the finite element displacement method. I. Fundamentals and static analysis,” Computers and Structures,26, 907–914 (1987).

  6. 6.

    R. A. Chaudhuri, “An equilibrium method for prediction of transverse shear stresses in a thick laminated plate,” Computers and Structures,23, 139–146 (1986).

  7. 7.

    D. H. Robbins, Jr. and J. N. Reddy, “Modelling of thick composites using a layerwise laminate theory,” Int. J. Numer. Meth. Eng.,36, 655–677 (1993).

  8. 8.

    F. Gruttmann and W. Wagner, “On the numerical analysis of local effects in composite structures,” Compos. Struct.,29, 1–12 (1994).

  9. 9.

    R. A. Chaudhuri and P. Seide, “An approximate semi-analytical method for prediction of interlaminar shear stress in an arbitrarily laminated thick plate,” Computers and Structures,25, 627–636 (1987).

  10. 10.

    J. J. Engblom and O. O. Ochoa, “Through-the-thickness stress predictions for laminated plates of advanced composite materials,” Int. J. Numer. Meth. Eng.,21, 1759–1776 (1985).

  11. 11.

    T. Kant and B. N. Pandya, “A simple finite element formulation of a higher-order theory for unsymmetrically laminated composite plates,” Compos. Struct.,9, 215–246 (1988).

  12. 12.

    J. N. Reddy, E. J. Barbero, and J. L. Teply, “A plate bending element based on a generalized laminate plate theory,” Int. J. Numer. Meth. Eng.,28, 2275–2292 (1989).

  13. 13.

    A. K. Noor, W. S. Burton, and J. M. Peters, “Predictor-corrector procedure for stress and free vibration analyses of multilayered composite plates and shells,” Comput. Meth. Appl. Mech. Eng.,82, 341–364 (1990).

  14. 14.

    C. W. Pryor and R. M. Barker, “A finite element analysis including transverse shear effects for application to laminated plates,” AIAA J.,9, 912–917 (1971).

  15. 15.

    B. S. Manjunatha and T. Kant, “On evaluation of transverse stresses in layered symmetric composite and sandwich laminates under flexure,” Eng.Comput., 10, 499–518 (1993).

  16. 16.

    A. K. Noor, Y. H. Kim, and J. M. Peters, “Transverse shear stresses and their sensitivity coefficients in multilayered composite panels,” AIAA J.,32, 1259–1269 (1994).

  17. 17.

    R. Rolfes and K. Rohwer, “Improved transverse shear stresses in composite finite elements based on first-order shear deformation theory,” Int. J. Numer. Meth. Eng., 40, 51–60 (1997).

  18. 18.

    R. Rolfes, K. Rohwer, “Eine einfache Methode zur Ermittlung aller Querspannungen in Faserverbundplatten als Voraussetzung einer verbesserten Versagensanalyse,” Technische Mechanik,18, 161–167 (1998).

  19. 19.

    R. Rolfes, A. K. Noor, and H. Sparr, “Evaluation of transverse thermal stresses in composite plates based on first-order shear deformation theory,” in: Y. W. Kwon, D. C. Davis, H. H. Chung, and L. Librescu (eds.), Recent Advances in Solids/Structures and Application of Metallic Materials, ASME International Mechanical Engineering Congress, Dallas (1997), pp. 167–184.

  20. 20.

    K. Rohwer, “Immproved transverse shear stiffnesses for layered finite elements,” Forschungsbericht DFVLR-FB 88-32, Braunschweig (1988).

  21. 21.

    M. Savoia and J. N. Reddy, “A variational approach to three-dimensional elasticity solutions of laminated composite plates,” J. Appl. Mech.,59, P. 2, S166-S175. (June, 1992).

  22. 22.

    A. K. Noor and W. S. Burton, “Three-dimensional solutions for the free vibrations and buckling of thermally stressed multilayered angle-ply composite plates,” J. Appl. Mech.,59, 868–877 (1992).

  23. 23.

    J. G. Ren, “Exact solutions for laminated cylindrical shells in cylindrical bending,” Compos. Sci. Technol.,29, 169–187 (1987).

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Deutsches Zentrum für Luft- und Raumfahrt e.V. (German Aerospace Center), Institute of Structural Mechanics, D-38022 BRAUNSCHWEIG, Germany. Published in Mekhanika Kompozitnykh Materialov, Vol. 34, No. 4, pp. 491–500, July–August, 1998.

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Rohwer, K., Rolfes, R. Calculating 3D stresses in layered composite plates and shells. Mech Compos Mater 34, 355–362 (1998). https://doi.org/10.1007/BF02257903

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  • Normal Stress
  • Stress Component
  • Fiber Composite
  • Composite Plate
  • Transverse Shear