×

Hierarchical models for failure analysis of plates bent by distributed and localized transverse loadings. (English) Zbl 1140.74534

Summary: The failure analysis of simply supported, isotropic, square plates is addressed. Attention focuses on minimum failure load amplitudes and failure locations. von Mises’ equivalent stress along the plate thickness is also addressed. Several distributed and localized loading conditions are considered. Loads act on the top of the plate. Bi-sinusoidal and uniform loads are taken into account for distributed loadings, while stepwise constant centric and off-centric loadings are addressed in the case of localized loadings. Analysis is performed considering plates whose length-to-thickness ratio \(a/h\) can be as high as 100 (thin plates) and as low as 2 (very thick plates). Results are obtained via several 2D plate models. Classical theories (CTs) and higher order models are applied. Those theories are based on polynomial approximation of the displacement field. Among the higher order theories (HOTs), HOTs\(^{\text d}\) models account for the transverse shear deformations, while HOTs models account for both transverse shear and transverse normal deformations. LHOTs represent a local application of the higher order theories. A layerwise approach is thus assumed: by means of mathematical interfaces, the plate is considered to be made of several fictitious layers. The exact 3D solution is presented in order to determine the accuracy of the results obtained via the 2D models. In this way a hierarchy among the 2D theories is established. CTs provide highly accurate results for \(a/h\) greater than 10 in the case of distributed loadings and greater than 20 for localized loadings. Results obtained via HOTs are highly accurate in the case of very thick plates for bi-sinusoidal and centric loadings. In the case of uniform and off-centric loadings a high gradient is present in the neighborhood of the plate top. In those cases, LHOTs yield results that match the exact solution.

MSC:

74R10 Brittle fracture
74K20 Plates
74E30 Composite and mixture properties
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Carrera, E., 2002. Theories and finite elements for multilayered plates and shells. Archives of Computational Methods in Engineering, 9(2):87-140. [doi:10.1007/BF02736649] · Zbl 1062.74048 · doi:10.1007/BF02736649
[2] Carrera, E., 2003. Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Archives of Computational Methods in Engineering, 10(3):215-296. [doi:10.1007/BF02736224] · Zbl 1140.74549 · doi:10.1007/BF02736224
[3] Carrera, E., Giunta, G., 2007. Hierarchical closed form solutions for plates bent by localized transverse loadings. Journal of Zhejiang University SCIENCE A, 8(7):1026-1037. [doi:10.1631/jzus.2007.A1026] · doi:10.1631/jzus.2007.A1026
[4] Cauchy, A.L., 1828. Sur l’euilibre et le mouvement d’une plaque solide. Exercises de Matematique, 3:328-355.
[5] Demasi, L., 2007. 3D closed form solution and exact thin plate theories for isotropic plates. Composites Structures, 80(2):183-195. [doi:10.1016/j.compstruct.2006.04.073] · doi:10.1016/j.compstruct.2006.04.073
[6] Kam, T.Y., Jan, T.B., 1995. First-ply failure analysis of laminated composite plates based on the layerwise linear displacement theory. Composites Structures, 32(1-4):583-591. [doi:10.1016/0263-8223(95)00069-0] · doi:10.1016/0263-8223(95)00069-0
[7] Kirchhoff, G., 1850. Über das Gleichgewicht und die Bewegung einer elastishen Sceibe. J. Reine Angew. Math., 40:51-88. · ERAM 040.1086cj · doi:10.1515/crll.1850.40.51
[8] Librescu, L., 1975. Elasto-statics and Kinematics of Anisotropic and Heterogeneous Shell-type Structures. Nordhoff Int., Leiden, The Netherlands. · Zbl 0335.73027
[9] Love, A.E.H., 1959. A Treatise on Mathematical Theory of Elasticity. Cambridge University Press, UK.
[10] Mindlin, E., 1951. Influence of the rotatory inertia and shear in flexural motions of isotropic elastic plates. J. Appl. Mech., 18:1031-1036. · Zbl 0044.40101
[11] Pandey, A.K., Reddy, J.N., 1987. A Post First-ply Failure Analysis of Composites Laminates. Structures, Structural Dynamics and Materials Conference, Monterey, CA, USA, p.788-797.
[12] Poisson, S.D., 1829. Memoire sur l’euilibre et le mouvement des corps elastique. Mem. Acad. Sci., 8:357.
[13] Reddy, J.N., 1984. A simple higher-order theory for laminated composite plates. J. Appl. Mech., 51:745-752. · Zbl 0549.73062 · doi:10.1115/1.3167719
[14] Reddy, J.N., 1997. Mechanics of Laminated Composites Plates. Theory and Analysis. CRC Press, Boca Raton, Florida. · Zbl 0899.73002
[15] Reddy, J.N., Pandey, A.K., 1987. A first-ply failure analysis of composites laminates. Computers and Structures, 25(3):371-393. [doi:10.1016/0045-7949(87)90130-1] · Zbl 0599.73055 · doi:10.1016/0045-7949(87)90130-1
[16] Reddy, Y.S.N., Reddy, J.N., 1987. Linear and Non Linear Failure Analysis of Composites Laminates with Transverse Shear. American Institute of Aeronautics and Astronautics, Inc.
[17] Reissner, E., 1945. The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech., 12:69-76. · Zbl 0063.06470
[18] Turvey, G.J., 1980a. An initial flexural failure analysis of symmetrically laminated cross-ply rectangular plates. International Journal of Solids and Structures, 16(5):451-463. [doi:10.1016/0020-7683 (80)90043-8] · Zbl 0446.73051 · doi:10.1016/0020-7683(80)90043-8
[19] Turvey, G.J., 1980b. Flexural failure analysis of angle-ply laminates of GFRP and CFRP. The Journal of Strain Analysis for Engineering Design, 15:43-49. [doi:10.1243/03093247V151043] · doi:10.1243/03093247V151043
[20] Turvey, G.J., 1980c. A study on the onset of flexural failure in cross-ply laminated strips. Fibre Science and Technology, 13(5):325-336. [doi:10.1016/0015-0568(80)90008-1] · doi:10.1016/0015-0568(80)90008-1
[21] Turvey, G.J., 1981. Initial flexural failure of square, simply supported, angle-ply plates. Fibre Science and Technology, 15(1):47-63. [doi:10.1016/0015-0568(81) 90031-2] · doi:10.1016/0015-0568(81)90031-2
[22] Turvey, G.J., 1982. Uniformly loaded, antisymmetric cross-ply laminated, rectangular plates: an initial flexural failure analysis. Fibre Science and Technology, 16(1):1-10. [doi:10.1016/0015-0568(82)90010-0]
[23] Turvey, G. J.; Marshall, I. H. (ed.), Effect of Shear Deformation on the Onset of Flexural Failure in Symmetric Cross-ply Laminated Rectangular Plates, 141-146 (1987), London
[24] Washizu, K., 1968. Variational Methods in Elasticity and Plasticity. Pergamon Press, NY. · Zbl 0164.26001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.