Analytical Solution for Bending Stress Intensity Factor from Reissner’S Plate Theory

Author(s)
Lalitha Chattopadhyay

Abstract

Plate-type structural members are commonly used in engineering applications like aircraft, ships nuclear reactors etc. These structural members often have cracks arising from manufacture or from material defects or stress concentrations. Designing a structure against fracture in service involves consideration of strength of the structure as a function of crack size, dimension and the applied load based on principles of fracture mechanics. In most of the engineering structures the plate thickness is generally small and in these cases though the classical plate theory has provided solutions, the neglect of transverse shear deformation leads to the limitation that only two conditions can be satisfied on any boundary whereas we have three physical boundary conditions on an edge of a plate. In this paper this incompatibility is eliminated by using Reissner plate theory where the transverse shear deformation is included and three physically natural boundary conditions of vanishing bending moment, twisting moment and transverse shear stress are satisfied at a free boundary. The problem of estimating the bending stress distribution in the neighbourhood of a crack located on a single line in an elastic plate of varying thickness subjected to out-of-plane moment applied along the edges of the plate is examined. Using Reissner’s plate theory and integral transform technique, the general formulae for the bending moment and twisting moment in an elastic plate containing cracks located on a single line are derived. The thickness depended solution is obtained in a closed form for the case in which there is a single crack in an infinite plate and the results are compared with those obtained from the literature.

Plate-type structural members are commonly used in engineering applications like aircraft, ships nuclear reactors etc. These structural members often have cracks arising from manufacture or from material defects or stress concentrations. Designing a structure against fracture in service involves consideration of strength of the structure as a function of crack size, dimension and the applied load based on principles of fracture mechanics. In most of the engineering structures the plate thickness is generally small and in these cases though the classical plate theory has provided solutions, the neglect of transverse shear deformation leads to the limitation that only two conditions can be satisfied on any boundary whereas we have three physical boundary conditions on an edge of a plate. In this paper this incompatibility is eliminated by using Reissner plate theory where the transverse shear deformation is included and three physically natural boundary conditions of vanishing bending moment, twisting moment and transverse shear stress are satisfied at a free boundary. The problem of estimating the bending stress distribution in the neighbourhood of a crack located on a single line in an elastic plate of varying thickness subjected to out-of-plane moment applied along the edges of the plate is examined. Using Reissner’s plate theory and integral transform technique, the general formulae for the bending moment and twisting moment in an elastic plate containing cracks located on a single line are derived. The thickness depended solution is obtained in a closed form for the case in which there is a single crack in an infinite plate and the results are compared with those obtained from the literature.

Keywords

Reissner Plate Theory, Integral Transform, Stress Intensity Factor, Singular Integral Equation

Reissner Plate Theory, Integral Transform, Stress Intensity Factor, Singular Integral Equation

Cite this paper

nullL. Chattopadhyay, "Analytical Solution for Bending Stress Intensity Factor from Reissner’S Plate Theory,"*Engineering*, Vol. 3 No. 5, 2011, pp. 517-524. doi: 10.4236/eng.2011.35060.

nullL. Chattopadhyay, "Analytical Solution for Bending Stress Intensity Factor from Reissner’S Plate Theory,"

References

[1] E. Reissner, “The Effect of Transverse Shear Deformation on the Bending of Elastic Plates,” ASME Journal of Applied Me-chanics, Vol. 12, 1945, pp. A68-A77.

[2] R. J. Hartranft and G. C. Sih, “Effect of Plate Thickness on the Bending Stress Distribution around Through Cracks,” Journal of Mathematics and Physics, Vol. 47, 276-291,1968

[3] S. Viswanath, “On the Bending of Plates with through Cracks from Higher Order Plate Theories,” Ph. D Thesis, Indian Institute of Science, 1985.

[4] M. L. Williams, “The Bending Stress Distribution at the Base of a Stationary Crack,” ASME Journal of Applied Mechanics, Vol. 28, 1961, pp. 78-82.

[5] G. C. Sih, P. C. Paris and F. Erdogan, “Crack-Tip, Stress-Intensity Factors for Plate Extension and Plate Bending Problems,” ASME Journal of Applied Mechanics, Vol. 9, 1962, pp. 306-312.

[6] S. Krenk, “The Stress Distribution in an Infinite Anisotropic Plate with Collinear Cracks,” International Journal of Solids and Structures, Vol. 11, No. 4, 1975, pp. 449-460.doi:10.1016/0020-7683(75)90080-3

[7] R. S. Alwar and K. N. Ramachandran, “Influence of Crack Closure on the Stress Intensity Factor for Plates Subjected to Bend-ing—A 3-D Finite Element Analysis,” Engineering Fracture Mechanics, Vol. 17, No. 4, 1983, pp. 323-333. doi:10.1016/0013-7944(83)90083-8

[8] M. J. Viz, D. O. Potyondy, A. T. Zehnder, C. C. Rankin and E. Riks, “Computa-tion of Membrane and Bending Stress Intensity Factors for Thin, Cracked Plates,” International Journal of Fracture, Vol. 72, No. 1, 1995, pp. 21-38. doi:10.1007/BF00036927

[9] A. Zucchini, C.-Y. Hui and A. T. Zehnder, “Crack Tip Stress Fields for Thin Plates in Bending, Shear and Twisting: A Three Di-mensional Finite Element Study,” International Journal of Fracture, Vol. 104, No. 4, 2000, pp. 387-407.

[10] A. T. Zehnder and C.-Y. Hui, “Stress Intensity Factors for Plate Bending and Shearing Problems,” Journal of Applied Mechan-ics, Vol. 61, No. 3, 1994, pp. 719-722.doi:10.1115/1.2901522

[11] G. R. Irwin, “Analysis of Stresses and Strains near the End of a crack Trav-ersing a Plate,” Journal of applied Mechanics, 1957, Vol. 24, pp. 361-364