Stress Function of a Rotating Variable-Thickness Annular Disk Using Exact and Numerical Methods
ABSTRACT
In this paper, the exact analytical and numerical solutions for rotating variable-thickness annular disk are presented. The inner and outer edges of the rotating variable-thickness annular disk are considered to have free boundary conditions. Two different annular disks for the radially varying thickness are given. The numerical Runge-Kutta solution as well as the exact analytical solution is available for the first disk while the exact analytical solution is not available for the second annular disk. Both exact and numerical results for stress function, stresses, strains and radial displacement will be investigated for the first annular disk of variable thickness. The accuracy of the present numerical solution is discussed and its ability of use for the second rotating variable-thickness annular disk is investigated. Finally, the distributions of stress function, displacement, strains, and stresses will be presented. The appropriate comparisons and discussions are made at the same angular velocity.
In this paper, the exact analytical and numerical solutions for rotating variable-thickness annular disk are presented. The inner and outer edges of the rotating variable-thickness annular disk are considered to have free boundary conditions. Two different annular disks for the radially varying thickness are given. The numerical Runge-Kutta solution as well as the exact analytical solution is available for the first disk while the exact analytical solution is not available for the second annular disk. Both exact and numerical results for stress function, stresses, strains and radial displacement will be investigated for the first annular disk of variable thickness. The accuracy of the present numerical solution is discussed and its ability of use for the second rotating variable-thickness annular disk is investigated. Finally, the distributions of stress function, displacement, strains, and stresses will be presented. The appropriate comparisons and discussions are made at the same angular velocity.
Cite this paper
nullA. Zenkour and D. Mashat, "Stress Function of a Rotating Variable-Thickness Annular Disk Using Exact and Numerical Methods," Engineering, Vol. 3 No. 4, 2011, pp. 422-430. doi: 10.4236/eng.2011.34048.
nullA. Zenkour and D. Mashat, "Stress Function of a Rotating Variable-Thickness Annular Disk Using Exact and Numerical Methods," Engineering, Vol. 3 No. 4, 2011, pp. 422-430. doi: 10.4236/eng.2011.34048.
References
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[1] S. P. Timoshenko and J. N. Goodier, “Theory of Elastici-ty”, McGraw-Hill, New York, 1970. [2] S. C. Ugral and S.K. Fenster, “Advanced Strength and Ap- plied Elasticity”, Elsevier, New York, 1987. [3] U. Güven, “Elastic-Plastic Stresses in a Rotating Annular Disk of Variable Thickness and Variable Density,” International Journal of Mechanical Sciences, Vol. 34, 1992, pp. 133-138. [4] U. Güven, “On the Stress in the Elas-tic-Plastic Annular Disk of Variable Thickness under Ex-ternal Pressure,” International Journal of Solids and Structures, Vol. 30, 1993, pp. 651-658. [5] U. Güven, “Stress Distribution in a Linear Hardening Annular Disk of Variable Thickness Subjected to External Pressure,” International Journal of Mechanical Sciences, Vol. 40, 1998, pp. 589-601. doi:10.1016/S0020-7403(97)00081-7 [6] A. M. Zen-kour, “Analytical Solutions for Rotating Exponential-ly-Graded Annular Disks with Various Boundary Condi-tions,” International Journal of Structural Stability and Dynamics, Vol. 5, 2005, pp. 557-577. doi:10.1142/S0219455405001726 [7] A. M. Zenkour and M. N. M. Allam, “On the Rotating Fiber-Reinforced Viscoelastic Composite Solid and Annular Disks of Va-riable Thickness,” International Journal for Computa-tional Methods in Engineering Science & Mechanics, Vol. 7, 2006, pp. 21-31. doi:10.1080/155022891009639 [8] A. M. Zenkour, “Thermoelastic Solutions for Annular Disks with Arbi-trary Variable Thickness,” Structural Engineering and Mechanics, Vol. 24, 2006, pp. 515-528. [9] O. C. Zien-kiewicz, “The Finite Element Method in Engineering Science,” McGraw-Hill, London, 1971. [10] P. K. Ba-nerjee and R. Butterfield, “Boundary Element Methods in Engineering Science,” McGraw-Hill, New York, 1981. [11] M. L. James, G. M. Smith and J. C. Wolford, “Applied Numerical Methods for Digital Computation,” Harper & Row, New York, 1985 [12] L. H. You, Y. Y. Tang, J. J. Zhang and C.Y. Zheng, “Numerical Analysis of Elastic-Plastic Rotating Disks with Arbitrary Variable Thickness and Density,” International Journal of Solids and Structures, Vol. 37, 2000, pp. 7809-7820. doi:10.1016/S0020-7683(99)00308-X [13] C. F. Gerald and P. O. Wheatley, “Applied Numerical Analysis,” 6th Edition, Addison-Wesley, California, 2002. [14] M. H. Hojjati and A. Hassani, “Theoretical and Numerical Analysis of Rotating Discs of Non-Uniform Thickness and Density,” International Journal of Pressure Vessels and Piping, Vol. 85, 2008, pp. 694-700. doi:10.1016/j.ijpvp.2008.02.010 [15] M. H. Hojjati and S. Jafari, “Semi-Exact Solution of Elastic Non-Uniform Thickness and Density Rotating Disks by Homotopy Perturbation and Adomian’s Decomposition Methods. Part I: Elastic Solution,” International Journal of Pressure Vessels and Piping, Vol. 85, 2008, pp. 871-878. doi:10.1016/j.ijpvp.2008.06.001 [16] M. Abramowitz and I. Stegun, “Handbook of Mathematical Functions,” 5th Printing, US Government Printing Office, Washington, DC, 1966.