Analytical and Numerical Solutions for a Rotating Annular Disk of Variable Thickness
ABSTRACT
In this paper, the analytical and numerical solutions for rotating variable-thickness solid disk and numerical solution for rotating variable-thickness annular disk are presented. The outer edge of the solid disk and the inner and outer edges of the annular disk are considered to have clamped boundary conditions. Two different cases for the radially varying thickness of the solid and annular disks are given. The numerical solution as well as the analytical solution is available for the first case of the solid disk while the analytical solution is not available for the second case of the annular disk. Both analytical and numerical results for displacement and stresses will be investigated for the first case of radially varying thickness. The accuracy of the present numerical solution is discussed and its ability of use for the second case of radially varying thickness is investigated. Finally, the distributions of displacement and stresses will be presented and the appropriate comparisons and discussions are made at the same angular velocity.
In this paper, the analytical and numerical solutions for rotating variable-thickness solid disk and numerical solution for rotating variable-thickness annular disk are presented. The outer edge of the solid disk and the inner and outer edges of the annular disk are considered to have clamped boundary conditions. Two different cases for the radially varying thickness of the solid and annular disks are given. The numerical solution as well as the analytical solution is available for the first case of the solid disk while the analytical solution is not available for the second case of the annular disk. Both analytical and numerical results for displacement and stresses will be investigated for the first case of radially varying thickness. The accuracy of the present numerical solution is discussed and its ability of use for the second case of radially varying thickness is investigated. Finally, the distributions of displacement and stresses will be presented and the appropriate comparisons and discussions are made at the same angular velocity.
Cite this paper
nullA. Zenkour and D. Mashat, "Analytical and Numerical Solutions for a Rotating Annular Disk of Variable Thickness," Applied Mathematics, Vol. 1 No. 5, 2010, pp. 430-437. doi: 10.4236/am.2010.15057.
nullA. Zenkour and D. Mashat, "Analytical and Numerical Solutions for a Rotating Annular Disk of Variable Thickness," Applied Mathematics, Vol. 1 No. 5, 2010, pp. 430-437. doi: 10.4236/am.2010.15057.
References
[1] S. P. Timoshenko and J. N. Goodier, “Theory of Elasticity,” McGraw-Hill, New York, 1970. [2] S. C. Ugral and S. K. Fenster, “Advanced Strength and Applied Elasticity,” Elsevier, New York, 1987. [3] U. Gamer, “Tresca’s Yield Condition and the Rotating Disk,” ASME Journal of Applied Mechanics, Vol. 50, No. 2, 1983, pp. 676-678. [4] U. Gamer, “Elastic-Plastic Deformation of the Rotating Solid Disk,” Ingenieur-Archiv, Vol. 45, No. 4, 1984, pp. 345-354. [5] A. M. Zenkour, “Analytical Solutions for Rotating Exponentially-Graded Annular Disks with Various Boundary Conditions,” International Journal of Structural Stability and Dynamics, Vol. 5, No. 4, 2005, pp. 557-577. [6] U. Güven, “Elastic-Plastic Stresses in a Rotating Annular Disk of Variable Thickness and Variable Density,” International Journal of Mechanical Sciences, Vol. 34, No. 2, 1992, pp. 133-138. [7] U. Güven, “On the Stress in the Elastic-Plastic Annular Disk of Variable Thickness under External Pressure,” International Journal of Solids and Structures, Vol. 30, No. 5, 1993, pp. 651-658. [8] U. Güven, “Stress Distribution in a Linear Hardening Annular Disk of Variable Thickness Subjected to External Pressure,” International Journal of Mechanical Sciences, Vol. 40, No. 6, 1998, pp. 589-601. [9] U. Güven, “Elastic-Plastic Stresses Distribution in a Rotating Hyperbolic Disk with Rigid Inclusion,” International Journal of Mechanical Sciences, Vol. 40, No. 1, 1998, pp. 97-109. [10] A. N. Eraslan, “Inelastic Deformation of Rotating Variable Thickness Solid Disks by Tresca and Von Mises Criteria,” International Journal for Computational Me- thods in Engineering Science, Vol. 3, 2000, pp. 89-101. [11] A. N. Eraslan and Y. Orcan, “On the Rotating Elastic-Plastic Solid Disks of Variable Thickness Having Concave Profiles,” International Journal of Mechanical Sciences, Vol. 44, No. 7, 2002, pp. 1445-1466. [12] A. N. Eraslan, “Stress Distributions in Elastic-Plastic Rotating Disks with Elliptical Thickness Profiles Using Tresca and von Mises Criteria,” Zurich Alternative Asset Management, Vol. 85, 2005, pp. 252-266. [13] A. M. Zenkour and M. N. M. Allam, “On the Rotating Fiber-Reinforced Viscoelastic Composite Solid and Annular Disks of Variable Thickness,” International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 7, No. 1, 2006, pp. 21-31. [14] A. M. Zenkour, “Thermoelastic Solutions for Annular Disks with Arbitrary Variable Thickness,” Structural Engineering and Mechanics, Vol. 24, No. 5, 2006, pp. 515-528. [15] O. C. Zienkiewicz, “The Finite Element Method in Engineering Science,” McGraw-Hill, London, 1971. [16] P. K. Banerjee and R. Butterfield, “Boundary Element Methods in Engineering Science,” McGraw-Hill, New York, 1981. [17] 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, No. 52, 2000, pp. 7809-7820.
[1] S. P. Timoshenko and J. N. Goodier, “Theory of Elasticity,” McGraw-Hill, New York, 1970. [2] S. C. Ugral and S. K. Fenster, “Advanced Strength and Applied Elasticity,” Elsevier, New York, 1987. [3] U. Gamer, “Tresca’s Yield Condition and the Rotating Disk,” ASME Journal of Applied Mechanics, Vol. 50, No. 2, 1983, pp. 676-678. [4] U. Gamer, “Elastic-Plastic Deformation of the Rotating Solid Disk,” Ingenieur-Archiv, Vol. 45, No. 4, 1984, pp. 345-354. [5] A. M. Zenkour, “Analytical Solutions for Rotating Exponentially-Graded Annular Disks with Various Boundary Conditions,” International Journal of Structural Stability and Dynamics, Vol. 5, No. 4, 2005, pp. 557-577. [6] U. Güven, “Elastic-Plastic Stresses in a Rotating Annular Disk of Variable Thickness and Variable Density,” International Journal of Mechanical Sciences, Vol. 34, No. 2, 1992, pp. 133-138. [7] U. Güven, “On the Stress in the Elastic-Plastic Annular Disk of Variable Thickness under External Pressure,” International Journal of Solids and Structures, Vol. 30, No. 5, 1993, pp. 651-658. [8] U. Güven, “Stress Distribution in a Linear Hardening Annular Disk of Variable Thickness Subjected to External Pressure,” International Journal of Mechanical Sciences, Vol. 40, No. 6, 1998, pp. 589-601. [9] U. Güven, “Elastic-Plastic Stresses Distribution in a Rotating Hyperbolic Disk with Rigid Inclusion,” International Journal of Mechanical Sciences, Vol. 40, No. 1, 1998, pp. 97-109. [10] A. N. Eraslan, “Inelastic Deformation of Rotating Variable Thickness Solid Disks by Tresca and Von Mises Criteria,” International Journal for Computational Me- thods in Engineering Science, Vol. 3, 2000, pp. 89-101. [11] A. N. Eraslan and Y. Orcan, “On the Rotating Elastic-Plastic Solid Disks of Variable Thickness Having Concave Profiles,” International Journal of Mechanical Sciences, Vol. 44, No. 7, 2002, pp. 1445-1466. [12] A. N. Eraslan, “Stress Distributions in Elastic-Plastic Rotating Disks with Elliptical Thickness Profiles Using Tresca and von Mises Criteria,” Zurich Alternative Asset Management, Vol. 85, 2005, pp. 252-266. [13] A. M. Zenkour and M. N. M. Allam, “On the Rotating Fiber-Reinforced Viscoelastic Composite Solid and Annular Disks of Variable Thickness,” International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 7, No. 1, 2006, pp. 21-31. [14] A. M. Zenkour, “Thermoelastic Solutions for Annular Disks with Arbitrary Variable Thickness,” Structural Engineering and Mechanics, Vol. 24, No. 5, 2006, pp. 515-528. [15] O. C. Zienkiewicz, “The Finite Element Method in Engineering Science,” McGraw-Hill, London, 1971. [16] P. K. Banerjee and R. Butterfield, “Boundary Element Methods in Engineering Science,” McGraw-Hill, New York, 1981. [17] 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, No. 52, 2000, pp. 7809-7820.