The Complementary Functions Method (CFM) Solution to the Elastic Analysis of Polar Orthotropic Rotating Discs

Document Type: Research Paper

Author

Department of Mechanical Engineering, Çukurova University, Adana, 01330, Turkey

Abstract

This study primarily deals with introducing an efficient numerical technique called the Complementary Functions Method (CFM) for the solutions of the initial value problem for the linear elastic analysis of anisotropic rotating uniform discs. To bring the performance of the method to light, first, closed form formulas are derived for such discs. The governing equation of the problem at stake is solved analytically with the help of the Euler-Cauchy technique under three types of boundary conditions namely free-free, fixed-free, and fixed-guided constraints. Secondly, the CFM is applied to the same problem. It was found that both numerical and analytical results coincide with each other up to a desired numerical accuracy. Third, after verifying the results with the literature, a parametric study with CFM on the elastic behavior of discs made up of five different materials which physically exist is performed. And finally, by using hypothetically chosen anisotropy degrees from 0.3 through 5, the effects of the anisotropy on the elastic response of such structures are investigated analytically. Useful graphs are provided for readers.

Keywords

Main Subjects

[1] Bidgoli, A.M.M., Daneshmehr, A.R., Kolahchi, R., Analytical Bending Solution of Fully Clamped Orthotropic Rectangular Plates Resting on Elastic Foundations by The Finite Integral Transform Method, Journal of Applied and Computational Mechanics, 1(2), 2015, 52-58.

[2] Akano, T.T., Fakinlede, O.A., Olayiwola, P.S., Deformation Characteristics of Composite Structures, Journal of Applied and Computational Mechanics, 2(3), 2016, 174-191.

[3] Katsikadelis, J.T., Tsiatas G.C., Saint-Venant Torsion of Non-Homogeneous Anisotropic Bars, Journal of Applied and Computational Mechanics, 2(1), 2016, 42-53.

[4] Tang, S., Elastic Stresses in Rotating Anisotropic Discs, International Journal of Mechanical Sciences, 11, 1969, 509–517.

[5] Murthy, D., Sherbourne, A., Elastic Stresses in Anisotropic Discs of Variable Thickness, International Journal of Mechanical Sciences, 12, 1970, 627-640.

[6] Reddy, T.Y., Srinath, H., Elastic Stresses in a Rotating Anisotropic Annular Disc of Variable Thickness and Variable Density, International Journal of Mechanical Sciences, 16(2), 1974, 85-89.

[7] Chang, C.I., A Closed-Form Solution for an Orthotropic Rotating Disc, Journal of Applied Mechanics, 41(4), 1974, 1122–1123.

[8] Chang, C.I., The Anisotropic Rotating Discs, International Journal of Mechanical Sciences, 17(6), 1975, 397-402.

[9] Bert, C.W., Centrifugal Stresses in Arbitrarily Laminated, Rectangular-Anisotropic Circular Discs, Journal of Strain Analysis for Engineering Design, 10, 1975, 84-92.

[10] Gurushankar, G.V., Thermal Stresses in a Rotating Nonhomogeneous, Anisotropic Disc of Varying Thickness and Density, Journal of Strain Analysis for Engineering Design, 10, 1975, 137-142.

[11] Christensen, R.M., Wu, E.M., Optimal Design of Anisotropic (Fiber-Reinforced) Flywheels, Journal of Composite Materials, 11, 1977, 395-404.

[12] Belingardi, G., Genta, G., Gola, M., A Study of the Stress Distribution in Rotating, Orthotropic Discs, Composites, 10(2), 1979, 77-80.

[13] Genta, G., Gola, M., The Stress Distribution in Orthotropic Rotating Discs, Journal of Applied Mechanics, 48, 1981, 559-562.

[14] Lekhnitskii, S.G., Theory of Elasticity of an Anisotropic Body, Mir Publishers, Moscow, 1981.

[15] Elishakoff, I., Buckling of Polar Orthotropic Circular Plates on Elastic Foundation by Computerized Symbolic Algebra, Computer Methods in Applied Mechanics and Engineering, 68(2), 1988, 229-247.

[16] Oilin, H., Oixuan, Z., Ping, W., Guicang, H., A New Method for Calculating Burst Speed of Aeroengine Disks, ASME-91-GT-121, 1991, 1-4.

[17] Tutuncu, N., Effect of Anisotropy on Stresses in Rotating Discs, International Journal of Mechanical Sciences, 37, 2000, 873–881.

[18] Arnoldi, S.M., Saleeb, A.F., Al-Zoubi, R., Deformation and Life Analysis of Composite Flywheel Disk and Multi-Disk Systems, NASAffM-2001-210578, 2001, 1-56.

[19] Zhou, F., Ogawa, A, Elastic Solutions for a Solid Rotating Disc with Cubic Anisotropy, Journal of Applied Mechanics, 69, 2002, 81-83.

[20] Callioglu, H., Stress Analysis of an Orthotropic Rotating Disc under Thermal Loading, Journal of Reinforced Plastics and Composites, 23(17), 2004, 1857–1869.

[21] Callioglu, H., Thermal Stress Analysis of Curvilinearly Orthotropic Rotating Discs, Journal of Thermoplastic Composite Materials, 20, 2007, 357-369.

[22] Callioglu, H., Topcu, M., Altan, G., Stress Analysis of Curvilinearly Orthotropic Rotating Discs under Mechanical and Thermal Loading, Journal of Reinforced Plastics and Composites, 24, 2005, 831-838.

[23] Callioglu, H., Topcu, M., Tarakçılar, A.R., Elastic-Plastic Stress Analysis of an Orthotropic Rotating Disc, International Journal of Mechanical Sciences, 48, 2006, 985-990.

[24] Sayer, M., Topcu, M., Bektas, N.B., Tarakcilar, A.R., Thermoelastic Stress Analysis in a Thermoplastic Composite Disc, Science and Engineering of Composite Materials, 12(4), 2005, 251–260.

[25] Tahani, M., Nosier, A., Zebarjad, S.M., Deformation and Stress Analysis of Circumferentially Fiber-Reinforced Composite Disks, International Journal of Solids and Structures, 42(9–10), 2005, 2741–2754.

[26] Koo, K.N., Vibration Analysis and Critical Speeds of Polar Orthotropic Annular Discs in Rotation, Composite Structures, 76, 2006, 67-72.

[27] Koo, K.N., Mechanical Vibration and Critical Speeds of Rotating Composite Laminate Discs, Microsystem Technologies, 14, 2008, 799-807.

[28] Alexandrova, N., Vila Real, P.M.M., Deformation and Stress Analysis of an Anisotropic Rotating Annular Disk, International Journal for Computational Methods in Engineering Science and Mechanics, 9(1), 2008, 43–50.

[29] Nie, G.J., Zhong, Z., Batra, R.C., Material Tailoring for Orthotropic Elastic Rotating Discs, Composites Science and Technology, 71, 2011, 406–414.

[30] Durodola, J., Attia, O., Deformation and Stresses in Functionally Graded Rotating Discs, Composites Science and Technology, 60, 2000, 987-995.

[31] Chen, J., Ding, H., Chen, W., Three-Dimensional Analytical Solution for a Rotating Disc of Functionally Graded Materials with Transverse Isotropy, Archive of Applied Mechanics, 77, 2007, 241-251.

[32] Wang, X., Sudak, L.J., Three-Dimensional Analysis of Multi-Layered Functionally Graded Anisotropic Cylindrical Panel under Thermomechanical Loading, Mechanics of Materials, 40(4), 2008, 235–254.

[33] Kansal, G., Parvez, M., Thermal Stress Analysis of Orthotropic Graded Rotating Discs, International Journal of Modern Engineering Research, 2(5), 2012, 3881-3885.

[34] Lubarda, V.A., On Pressurized Curvilinearly Orthotropic Circular Disc, Cylinder and Sphere Made of Radially Nonuniform Material, Journal of Elasticity, 109, 2012, 103-133.

[35] Peng, X.L., Li, X.F., Elastic Analysis of Rotating Functionally Graded Polar Orthotropic Discs, International Journal of Mechanical Sciences, 60, 2012, 84-91.

[36] Boga, C., Analytical and Numerical Axisymmetric Elastic Stress Analyses of Stationary/Rotating Discs Made of Isotropic/Orthotropic Functionally Graded Materials by The Transfer Matrix Method, Ph.D. Thesis, Department of Mechanical Engineering, Çukurova University: No 1698, 2016.

[37] Kacar, I., Yıldırım, V., Effect of the Anisotropy Ratios on the Exact Elastic Behavior of Functionally Power-Graded Polar Orthotropic Rotating Uniform Discs Under Various Boundary Conditions, Digital Proceeding of ICOCEE – Cappadocia, Nevsehir, Turkey, 1743-1752, May 8-10, 2017.

[38] Zheng, Y., Bahaloo, H., Mousanezhad, D., Vaziri, A., Nayeb-Hashemi, H., Displacement and Stress Fields in a Functionally Graded Fiber-Reinforced Rotating Disk with Nonuniform Thickness and Variable Angular Velocity, Journal of Engineering Materials and Technology, 39, 2017, 031010-1-9.

[39] Aktas, Z., Numerical Solutions of Two-Point Boundary Value Problems, Ankara, Turkey, METU, Department of Computer Engineering, 1972.

[40] Roberts, S., Shipman, J., Fundamental Matrix and Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, 28(1), 1979, 77-88.

[41] Haktanır, V., Kıral, E., Direct Application of Complementary Functions Method to Axisymmetrical Shells and Cylindrical Vaults (Barrels), Journal of Isparta Engineering Faculty of Akdeniz University, 6, 1991, 220-239.

[42] Haktanır, V., A New Method for the Element Stiffness Matrix of Arbitrary Planar Bars, Composite Structures, 52(4), 1994, 679–691.

[43] Haktanır, V., The Complementary Functions Method for the Element Stiffness Matrix of Arbitrary Spatial Bars of Helicoidal Axes, International Journal for Numerical Methods in Engineering, 38(6), 1995, 1031–1056.

[44] Yıldırım, V., Free Vibration Analysis of Non-Cylindrical Coil Springs by Combined Use of the Transfer Matrix and the Complementary Functions Methods, Communications in Numerical Methods in Engineering, 13(6), 1997, 487–94.

[45] Kacar, İ., Yıldırım, V., Free Vibration/Buckling Analyses of Non-Cylindrical Initially Compressed Helical Composite Springs, Mechanics Based Design of Structures and Machines, 44 (4), 2016, 340-353.

[46] Yıldırım, V., Kacar, I., Introducing a Computer Package Program for Elastic Analysis of Functionally Graded Rotating Thick-Walled Annular Structures, Digital Proceeding of ICOCEE – CAPPADOCIA, Nevşehir, Turkey, 1733-1742, May 8-10, 2017.