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Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads

Year 2018, , 73 - 92, 15.08.2018
https://doi.org/10.24107/ijeas.443239

Abstract

Using the true
temperature distribution along the radial coordinate, closed-form formulas are
offered for readers to study the thermo-mechanical behavior of variable
thickness disks having both convergent and divergent hyperbolic thickness
profiles made of conventional materials. Internal and external pressures,
centrifugal forces and thermal loads due to the differences in prescribed surface
temperatures are all considered with three boundary conditions: free-free
(circular annulus), fixed-free (a disk mounted on a rotating shaft at the inner
surface), and fixed-fixed (mounted on a rotating shaft at the inner surface and
cased at the outer surface) boundary conditions. A parametric study is also
conducted in almost real working environment in which the outer surface of the
disk has considerably higher temperature rather than the inner surface. The
thermomechanical linear elastic response of a hyperbolic mounted rotating disk
subjected to the external pressure induced by blades is originally handled by
those proposed formulas. 

References

  • [1] Güven, U., Altay O., Elastic–plastic solid disk with nonuniform heat source subjected to external pressure, Int J Mech Sci, 42(5), 831-842, 2000.
  • [2] Jahed, H., Shirazi, R., Loading and unloading behaviour of a thermoplastic disc, Int. J Pressure Vessels Piping, 78, 637–645, 2001.
  • [3] Gogulwar, V.S., Deshmukh, K.C., An inverse quasi-static thermal stresses in an annular disc, Proceeding of ICADS, Narosa Publishing House, New Delhi, 2002.
  • [4] Kulkarni, V.S., Deshmukh, K.C., Thermal stresses in a thick annular disc, Journal of Thermal Stresses, 31(4), 331-342, 2008. [5] Genç, M.S., Özşik, G., Yapıcı, H., A numerical study of the thermally induced stress distribution in a rotating hollow disc heated by a moving heat source acting on one of the side surfaces, P I Mech Eng C-J Mec, 223(8), 1877-1887, 2009.
  • [6] Nejad, M.Z., Afshin, A., Transient thermoelastic analysis of pressurized rotating disks subjected to arbitrary boundary and initial conditions, Chinese Journal of Engineering, Article ID 894902, 13 pages, 2014. http://dx.doi.org/10.1155/2014/894902
  • [7] Rattan, M., Kaushik, A., Chamoli, N., Steady state creep behavior of thermally graded isotropic rotating disc of composite taking into account the thermal residual stress, European Journal of Mechanics A/Solids, 60, 315-326, 2016.
  • [8] Kaur, J., Thakur, P., Singh, S.B., Steady thermal stresses in a thin rotating disc of finitesimal deformation with mechanical load, Journal of Solid Mechanics, 8(1), 204-211, 2016.
  • [9] Nayak, P., Saha, K., Elastic limit angular speed of solid and annular disks under thermomechanical loading, International Journal of Engineering, Science and Technology, 8(2), 30-45, 2016.
  • [10] Yıldırım, V., Heat-induced, pressure-induced and centrifugal-force-induced exact axisymmetric thermo-mechanical analyses in a thick-walled spherical vessel, an infinite cylindrical vessel, and a uniform disc made of an isotropic and homogeneous material, Int J Eng Appl Sci (IJEAS), 9(2), 66-87, 2017. Doi: 10.24107/ijeas.309786.
  • [11] Sayman, O. Thermal stress analysis in an aluminum metal-matrix orthotropic disc, Journal of Reinforced Plastics and Composites, 23, 1473-1479, 2004.
  • [12] Çallıoğlu, H., Topçu, M., Altan, G., Stress analysis of curvilinearly orthotropic rotating discs under mechanical and thermal loading, Journal of Reinforced Plastics and Composites, 24(8), 831-838, 2005.
  • [13] Çallıoğlu, H., Thermal stress analysis of curvilinear orthotropic rotating disks, Journal of Thermoplastic Composite Materials, 20, 357-369, 2007.
  • [14] Sen, F., Sayer, M., Elasto-plastic thermal stress analysis in a thermoplastic composite disc under uniform temperature using FEM, Mathematical and Computational Applications, 11(1), 31-39, 2006.
  • [15] Altan, G., Topçu, M., Bektaş, N.B., Altan, B.D., Elastic-plastic thermal stress analysis of an aluminum composite disc under parabolic thermal load distribution, J. Mech. Sci. Technol., 22(12), 2318–2327, 2008.
  • [16] Mohammadi, F., Hadadian, A., Singh, G.J., Analytical solution of pressurized rotating composite disk under thermal loading, Proceedings of the World Congress on Engineering II: WCE 2010, London, U.K. (4 pages) 2010.
  • [17] Mustafa, M.T., Zubair, S.M., Arif, A.F.M., Thermal analysis of orthotropic annular fins with contact resistance: A closed-form analytical solution, Appl. Therm. Eng., 31, 937–945, 2011.
  • [18] Kansal, G., Parvez, M., Thermal stress analysis of orthotropic graded rotating discs, International Journal of Modern Engineering Research (IJMER), 2(5), 3381-3885, 2012.
  • [19] Stampouloglou, I.H., Theotokoglou, E.E., The radially nonhomogeneous thermoelastic axisymmetric problem, Int J Mech Sci, 120, 311–321, 2017.
  • [20] Bayat, M., Saleem, M., Sahari, B.B., Hamouda, A.M.S., Mahdi, E., Thermo elastic analysis of a functionally graded rotating disk with small and large deflections, Thin-Wall Struct, 45, 677–691, 2007.
  • [21] Kordkheili, S.A.H., Naghdabadi, R., Thermoelastic analysis of a functionally graded rotating disk, Compos Struct, 79, 508–516, 2007.
  • [22] Arani, A.G., Mozdianfard, M.R., Maraghi, Z.K., Shajari, A.R., Thermo-piezo-magneto-mechanical stresses analysis of FGPM hollow rotating thin disk, Int. J. Mech. Mater. Des., 6, 341–349, 2010.
  • [23] Afsar, A.M., Go, J., Finite element analysis of thermoelastic feld in a rotating FGM circular disk, Appl. Math. Model, 34, 3309–3320, 2010.
  • [24] Peng, X.L., Li, X.F., Thermal stress in rotating functionally graded hollow circular discs, Compos Struct, 92(8), 1896–1904, 2010.
  • [25] Kursun, A., Topçu, M., Tetik, T., Stress analysis of functionally graded disc under thermal and mechanical loads, ICM11, Engineering Procedia, 10, 2949–2954, 2011.
  • [26] Gong, J.F., Ming, P.J., Xuan, L.K., Zhang, W.P., Thermoelastic analysis of three-dimensional functionally graded rotating disks based on finite volume method, P I Mech Eng C-J Mec, 228(4), 583–598, 2014.
  • [27] Gonczi, D., Ecsedi, I., Thermoelasic analysis of functionally graded hollow circular disk, Arch Mech Eng, LXII, 5–18, 2015.
  • [28] Yıldırım, V., Thermomechanical characteristics of a functionally graded mounted uniform disc with/without rigid casing, The Journal of Aerospace Technology and Management (JATM), 2018. (to be published)
  • [29] Chiba, R., Stochastic thermal stresses in an FGM annular disc of variable thickness with spatially random heat transfer coefficients, Meccanica, 44, 159–176, 2009.
  • [30] Bayat, M., Saleem, M., Sahari, B.B., Hamouda ,A.M.S, Mahdi E., Mechanical and thermal stresses in a functionally graded rotating disk with variable thickness due to radially symmetry loads, Int. J Pressure Vessels Piping, 86, 357–372, 2009.
  • [31] Bayat, M., Sahari, B.B., Saleem, M., Ali, A., Wong, S.V., Thermoelastic solution of a functionally graded variable thickness rotating disk with bending based on the first-order shear deformation theory, Thin-Wall Struct, 47, 568–582, 2009.
  • [32] Bayat, M., Sahari, B.B., Saleem, M., Hamouda, A.M.S., Reddy, J.N., Thermo elastic analysis of functionally graded rotating disks with temperature-dependent material properties: uniform and variable thickness, Journal of Mechanics and Materials in Design, 5(3), 263–279, 2009.
  • [33] Bayat, M., Mohazzab, A.H., Sahari, B.B., Saleem, M., Exact solution for functionally graded variable-thickness rotating disc with heat source, P I Mech Eng C-J Mec, 224(11), 2316–2331, 2010.
  • [34] Arani, G., Loghman, A., Shajari, A., Amir, S.A.R., Semi-analytical solution of magneto-thermo-elastic stresses for functionally graded variable thickness rotating disks, J Mech Sci Technol, 24(10), 2107–2117, 2010.
  • [35] Nie, G.J., Batra, R.C., Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness, Compos Struct, 92, 720–729, 2010.
  • [36] Damircheli, M., Azadi, M., Temperature and thickness effects on thermal and mechanical stresses of rotating FG-disks, J Mech Sci Technol, 25(3), 827-836, 2011.
  • [37] Hassani, A., Hojjati, M.H., Farrahi, G., Alashti, R.A., Semi-exact elastic solutions for thermo-mechanical analysis of functionally graded rotating disks, Compos Struct, 93(12), 3239–3251, 2011.
  • [38] Tutuncu, N., Temel, B., An efficient unified method for thermoelastic analysis of functionally graded rotating disks of variable thickness, Mechanics of Advanced Materials and Structures, 20(1), 38-46, 2013.
  • [39] Golmakani, M.E., Large deflection thermoelastic analysis of shear deformable functionally graded variable thickness rotating disk, Composites: Part B, 45, 1143-1155, 2013.
  • [40] Kurşun, A., Topçu, M., Thermal stress analysis of functionally graded disc with variable thickness due to linearly increasing temperature load, Arab J Sci Eng, 38, 3531–3549, 2013.
  • [41] Mahdavi, E., AkbariAlashti, R., Darabi, A.C., Alizadeh, M., Linear thermoplastic analysis of FGM rotating discs with variable thickness, Iranian Journal of Mechanical Engineering, 14(2), 73-87, 2013.
  • [42] Jabbari, M., Ghannad, M., Nejad, M.Z., Effect of thickness profile and FG function on rotating disks under thermal and mechanical loading, Journal of Mechanics, 32(1), 35-46, 2016.
  • [43] Vivio, F., Vullo, V., Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius, Int J Solids Struct, 44, 7767–7784, 2007.
  • [44] Vullo, V., Vivio, F., Elastic stress analysis of non-linear variable thickness rotating disks subjected to thermal load and having variable density along the radius, Int J Solids Struct, 45, 5337–5355, 2008.
  • [45] Garg, M., Salaria, B.S., Gupta, V.K., Effect of thermal gradient on steady state creep in a rotating disc of variable thickness, Procedia Eng., 55, 542e547. 2013.
  • [46] Çetin, E., Kurşun, A., Aksoy, Ş., Çetin, M.T., Elastic stress analysis of annular bi-material discs with variable thickness under mechanical and thermomechanical loads, World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 8(2), 288-292, 2014.
  • [47] Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.
  • [48] Yıldırım, V., Effects of inhomogeneity and thickness parameters on the elastic response of a pressurized hyperbolic annulus/disc made of functionally graded material, Int J Eng Appl Sci, 9(3), 2017. Doi: 10.24107/ijeas.329433
  • [49] Young, W.C., Budynas, R.G., Roark’s Formulas for Stress and Strain; McGrawHill, Seventh Edition, New York. 2002.
  • [50] Yıldırım, V., Analytic solutions to power-law graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS), 8(1), 38-52, 2016.
  • [51] Yıldırım V., A parametric study on the centrifugal force-induced stress and displacements in power-law graded hyperbolic discs, Latin American Journal of Solids and Structures, LAJSS, 15(4) e34, 1-16, 2018. Doi: 10.1590/1679-78254229
Year 2018, , 73 - 92, 15.08.2018
https://doi.org/10.24107/ijeas.443239

Abstract

References

  • [1] Güven, U., Altay O., Elastic–plastic solid disk with nonuniform heat source subjected to external pressure, Int J Mech Sci, 42(5), 831-842, 2000.
  • [2] Jahed, H., Shirazi, R., Loading and unloading behaviour of a thermoplastic disc, Int. J Pressure Vessels Piping, 78, 637–645, 2001.
  • [3] Gogulwar, V.S., Deshmukh, K.C., An inverse quasi-static thermal stresses in an annular disc, Proceeding of ICADS, Narosa Publishing House, New Delhi, 2002.
  • [4] Kulkarni, V.S., Deshmukh, K.C., Thermal stresses in a thick annular disc, Journal of Thermal Stresses, 31(4), 331-342, 2008. [5] Genç, M.S., Özşik, G., Yapıcı, H., A numerical study of the thermally induced stress distribution in a rotating hollow disc heated by a moving heat source acting on one of the side surfaces, P I Mech Eng C-J Mec, 223(8), 1877-1887, 2009.
  • [6] Nejad, M.Z., Afshin, A., Transient thermoelastic analysis of pressurized rotating disks subjected to arbitrary boundary and initial conditions, Chinese Journal of Engineering, Article ID 894902, 13 pages, 2014. http://dx.doi.org/10.1155/2014/894902
  • [7] Rattan, M., Kaushik, A., Chamoli, N., Steady state creep behavior of thermally graded isotropic rotating disc of composite taking into account the thermal residual stress, European Journal of Mechanics A/Solids, 60, 315-326, 2016.
  • [8] Kaur, J., Thakur, P., Singh, S.B., Steady thermal stresses in a thin rotating disc of finitesimal deformation with mechanical load, Journal of Solid Mechanics, 8(1), 204-211, 2016.
  • [9] Nayak, P., Saha, K., Elastic limit angular speed of solid and annular disks under thermomechanical loading, International Journal of Engineering, Science and Technology, 8(2), 30-45, 2016.
  • [10] Yıldırım, V., Heat-induced, pressure-induced and centrifugal-force-induced exact axisymmetric thermo-mechanical analyses in a thick-walled spherical vessel, an infinite cylindrical vessel, and a uniform disc made of an isotropic and homogeneous material, Int J Eng Appl Sci (IJEAS), 9(2), 66-87, 2017. Doi: 10.24107/ijeas.309786.
  • [11] Sayman, O. Thermal stress analysis in an aluminum metal-matrix orthotropic disc, Journal of Reinforced Plastics and Composites, 23, 1473-1479, 2004.
  • [12] Çallıoğlu, H., Topçu, M., Altan, G., Stress analysis of curvilinearly orthotropic rotating discs under mechanical and thermal loading, Journal of Reinforced Plastics and Composites, 24(8), 831-838, 2005.
  • [13] Çallıoğlu, H., Thermal stress analysis of curvilinear orthotropic rotating disks, Journal of Thermoplastic Composite Materials, 20, 357-369, 2007.
  • [14] Sen, F., Sayer, M., Elasto-plastic thermal stress analysis in a thermoplastic composite disc under uniform temperature using FEM, Mathematical and Computational Applications, 11(1), 31-39, 2006.
  • [15] Altan, G., Topçu, M., Bektaş, N.B., Altan, B.D., Elastic-plastic thermal stress analysis of an aluminum composite disc under parabolic thermal load distribution, J. Mech. Sci. Technol., 22(12), 2318–2327, 2008.
  • [16] Mohammadi, F., Hadadian, A., Singh, G.J., Analytical solution of pressurized rotating composite disk under thermal loading, Proceedings of the World Congress on Engineering II: WCE 2010, London, U.K. (4 pages) 2010.
  • [17] Mustafa, M.T., Zubair, S.M., Arif, A.F.M., Thermal analysis of orthotropic annular fins with contact resistance: A closed-form analytical solution, Appl. Therm. Eng., 31, 937–945, 2011.
  • [18] Kansal, G., Parvez, M., Thermal stress analysis of orthotropic graded rotating discs, International Journal of Modern Engineering Research (IJMER), 2(5), 3381-3885, 2012.
  • [19] Stampouloglou, I.H., Theotokoglou, E.E., The radially nonhomogeneous thermoelastic axisymmetric problem, Int J Mech Sci, 120, 311–321, 2017.
  • [20] Bayat, M., Saleem, M., Sahari, B.B., Hamouda, A.M.S., Mahdi, E., Thermo elastic analysis of a functionally graded rotating disk with small and large deflections, Thin-Wall Struct, 45, 677–691, 2007.
  • [21] Kordkheili, S.A.H., Naghdabadi, R., Thermoelastic analysis of a functionally graded rotating disk, Compos Struct, 79, 508–516, 2007.
  • [22] Arani, A.G., Mozdianfard, M.R., Maraghi, Z.K., Shajari, A.R., Thermo-piezo-magneto-mechanical stresses analysis of FGPM hollow rotating thin disk, Int. J. Mech. Mater. Des., 6, 341–349, 2010.
  • [23] Afsar, A.M., Go, J., Finite element analysis of thermoelastic feld in a rotating FGM circular disk, Appl. Math. Model, 34, 3309–3320, 2010.
  • [24] Peng, X.L., Li, X.F., Thermal stress in rotating functionally graded hollow circular discs, Compos Struct, 92(8), 1896–1904, 2010.
  • [25] Kursun, A., Topçu, M., Tetik, T., Stress analysis of functionally graded disc under thermal and mechanical loads, ICM11, Engineering Procedia, 10, 2949–2954, 2011.
  • [26] Gong, J.F., Ming, P.J., Xuan, L.K., Zhang, W.P., Thermoelastic analysis of three-dimensional functionally graded rotating disks based on finite volume method, P I Mech Eng C-J Mec, 228(4), 583–598, 2014.
  • [27] Gonczi, D., Ecsedi, I., Thermoelasic analysis of functionally graded hollow circular disk, Arch Mech Eng, LXII, 5–18, 2015.
  • [28] Yıldırım, V., Thermomechanical characteristics of a functionally graded mounted uniform disc with/without rigid casing, The Journal of Aerospace Technology and Management (JATM), 2018. (to be published)
  • [29] Chiba, R., Stochastic thermal stresses in an FGM annular disc of variable thickness with spatially random heat transfer coefficients, Meccanica, 44, 159–176, 2009.
  • [30] Bayat, M., Saleem, M., Sahari, B.B., Hamouda ,A.M.S, Mahdi E., Mechanical and thermal stresses in a functionally graded rotating disk with variable thickness due to radially symmetry loads, Int. J Pressure Vessels Piping, 86, 357–372, 2009.
  • [31] Bayat, M., Sahari, B.B., Saleem, M., Ali, A., Wong, S.V., Thermoelastic solution of a functionally graded variable thickness rotating disk with bending based on the first-order shear deformation theory, Thin-Wall Struct, 47, 568–582, 2009.
  • [32] Bayat, M., Sahari, B.B., Saleem, M., Hamouda, A.M.S., Reddy, J.N., Thermo elastic analysis of functionally graded rotating disks with temperature-dependent material properties: uniform and variable thickness, Journal of Mechanics and Materials in Design, 5(3), 263–279, 2009.
  • [33] Bayat, M., Mohazzab, A.H., Sahari, B.B., Saleem, M., Exact solution for functionally graded variable-thickness rotating disc with heat source, P I Mech Eng C-J Mec, 224(11), 2316–2331, 2010.
  • [34] Arani, G., Loghman, A., Shajari, A., Amir, S.A.R., Semi-analytical solution of magneto-thermo-elastic stresses for functionally graded variable thickness rotating disks, J Mech Sci Technol, 24(10), 2107–2117, 2010.
  • [35] Nie, G.J., Batra, R.C., Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness, Compos Struct, 92, 720–729, 2010.
  • [36] Damircheli, M., Azadi, M., Temperature and thickness effects on thermal and mechanical stresses of rotating FG-disks, J Mech Sci Technol, 25(3), 827-836, 2011.
  • [37] Hassani, A., Hojjati, M.H., Farrahi, G., Alashti, R.A., Semi-exact elastic solutions for thermo-mechanical analysis of functionally graded rotating disks, Compos Struct, 93(12), 3239–3251, 2011.
  • [38] Tutuncu, N., Temel, B., An efficient unified method for thermoelastic analysis of functionally graded rotating disks of variable thickness, Mechanics of Advanced Materials and Structures, 20(1), 38-46, 2013.
  • [39] Golmakani, M.E., Large deflection thermoelastic analysis of shear deformable functionally graded variable thickness rotating disk, Composites: Part B, 45, 1143-1155, 2013.
  • [40] Kurşun, A., Topçu, M., Thermal stress analysis of functionally graded disc with variable thickness due to linearly increasing temperature load, Arab J Sci Eng, 38, 3531–3549, 2013.
  • [41] Mahdavi, E., AkbariAlashti, R., Darabi, A.C., Alizadeh, M., Linear thermoplastic analysis of FGM rotating discs with variable thickness, Iranian Journal of Mechanical Engineering, 14(2), 73-87, 2013.
  • [42] Jabbari, M., Ghannad, M., Nejad, M.Z., Effect of thickness profile and FG function on rotating disks under thermal and mechanical loading, Journal of Mechanics, 32(1), 35-46, 2016.
  • [43] Vivio, F., Vullo, V., Elastic stress analysis of rotating converging conical disks subjected to thermal load and having variable density along the radius, Int J Solids Struct, 44, 7767–7784, 2007.
  • [44] Vullo, V., Vivio, F., Elastic stress analysis of non-linear variable thickness rotating disks subjected to thermal load and having variable density along the radius, Int J Solids Struct, 45, 5337–5355, 2008.
  • [45] Garg, M., Salaria, B.S., Gupta, V.K., Effect of thermal gradient on steady state creep in a rotating disc of variable thickness, Procedia Eng., 55, 542e547. 2013.
  • [46] Çetin, E., Kurşun, A., Aksoy, Ş., Çetin, M.T., Elastic stress analysis of annular bi-material discs with variable thickness under mechanical and thermomechanical loads, World Academy of Science, Engineering and Technology, International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 8(2), 288-292, 2014.
  • [47] Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.
  • [48] Yıldırım, V., Effects of inhomogeneity and thickness parameters on the elastic response of a pressurized hyperbolic annulus/disc made of functionally graded material, Int J Eng Appl Sci, 9(3), 2017. Doi: 10.24107/ijeas.329433
  • [49] Young, W.C., Budynas, R.G., Roark’s Formulas for Stress and Strain; McGrawHill, Seventh Edition, New York. 2002.
  • [50] Yıldırım, V., Analytic solutions to power-law graded hyperbolic rotating discs subjected to different boundary conditions, International Journal of Engineering & Applied Sciences (IJEAS), 8(1), 38-52, 2016.
  • [51] Yıldırım V., A parametric study on the centrifugal force-induced stress and displacements in power-law graded hyperbolic discs, Latin American Journal of Solids and Structures, LAJSS, 15(4) e34, 1-16, 2018. Doi: 10.1590/1679-78254229
There are 50 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Vebil Yıldırım

Publication Date August 15, 2018
Acceptance Date July 24, 2018
Published in Issue Year 2018

Cite

APA Yıldırım, V. (2018). Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads. International Journal of Engineering and Applied Sciences, 10(2), 73-92. https://doi.org/10.24107/ijeas.443239
AMA Yıldırım V. Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads. IJEAS. August 2018;10(2):73-92. doi:10.24107/ijeas.443239
Chicago Yıldırım, Vebil. “Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads”. International Journal of Engineering and Applied Sciences 10, no. 2 (August 2018): 73-92. https://doi.org/10.24107/ijeas.443239.
EndNote Yıldırım V (August 1, 2018) Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads. International Journal of Engineering and Applied Sciences 10 2 73–92.
IEEE V. Yıldırım, “Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads”, IJEAS, vol. 10, no. 2, pp. 73–92, 2018, doi: 10.24107/ijeas.443239.
ISNAD Yıldırım, Vebil. “Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads”. International Journal of Engineering and Applied Sciences 10/2 (August 2018), 73-92. https://doi.org/10.24107/ijeas.443239.
JAMA Yıldırım V. Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads. IJEAS. 2018;10:73–92.
MLA Yıldırım, Vebil. “Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads”. International Journal of Engineering and Applied Sciences, vol. 10, no. 2, 2018, pp. 73-92, doi:10.24107/ijeas.443239.
Vancouver Yıldırım V. Closed-Form Formulas for Hyperbolically Tapered Rotating Disks Made of Traditional Materials under Combined Thermal and Mechanical Loads. IJEAS. 2018;10(2):73-92.

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