Closed-Form Solution of Axisymmetric Slender Elastic Toroidal Shells
Publication: Journal of Engineering Mechanics
Volume 136, Issue 10
Abstract
Toroidal shells are widely used in structural engineering. The governing equations of toroidal shells are very complicated because of its variable coefficients with singularity. To find their analytical solution, traditionally, the complex form governing equations were proposed and some useful solutions were obtained. Unfortunately, no any closed-form solution has even been obtained for either general or slender toroidal shells. This paper focus on a special case of toroidal shells, i.e., slender symmetrical toroidal shells. For the first time, the closed-form solution of this kind of shell has been successfully obtained from displacement form governing equations. The closed-form solution is demonstrated for the example of thermal compensation devices. The correction of well-known Dahl formula for slender toroidal shell has been proposed based on the solution obtained in this paper.
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References
Baker, W. R., Bello, S. D., Marcuzzi, D., Sonato, P., and Zaccaria, P. (2002). “Design of a new toroidal shell and support for RFX.” Fusion Eng. Des., 63–64, 461–466.
Balderes, T., and Armenakas, A. E. (1973). “Free vibrations of ring-stiffened toroidal shells.” AIAA J., 11, 1637–1644.
Bessarabov, Y. D., and Rudis, M. A. (1966). “On the symmetrical deformation of an orthotropic toroidal shell.” Proc., 4th All-Union Conf. on Shells and Plates, S. M. Dugar’yan, ed., Israel Program for Scientific Translations, Jerusalem, 207–215.
Blachut, J. (2003). “Collapse tests on externally pressurised toroids.” ASME J. Pressure Vessel Technol., 125, 91–96.
Blachut, J. (2004). “Buckling and first ply failure of a composite toroidal pressure hull.” Comput. Struct., 82, 1981–1992.
Blachut, J. (2005). “Plastic loads for internally pressurised toroidal shells.” ASME J. Pressure Vessel Technol., 127, 151–156.
Blachut, J., and Jaiswal, O. R. (2000). “On the buckling of toroidal shells under external pressure.” Comput. Struct., 77, 233–251.
Buchanan, G. R., and Liu, Y. J. (2005). “An analysis of the free vibration of thick-walled isotropic toroidal shells.” Int. J. Mech. Sci., 47, 277–292.
Chien, W. Z. (1979). Selected papers in applied math and mechanics, Jiangsu Sciences and Technology, Jiangsu, China.
Clark, R. A. (1950). “On the theory of thin elastic toroidal shells.” J. Mech. Phys., 29(3), 146–178.
Clark, R. A. (1958). “Asymptotic solutions of toroidal shell problems.” Q. Appl. Math., 16, 47–60.
Clark, R. A., and Reissner, E. (1951). “Bending of curved tubes.” Adv. Appl. Mech., 2, 93–122.
Combescure, A., and Galletly, G. D. (1999). “Plastic buckling of complete toroidal shells of elliptical cross-section subjected to internal pressure.” Thin-Walled Struct., 34, 135–146.
Dahl, N. C. (1953). “Toroidal-shell expansion joints.” ASME J. Appl. Mech., 20, 497–503.
Galletly, G. D. (1998). “Elastic buckling of complete toroidal shells of elliptical cross-section subjected to uniform internal pressure.” Thin-Walled Struct., 30, 23–34.
Galletly, G. D., and Galletly, D. A. (1996). “Buckling of complex toroidal shell structures.” Thin-Walled Struct., 26, 195–212.
Hoff, N. J. (1955). “The accuracy of Donnell’s equations.” J. Appl. Mech., 22, 329–334.
Jiang, W., and Redekop, D. (2003). “Static and vibration analysis of orthotropic toroidal shells of variable thickness by differential quadrature.” Thin-Walled Struct., 41, 461–478.
Kar’yagdyev, N. Y., and Mukoed, A. P. (1993). “Axisymmetric deformation of flexible orthotropic toroidal shells.” J. Math. Sci. (N.Y.), 66(5), 2502–2506.
Korovaitsev, A. V., and Evkin, A. I. (1992). “Axisymmetrical deformation of a toroidal shell under strong bending.” Prikladnaia Mekhanika, 28(4), 16–23 (in Russian).
Leung, A. Y. T., and Kwok, N. T. C. (1994). “Free vibration analysis of a toroidal shell.” Thin-Walled Struct., 18, 317–332.
Meissner, E. (1915). “Uber und Elastizitat Festigkeit dunner Schalen.” Viertelschr. D. nature.Ges., Bd.60, Zurich, Switzerland.
Ming, R. S., Pan, J., and Norton, M. P. (2002). “Free vibrations of elastic circular toroidal shells.” Appl. Acoust., 63, 513–528.
Morley, L. S. D. (1959). “An improvement on Donnell’s approximation for thin-walled circular cylinders.” Q. J. Mech. Appl. Math., 12, 89–99.
Novozhilov, V. V. (1959). The theory of thin shells, Noordhoff, Groningen, The Netherlands.
Novozhilov, V. V. (1966). “Development of the method of complex transformation in the linear theory of shells during the last fifty years.” Proc., 4th All-Union Conf. on Shells and Plates, S. M. Dugar’yan, ed., Israel Program for Scientific Translations, Jerusalem, 88–95.
Pomares, R. J., and Durlofsky, H. (1988). “Collapse analysis of toroidal shell.” PVP (Am. Soc. Mech. Eng.), 199, 23–33.
Redekop, D. (2004). “Free vibration of hollow bodies of revolution.” J. Sound Vib., 273, 415–420.
Redekop, D. (2005). “Buckling analysis of an orthotropic thin shell of revolution using differential quadrature.” Int. J. Pressure Vessels Piping, 82, 618–624.
Redekop, D. (2006). “Three-dimensional free vibration analysis of inhomogenous thick orthotropic shells of revolution using differential quadrature.” J. Sound Vib., 291, 1029–1040.
Redekop, D. (2009). “Buckling analysis of an orthotropic elliptical toroidal shell.” Proc., ASME 2009 Pressure Vessels and Piping Div. Conf., PVP2009, ASME, New York, 1–8.
Redekop, D., and Muhammad, T. (2003). “Analysis of toroidal shells using the differential quadrature method.” Int. J. Struct. Stab. Dyn., 3, 215–226.
Reissner, H. (1912). Spannungen in Kugelschalen (Kuppeln), Müller-Breslau Festschrift, Leipzig, Germany, 181–193.
Ren, W., Liu, W., Zhang, W., Reimerdes, H. G., and Oery, H. (1999). “A survey of works on the theory of toroidal shells and curved tubes.” Acta Mech. Sin., 15(3), 225–234.
Ross, C. T. F. (2005). “A conceptual design of an underwater missile launcher.” Ocean Eng., 32, 85–99.
Ruggiero, E. J., Jha, A., Park, G., and Inman, D. J. (2003). “A literature review of ultra-light and inflated toroidal satellite components.” Shock Vib. Dig., 35, 171–181.
Sanders, J. L., Jr. (1959). “An improved first-approximation theory for thin shells.” NASA Tech. Rep. No. R-24, National Aeronautics and Space Administration (NASA), Washington, D.C., 1–11.
Sanders, J. L., Jr. (1963). “Nonlinear theories for thin shells.” Q. Appl. Math., 21, 21–36.
Shamina, V. A. (1962). “Calculation of a ribbed toroidal shell under a symmetrical road.” Proc., 4th All-Union Conf. on Shells and Plates, S. M. Dugar’yan, ed., 907–911.
Steele, C. R. (1959). “Toroidal shells with nonsymmetric loading.” Ph.D. dissertation, Stanford Univ., Stanford, CA.
Stein, M., and Mcelman, J. A. (1965). “Buckling of segments of toroidal shells.” AIAA J., 3, 1704–1709.
Sun, B. (1991). “Progress in applied mechanics.” Postdoctoral Research Rep., Tsinghua Univ., Beijing.
Sutcliffe, W. J. (1971). “Stress analysis of toroidal shells of elliptical cross section.” Int. J. Mech. Sci., 13, 951–958.
Vu, V. T., and Blachut, J. (2009). “Plastic instability pressure of toroidal shells.” ASME J. Pressure Vessel Technol., 131(5), 051203.
Wang, A., and Zhang, W. (1990). “Solution for postbuckling of toroidal shells.” Sci. China, Ser. A: Math., Phys., Astron. Technol. Sci., 33(10), 1220–1229.
Wang, A., and Zhang, W. (1991). “Asymptotic solution for buckling of toroidal shells.” Int. J. Pressure Vessels Piping, 45, 61–72.
Wang, X. H., and Redekop, D. (2005). “Natural frequencies and mode shapes of an orthotropic thin shell of revolution.” Thin-Walled Struct., 43, 735–750.
Wang, X. H., Xu, B., and Redekop, D. (2006a). “FEM free vibration and buckling analysis of stiffened toroidal shells.” Thin-Walled Struct., 44, 2–9.
Wang, X. H., Xu, B., and Redekop, D. (2006b). “Theoretical natural frequencies and mode shapes for thin and thick curved pipes and toroidal shells.” J. Sound Vib., 292, 424–434.
Weingarten, V. I., Veronda, D. R., and Saghera, S. S. (1973). “Buckling of segments of toroidal shells.” AIAA J., 11, 1422–1424.
Widera, G. E. O. (1986). “Validity of various shell theories applicable in the design and analysis of cylindrical pressure vessels.” PVP (Am. Soc. Mech. Eng.), 100, 11–19.
Wissler, H. (1916). Festigkeiberechung von Ringsflachen, Promotionarbeit, Zurich, Switzerland.
Xia, Z., and Ren, W. (1986). “An analysis of stress and strain for orthotropic toroidal shells.” Nucl. Eng. Des./Fusion, 3, 309–318.
Xia, Z. H., and Zhang, W. (1986). “The general solution for thin-walled curved tubes with arbitrary loadings and various boundary conditions.” Int. J. Pressure Vessels Piping, 26, 129–144.
Xu, B., and Redekop, D. (2006). “Natural frequencies of an orthotropic thin toroidal shell of elliptical cross-section.” J. Sound Vib., 293, 440–448.
Yamada, G., Kobayahsi, Y., Ohta, Y., and Yokota, S. (1989). “Free vibration of a toroidal shell with elliptical cross-section.” J. Sound Vib., 135, 411–425.
Young, W. C., and Budynas, R. G. (2002). Roark’s formulas for stress and strain, 7th Ed., McGraw-Hill, New York.
Zhan, H. J., and Redekop, D. (2008). “Vibration, buckling and collapse of ovaloid toroidal tank.” Thin-Walled Struct., 46, 380–389.
Zhang, W. (1949). “Toroidal shells.” Sci. Rep. Natl. Tsing Hua Univ., Ser. A, 259–349.
Zhang, R. J., and Zhang, W. (1994). “Toroidal shells under nonsymmetric loading.” Int. J. Solids Struct., 31(19), 2735–2750.
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Received: Dec 21, 2009
Accepted: Mar 24, 2010
Published online: Mar 25, 2010
Published in print: Oct 2010
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