Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Simplifications and reduction of the Sanders-Koiter linear shell equations for various midsurface geometries

Author: James G. Simmonds
Journal: Quart. Appl. Math. 28 (1970), 259-275
MSC: Primary 73.35
DOI: https://doi.org/10.1090/qam/267821
MathSciNet review: 267821
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the twelve linear field equations of the Sanders-Koiter first approximation shell theory, when specialized to shells with midsurfaces of constant mean curvature, can be replaced by eight equations of particularly simple form. This is accomplished by adding certain negligibly small terms to the conventional uncoupled stress-strain relations. On the basis of order of magnitude estimates, it is argued that these equations are actually adequate for shells of arbitrary geometry. For shells of nonzero Gaussian curvature, the simplified field equations are reduced to four coupled scalar equations. For catenoidal and helicoidal shells, these equations are further reduced to two coupled fourth-order equations. Various special forms of these equations are shown to agree with results obtained by Lardner, Reissner, and Wan for shallow shells, shells of revolution undergoing axisymmetric and lateral deformation, and helicoidal shells undergoing axisymmetric deformation.

References [Enhancements On Off] (What's this?)

  • [1] J. L. Sanders, Jr., An improved first approximation theory for thin shells, N.A.S.A. Rept. no. 24, 1959.
  • [2] W. T. Koiter, A consistent first approximation in the general theory of thin elastic shells, Proc. Sympos. Thin Elastic Shells (Delft, 1959) North-Holland, Amsterdam, 1960, pp. 12–33. MR 0142241
  • [3] B. Budiansky and J. L. Sanders Jr., On the “best” first-order linear shell theory, Progress in Applied Mechanics, Macmillan, New York, 1963, pp. 129–140. MR 0158595
  • [4] A. Libai, Invariant stress and deformation functions for doubly curved shells, J. Appl. Mech. 34, 43-48(1967).
  • [5] J. G. Simmonds, Further reduction of equation (20) for arbitrary shells of non-zero Gaussian curvature (Appendix to [6]), Proc. Second Sympos. Thin Elastic Shells, Springer-Verlag, Berlin and New York, 1969, pp. 157-160
  • [6] J. L. Sanders, Jr., On the shell equations in complex form, Proc. Second Sympos. Thin Elastic Shells, Springer-Verlag, Berlin and New York, 1969, pp. 135-156
  • [7] V. V. Novozhilov, Thin shell theory, Second augmented and revised edition. Translated from the second Russian edition by P. G. Lowe. Edited by J. R. M. Radok, P. Noordhoff, Ltd., Groningen, 1964. MR 0208886
  • [8] W. T. Koiter, A systematic simplification of the general equations in the linear theory of thin shells Proc. Nederl. Akad. Wetensch. B64, 612-619 (1961)
  • [9] A. L. Gol′denveĭzer, Equations of the theory of shells in terms of displacements and stress functions, Prikl. Mat. Meh. 21 (1957), 801–814 (Russian). MR 0099781
  • [10] F. Y. M. Wan, The exact reduction of equations of elastic shells of revolution, Studies in Appl. Math. 48, 361-375 (1969)
  • [11] W. T. Koiter, A spherical shell under point loads at its poles, Progress in Applied Mechanics, Macmillan, New York, 1963, pp. 155–169. MR 0158591
  • [12] J. G. Simmonds, Green's functions for closed elastic spherical shells: exact and accurate approximate solutions, Nederl. Akad. Wetensch. Proc. B71, 236-249 (1968)
  • [13] J. G. Simmonds, A set of simple, accurate equations for circular cylindrical elastic shells, Int. J. Solids and Structures 2, 523-541 (1966)
  • [14] W. T. Koiter, A comparison between John's refined interior shell equations and classical shell theory Z.A.M.P. 20, 642-652 (1969)
  • [15] A. L. Gol′denveĭzer, Theory of elastic thin shells, Translation from the Russian edited by G. Herrmann. International Series of Monographs on Aeronautics and Astronautics, Published for the American Society of Mechanical Engineers by Pergamon Press, Oxford-London-New York-Paris, 1961. MR 0135763
  • [16] I. N. Vekua, \cyr Obobshchennye analiticheskie funktsii., Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959 (Russian). MR 0108572
  • [17] Thomas J. Lardner, On the strss distribution in a shallow logarithmic shell of revolution, J. Math. and Phys. 45 (1966), 23–34. MR 0191176
  • [18] E. Reissner and F. Y. M. Wan, Rotationally symmetric stress and strain in shells of revolution, Studies Appl. Math. 48, 1-17 (1969)
  • [19] V. S. Chernin, On the system of differential equations of equilibrium of shells of revolution under bending loads, J. Appl. Math. Mech. 23 (1959), 372–382. MR 0149732, https://doi.org/10.1016/0021-8928(59)90093-0
  • [20] F. Y. M. Wan, Circumferentially sinusoidal stress and strain in shells of revolution, to appear in Int. J. Solids and Structures.
  • [21] E. Reissner, On the equations for finite symmetrical deflections of thin shells of revolution, Progress in Applied Mechanics, Macmillan, New York, 1963, pp. 171–178. MR 0159480
  • [22] Eric Reissner, Small rotationally symmetric deformations of shallow helicoidal shells, J. Appl. Mech. 22 (1955), 31–34. MR 0069714
  • [23] James K. Knowles and Eric Reissner, Torsion and extension of helicoidal shells, Quart. Appl. Math. 17 (1959/1960), 409–422. MR 0128149, https://doi.org/10.1090/S0033-569X-1960-0128149-8
  • [24] E. Reissner, On twisting and stretching of helicoidal shells, Proc. Sympos. Thin Elastic Shells (Delft, 1959) North-Holland, Amsterdam, 1960, pp. 434–466. MR 0135764
  • [25] E. Reissner, Note on axially symmetric stress distributions in helicoidal shells, Miszellaneen der Angewandten Mechanik (Tollmien Festschrift), Akademie-Verlag, Berlin 1962, pp. 257-266
  • [26] Diarmuid O’Mathuna, Rotationally symmetric deformations in helicoidal shells, J. Math. and Phys. 42 (1963), 85–111. MR 0151029
  • [27] E. Reissner and F. Y. M. Wan, On axial extension and torsion of helicoidal shells, J. Math and Phys. 47, 1-31 (1968)
  • [28] J. W. Cohen, The inadequacy of the classical stress-strain relations for the right helicoidal shell, Proc. Sympos. Thin Elastic Shells, North Holland, Amsterdam, 1960, pp. 415-433
  • [29] F. Y. M. Wan, The side-force problems for shallow helicoidal shells, J. Appl. Mech. 36, 292-295 (1969)
  • [30] F. Y. M. Wan, Pure bending of shallow helicoidal shells, J. Appl. Mech. 35, 387-392 (1968)
  • [31] F. Y. M. Wan, A class of unsymmetric stress distributions in helicoidal shells, Quart. Appl. Math. 24, 374-379 (1967)
  • [32] F. Y. M. Wan, St. Venant flexure of pretwisted rectangular plates, Int . J. Engr. Sci. 7, 351-360 (1969)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73.35

Retrieve articles in all journals with MSC: 73.35

Additional Information

DOI: https://doi.org/10.1090/qam/267821
Article copyright: © Copyright 1970 American Mathematical Society

American Mathematical Society