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The static–geometric analogy in the equations of thin shell structures

Published online by Cambridge University Press:  24 October 2008

C. R. Calladine
C. R. Calladine
Affiliation:
Peterhouse, Cambridge
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Abstract

The ‘static-geometric analogy’ in thin shell structures is a formal correspondence between equilibrium equations on the one hand and geometric compatibility equations on the other. It is well known as a fact, but no satisfactory explanation of its basis has been given. The paper gives an explanation for the analogy, within the framework of shallow-shell theory. The explanation is facilitated by two innovations: (i) separation of the shell surface conceptually into separate stretching (S) and bending (B) surfaces; (ii) use of change of Gaussian curvature as a prime variable. Various limitations of the analogy are pointed out, and a scheme for numerical calculation which embodies the most useful features of the analogy is outlined.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 2 , September 1977 , pp. 335 - 351
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

(1)Aleksandrov, A. D. and Zalgaller, V. A.Intrinsic geometry of surfaces (transl. by Danskin, J. H.) (Providence: American Mathematical Society, 1967).Google Scholar
(2)Budiansky, B. and Sanders, J. L. On the ‘best’ first-order linear shell theory. In Progress in applied mechanics; the Prager anniversary volume, pp. 129–140. (New York: Macmillan, 1963).Google Scholar
(3)Calladine, C. R. Creep in torispherical pressure vessel heads. In Creep in structures: Proc. IUTAM Symposium, Gothenburg, pp. 247–268. (Berlin: Springer-Verlag, 1972).Google Scholar
(4)Calladine, C. R.Structural consequences of small imperfections in elastic thin shells of revolution. Int. J. Solids Structures 8 (1972), 679–697.CrossRefGoogle Scholar
(5)Calladine, C. R.Thin-walled elastic shells analysed by a Rayleigh method. Int. J. Solids Structures (in the Press).Google Scholar
(6)Caspar, D. L. D. and Klug, A.Physical principles in the construction of regular viruses. Cold Spring Harbour Symposia on Quantitative Biology 27 (1962), 1–24.CrossRefGoogle ScholarPubMed
(7)Chernyshev, G. N.On the action of concentrated forces and moments on an elastic thin shell of arbitrary shape. J. Appl. Math. Mech. (Prikl. Mat. Mekh.) 27 (1963), 172–184.CrossRefGoogle Scholar
(8)Collins, I. F.On an analogy between plane strain and plate bending solutions in rigid/perfect plasticity theory. Int. J. Solids Structures 7 (1971), 1057–1073.CrossRefGoogle Scholar
(9)Donnell, L. H. Stability of thin-walled tubes under torsion. National Advisory Committee for Aeronautics, report 497 (Washington, 1933).Google Scholar
(10)Flügge, W.Statik und Dynamik der Schalen, 3rd ed., pp. 202–204 (Berlin: Springer-Verlag, 1962).CrossRefGoogle Scholar
(11)Flügge, W.Stresses in Shells, 2nd ed., pp. 414–422 (Berlin: Springer-Verlag, 1973).CrossRefGoogle Scholar
(12)Flügge, W. and Elling, R. E.Singular solutions for shallow shells. Int. J. Solids Structures 8 (1972), 227–247.CrossRefGoogle Scholar
(13)Gauss, K. F. General investigation of curved surfaces (in Latin) (transl. by J. C. Morehead, and A. M. Hiltebeitel), (Hewlett, New York: Raven Press, 1965).Google Scholar
(14)Hilbert, D. and Cohn-Vossen, S.Geometry and the imagination, pp. 193–204 (New York: Chelsea Publ. Co., 1952).Google Scholar
(15)Gol'denweiser, A. L.Equations of the theory of thin shells (in Russian). Prikl. Mat. Mekh. 4 (1940), 35–42.Google Scholar
(16)Gol'denweiser, A. L.Theory of elastic thin shells (transl. ed. Herrmann, A.), pp. 92–96 (Oxford: Pergamon for A.S.M.E., 1961).Google Scholar
(17)Johnson, W.Upper bounds to the load for the transverse bending of flat rigid-perfectly plastic plates. Part 2. An analogy: slip-line fields for analysing the bending and torsion of plates. Int. J. Mech. Sci. 11 (1969), 913–938.CrossRefGoogle Scholar
(18)Koiter, W. T.The effect of axisymmetric imperfections on the buckling of cylindrical shells under axial compression. Proc. Kon. Ned. Acad. Wet. (B) 66 (1963), 265–279.Google Scholar
(19)Koiter, W. T.On the nonlinear theory of thin elastic shells. Proc. Kon. Ned. Acad. Wet. (B) 69 (1966), 1–54.Google Scholar
(20)Lur'e, A. I.General theory of elastic shells (in Russian). Prikl. Mat. Mekh. 4 (1940), 7–34.Google Scholar
(21)Lur'e, A. I. On the static geometric analogue of shell theory. In Problems of continuum mechanics; the Muskhelisvili anniversary volume, pp. 267–274 (Philadelphia: Society for Industrial and Applied Mathematics, 1961).Google Scholar
(22)Marguerre, K.Zur Theorie der gekrümmten Platte groβer Formänderung. Proc. 5th Intern. Cong. Appl. Mech., Cambridge, Mass. IUTAM (1939), 93–101.Google Scholar
(23)Maxwell, J. C.On the transformation of surfaces by bending. Trans. Cambridge Philos. Soc. 9 (1856), 445–470.Google Scholar
(24)Naghdi, P. M. The theory of shells and plates. Handbuch der Physik, Band VIaa/2, pp. 425–640 (esp. p. 613) (Berlin: Springer-Verlag, 1972).Google Scholar
(25)Novozhilov, V. V.The theory of thin shells (transl. Lowe, P. G., ed. Radok, J. R. M.) (Groningen: P. Noordhoff, 1959).Google Scholar
(26)Reissner, E.Stresses and small displacements of shallow shells, II. J. Math. Phys. 25 (1946), 279–300.CrossRefGoogle Scholar
(27)Sanders, J. L. An improved first-approximation theory for thin shells. National Aeronautics and Space Administration Technical Report R24 (Washington, 1959).Google Scholar
(28)Sanders, J. L.Singular solutions to the shallow shell equations. J. Appl. Mech. 37 (1970), 361–364.CrossRefGoogle Scholar
(29)Sanders, J. L. and Simmonds, J. G.Concentrated forces on shallow cylindrical shells. J. Appl. Mech. 37 (1970), 367–373.CrossRefGoogle Scholar
(30)Terzaghi, K.Theoretical soil mechanics, pp. 11–15 (New York: Wiley, 1943).CrossRefGoogle Scholar
(31)Timoshenko, S. P. and Goodier, J. N.Theory of elasticity, 2nd ed., pp. 24, 26–27 (New York: McGraw-Hill, 1951).Google Scholar
(32)Timoshenko, S. P. and Woinowsky-Krieger, S.Theory of plates and shells, 2nd ed. (New York: McGraw-Hill, 1959).Google Scholar