Generalized Three-Dimensional Theory of Elastic Body Stability. Part 3. Theory of Shell Stability

Based on three-dimensional equations of theory of elastic body stability with small deformations, the equations of theory of stability of thin shells of Timoshenko type are deduced. These equations differ from the known empirically derived equations of stability theory in different expressions for coefficients at efforts of the basic (stable) state as well as in presence of moments offictitious forces of the basic state, which typically are assumed to be zero. It is shown that for the classical problem on rod stability, the deduced equations of stability theory are reduced to the classical eigenvalue equation. However for more elaborate shell structures, the distinctions are possible in equations of the stability theory and in the expression for critical loads.

References

[1] Timoshenko S.P. Ustoychivost’ sterzhney, plastin i obolochek. Izbrannye raboty [Stability of rods, plates and shells]. Moscow, Nauka Publ., 1971. 808 p.

[2] Vol’mir A.S. Ustoychivost’ deformiruemykh system [Stability of deformable systems]. Moscow, Nauka Publ., 1967. 964 p.

[3] Panovko Ya.G., Gubanova I.I. Ustoychivost’ i kolebaniya uprugikh system [Stability of deformable systems]. Nauka Publ., 1967. 420 p.

[4] Alfutov N.A. Osnovy rascheta na ustoychivost’ uprugikh system [Basis of calculation for stability of elastic systems]. Moscow, Mashinostroenie Publ., 1978. 312 p.

[5] Bolotin V.V. Nekonservativnye zadachi teorii uprugoy ustoychivosti [Nonconservative problems of the elastic stability theory]. Moscow, Fizmatlit Publ., 1961. 340 p.

[6] Vasil’ev V.V. Mekhanika kompozitsionnykh materialov [Mechanics of composite materials]. Moscow, Mashinostroyeniye Publ., 1984. 272 p.

[7] Grigolyuk E.I., Chulkov P.P. Ustoychivost’ i kolebaniya trekhsloynykh obolochek [Stability and vibrations of sandwich shells]. Moscow, Mashinostroyeniye Publ., 1973. 172 p.

[8] Simitses G.J. An introduction to the elastic stability of structures. NJ, Prentice Hall, 1976. 256 p.

[9] Bazant Z.P., Cedolin L. Stability of structures. Oxford, Oxford University Press, 1990. 316 p.

[10] Iyengar N.G.R. Structural stability of columns and plates. New Delhi, Affiliated East-West Press, 1986. 284 p.

[11] Tomczyk B. Dynamic Stability of Micro-Periodic Cylindrical Shell. Mechanics and Mechanical Engineering, 2010, vol. 14, no. 2, pp. 351-374.

[12] Bazant Z.P. Stability of elastic, an elastic and disintegrating structures: a conspectus of main results. ZAMM (Z. Angew. Math. Mech.), 2000, vol. 80, no. 11-12, pp. 709732.

[13] Dimitrienko Yu.I. Generalized three-dimensional stability theory of elastic bodies. Part 1. Finite deformations. Vestn. Mosk. Gos. Tekh. Univ. im.N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2013, no. 4 (51), pp. 79-95 (in Russ.).

[14] Dimitrienko Yu.I. Generalized three-dimensional stability theory of elastic bodies. Part 2. Small deformation. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2014, no. 1, pp. 17-26 (in Russ.).

[15] Dimitrienko Yu.I. Mekhanika kompozitsionnykh materialov pri vysokikh temperaturakh [Mechanics of composite materials at high temperatures]. Moscow, Mashinostroenie Publ., 1997. 356 p.

[16] Dimitrienko Yu.I. Mekhanika sploshnoy sredy. T. 2. Universal’nye zakony mekhaniki i elektrodinamiki sploshnoy sredy [Continuum mechanics. Vol. 2. Universal laws of mechanics and electrodynamics of continuous media], Moscow, MGTU im. N.E. Baumana Publ., 2011. 464 p.

[17] Dimitrienko Yu.I. Nelineynaya mekhanika sploshnoy sredy [Nonlinear continuum mechanics]. Moscow, Fizmatlit Publ., 2009. 624 p.

[18] Ambartsumyan S.A. Obshchaya teoriya anizotropnykh obolochek [General theory of anisotropic shells]. Moscow, Nauka Publ., 1974. 446 p.