ISSN 0025-6544, Mechanics of Solids, 2008, Vol. 43, No. 4, pp. 539–544. c  Allerton Press, Inc., 2008. Original Russian Text c  S.M. Bauer, O.G. Klets, N.F. Morozov, 2008, published in Izvestiya Akademii Nauk. Mekhanika Tverdogo Tela, 2008, No. 4, pp. 19–25. Behavior of Transversally Isotropic Cylindrical Shells under Dynamic Application of Radial Pressure S. M. Bauer * , O. G. Klets ** , and N. F. Morozov *** St.-Petersburg State University, Universitetskaya nab. 7–9, St. Petersburg, 199034 Russia Received May 8, 2008 Abstract—We consider the problem on the buckling of a transversally isotropic long cylindrical shell under dynamic external pressure. For the cases of a sudden load [1] and a load rapidly increasing with time [2], we compare the results obtained from the two-dimensional Kirchhoff–Love and Timoshenko–Reissner theories. DOI: 10.3103/S0025654408040031 Several problems of statics and free vibrations of homogeneous transversally isotropic thin-walled structures were considered in [3, 4], where the results obtained for these problems from approximate models based on the Kirchhoff–Love and Timoshenko–Reissner kinematic hypotheses were compared with the asymptotic solutions of the three-dimensional elasticity equations. In [3], it was shown that for thin-walled structures the Kirchhoff–Love theory is the first asymptotic approximation to the three-dimensional theory. The Timoshenko theory, which takes into account shear, is asymptotically inconsistent for bodies made of isotropic materials and gives only an insignificant refinement of the two- dimensional Kirchhoff–Love model. But it was shown in [4] that the situation is different for bodies made of transversally isotropic materials with small shear rigidity in the shell or plate thickness direction, and in this case the Timoshenko–Reissner theory refines the Kirchhoff–Love theory and gives the next asymptotic approximation to the three-dimensional theory. Consider the stability of a long cylindrical shell compressed by a dynamic radial load. For the case of a sudden load exceeding the static critical value, the Kirchhoff–Love theory was used to study this problem in [1], where a new approach to solving stability problems for structures under dynamic loads was proposed. It was assumed that a load exceeding the static critical load is suddenly applied to a system with initial imperfections and a motion starts such that the system does not return to its initial position. It was also assumed that, at the initial stage of this motion, the displacements of all elements of the system are proportional to a same function of time. If the load is small, then this function is harmonic and the system exhibits normal vibrations. If the load is sufficiently large, then the motion is described by an exponential function and corresponds to buckling. The number of dynamic buckling modes increases with the value of the sudden load, because loads under which, along with the main equilibrium mode, some close curved modes are possible are known to form an increasing sequence. It was assumed that the initial imperfections corresponding to different normal motion modes are of the same order of magnitude. It was shown in [1] that the most rapidly varying modes are those associated with the largest coefficient in the exponent of the time function for the corresponding motion. It was also noted that the buckling modes may differ from the lowest and highest equilibrium modes possible for a given load. In [1], the equation of small motions of the tube was obtained by the classical model of the theory of shells based on the Kirchhoff–Love hypotheses with taking into account the dependence of the normal deflection w on the angle θ and time t and the shell initial deviation w 0 (θ) from the circular shape. This * E-mail: s_bauer@mail.ru ** E-mail: oklec@yandex.ru *** E-mail: morozov@nm1016.spb.edu 539