ELASTIC PROBLEMS IN SHELLS

Abstract

In the present thesis free vibrations of isotropic as well as anisotropic shells of various shapes are studied. The thesis can be divided into three parts. The first part deals with the vibrations of isotropic spher ical shells; the second part considers the vibrations of cylindrical shells; whereas the third part studies the vibrations of orthotropic eonioal shells. The vibrations of isotropic spherical shells have been investigated in the first three chapters. The fre quency equation of the free vibrations of a moderately thick non-shallow spherical shell clamped at the edge has been derived in the first chapter. The numerical values of the frequency parameter for free axisymmetric as well as asymmetric vibrations have been computed for various opening angles of the shell. For the sake of comparison, the frequencies of vibration of these shells neglecting the effects of rotatory inertia and transverse shear have also been computed. In addition, the effect of Poisson's ratio on the vibrations of spherioal shells is also studied in this chapter. Equations of motion governing the axi symmetric vibrations of thick non-shallow spherioal shells, in which the effeot of transverse normal stress is included in addition to the transverse shear and rotatory inertia -iiare derived in the second chapter by applying Hamilton's principle of energy. In place of the three equations governing the axi symmetric motion of moderately thick spherical shell, we obtain a set of four equations in the present case. The differential equations have been solved to obtain the solutions for the displacement comp onents in terms of Legendre functions. Frequency equation for the free vibrations of a shell closed at the apex and clamped at the edge e^e0 is obtained by applying the appropriate edge conditions. Numerical values of the frequency parameter have been computed for a hemispherical shell for various thicknesses. The frequency parameter has also been computed for different opening angles. The results obtained from the present thick shell theory are compared with those obtained from the theories of chapter I, The third chapter considers the free torsional vibrations of a spheric .1 shell on the basis of three dimensional theory of elasticity. The single differential equation governing such vibrations is solved by the method of separation of variables. The tangential component v, which is the only non-zero component of displacement is expressed as a function of radial and meridional coord inates r and 9 respectively. The amplitudes and freq uency equation of vibration are obtained by applying the -iiisurface and the edge conditions. The frequency parameter has been computed for various values of the shell thick ness. The mode shapes are also studied. The vibrations of cylindrical shells of finite length are investigated in chapter IV. This is an invest igation of the vibrations of a transversely isotropic cylindrical shell of finite length, simply supported at its edges, on the basis of the three dimensional theory of elasticity. The three equations of motion governing the asymmetric motions are simplified by substituting three auxiliary variables in place of T*t , -*9 , ^ _ the three displacement components. The resulting differential equa tions are solved in terms of Bessel functions of first and second kind. Frequency equations for asymmetric, axisymmetric and torsional vibrations of a cylindrical shell are derived by applying the appropriate surface conditions. It is further shown that pure torsional motions are un coupled from the torsionless axisymmetric motions. The vibrations of conical shells are considered in chapters Vand VI. m chapter V, the frequency equation for the free axisymmetrio vibrations of orthotopic trun cated conical shell, simply supported at the two ends is derived by applying iialeigh-Ritz procedure. In the present analysis, the variation of transverse normal strain is considered in addition to the secondary effects of trans verse shear and rotatory inert! a. The displacements are -ivexpressed as an infinite series in the meridional coordinate ; and while computing the values of frequenoy parameter we have truncated the series in view of the limited capacity of the computer available here. It is observed that the results obtained from one term and two terra truncation of series are practioally the same even for short and thick shells. The values of frequenoy parameter on the basis of the present analysis has also been computed for isotropic conical shells. It is found that the results are in satisfactory agreement with the results obtained by earlier workers. In the last chapter, torsional vibrations of coni cal shells are studied. Secondary effects of transverse shear and rotatory inertia, neglected by the earlier workers, have been included. The frequency equation of vibrations of a conical shell clamped at the two edges is derived in a way similar to that of the previous chapter. Numerical values of the frequencies are computed for various values of length and thickness parameters. It is observed that the thickness of the shell does not affect its frequencies appreciably. The extensive numerical work involved in this thesis was done on IBM 1620 computer at the Computer Centre, Structural Engineering Research Centre, Roorkee. The work presented in this thesis is original research -Vby the author exoept sections 1-3 to 1-6 which have been included to present a connected account of the whole. The material of chapter IV has been published in the Journal of Acoustical Society of America, In December , 1965.