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The thickness functions of the shells of revolution subjected to axisymmetrical loads

In this paper, the problem for determining the thickness function of shells of revolution such as a parabola, sphere arc! under axisymmetrical loads is considered. The general integro-differential equations for determination of the meridian form and shell thickness are obtained. | Vietnam Journal of Mechanics, VAST, Vol. 27, No. 2 (2005) , pp. 66 - 73 THE THICKNESS FUNCTIONS OF THE SHELLS OF REVOLUTION SUBJECTED TO AXISYMMETRICAL LOADS NGO HUONG NHU1, PHAM HONG NGA 2 1 Institute 2 of Mechanics University of Transport and Communication Abstract. The inverse problems for determining the meridian shape or varying thickness function of momentless shells of revolution under given loads were concerned in many works [2, 3, 4]. However, for the complexity of loads or configuration of a shell these problems haven ' t bee.n solved perfectly because of its mathematical difficulties. In this paper, the problem for determining the thickness function of shells of revolution such as a parabola, sphere arc! under axisymmetrical loads is considered. The general integro-differential equations for determination of the meridian form and shell thickness are obtained. A solution of differential equations by semi-analytical and numerical methods for the thickness is presented. The numerical solutions are given for the parabola under external pressure, the sphere immerged in the fluid and t he sphere arc. Obtained results may be used in the thin shell design. 1. THE MOMENTLESS THEORY OF SHELLS OF REVOLUTION SUBJECTED TO AXISYMETRICAL LOADS The equilibrium equations are of the form [1]: ddTs s l · + (Tc: _ Ts) sine + X = O, r (1.1) Ts+ T'P = Ri R2 z. ' · where Ts and T'P are membrane forces , R 1 , R2 are curvature radii of the shell, e is the angle between the tangential line of the meridian and the Oz axis , X, Z - the external load components, r - radius of the hoop circle. For the shell of revolution we have following useful relations [1]: de ds' 1 R2 cos r e dr = - sine· ds ' (1.2) d ( 1 ) ( 1 1 ) sin e ds R2 = R<2 ~ R1 - r -. 67 The Thickness Functions of the Revolution Parabola, . Then, t he relations between small deformations and displacements are: du cs = ds w + R1 ' (1.3) - sin() ccp = - - - U w +- . R2 Corresponding to .