Displacement Analysis of Spherical Thin Shells under the Concentrated Load

Abstract

The action of concentrated loads on spherical shell has been widely studied. The solution for a concentrated normal force was first obtained by E. Reissner in 1946 by using the classical theory of shallow shells. This problem was also considered by S.P. Timoshenko (1946), Stanislaw Lukasiewic (1979), W.T. Koiter (1963), W.C. Young (1989) etc. In deriving these solutions, it is assumed that the stresses and displacements are very small at some distance from the loading point. In this paper, the displacement of non-shallow spherical thin shell under a concentrated load has been considered. Appling the above all kinds of methods, we have calculated the deflections under the point of application of concentrated loads on spherical thin shells with R/t=10~50 and R/t=100~500. The results are analyzed and all kinds of methods are compared, the calculation precisions and covering ranges of all kinds of methods are discussed. Introduction The action of concentrated loads on spherical thin shell has been widely studied. The solution for a concentrated normal force was first obtained by E. Reissner in 1946 by using the classical theory of shallow shells. This problem was also considered by S.P. Timoshenko (1946), Stanislaw Lukasiewicz (1979), W.T. Koiter (1963), W.C. Young (1989) etc. In deriving these solutions it is assumed that the stresses and displacements are very small at some distance from the loading point. The results are analyzed and all kinds of methods are compared, the calculation precisions and covering ranges of all kinds of methods are discussed. Deflection Solutions Timoshenko (1946) Timoshenko (1946) take the fact into account that the effect of transverse shear Qr on membrane forces can be neglected in the case of a shallow shell and the deflection at the center of such a shell is affected very little by the respective conditions on the outer edge. He considered a shallow shell with a very large radius subjected to a point load P at the apex r=0 (Figure 1), while the normal displaces w must be finite at r=0, and w must vanish for r=∞. He obtained for the deflection of the shell at the point of the application of the load the value (Figure 1): w0 4 ) 1 ( 3 2    2 Et PR Timoshenko(1946)a When the central load P is uniformly distributed over a circular area of a small radius c (Figure 2), the following results hold at the center of the loaded area r=0: w0   ) 1 ( 12 2           ) ( 2 1 4 2 2      Et PR Timoshenko(1946)b International Conference on Mechanics and Civil Engineering (ICMCE 2014) © 2014. The authors Published by Atlantis Press 778 Where l c   , ) 1 ( 12 4 2    Rt l ; the function     ker ) / 2 ( ) ( 4   , numerical values of the function 4   are given in Table 86 of Timoshenko(1946). Fig. 1 The Shell at the Point of the Load Fig. 2 P Uniformly Distributed Koiter (1963) The non-shallow spherical shell loaded at the vertex has been considered by Koiter(1963) in spherical coordinates, and based on the equations of the theory of non-shallow spherical shell. Koiter obtained the following expression for the normal deflection under the point of application of the load P (Figure 1): w0 4 ) 1 ( 3 2    2 Et PR                     ) ( 3 4 2 2 ln 1 ln ) 1 ( 2 1 3 2 0 2       O Koiter(1963)