The Buckling of Spherical Shells by External Pressure

The general theory of thin shells was developed by A. E. H. Love. He assumed small deflections and for that reason neglected in the energy expression all terms higher than quadratic, and thus obtained linear differential equations for the determination of the equilibrium position of a shell under given forces. The theory of buckling of thin shells is also based essentially on Love's equations. The buckling of cylindrical shells of uniform thickness under the action of a uniformly distributed axial load was calculated by R. Lorentz, R. V. Southwell, S. Timoshenko, W. Flugge, L. H. Donnell, and others. The same problem was also investigated experimentally by many authors, especially by E. E. Lundquist and L. H. Donneil. Unfortunately, a systematic discrepancy was found between the theoretically calculated and experimentally obtained buckling loads; the theoretical values are as much as 3 to 4 times higher than the experimental values. To remedy this situation W. Flugge [1] first considered the deviation of the assumed end conditions of the cylindrical shell from that realized in the laboratory. However, this effect is not sufficient to explain the discrepancy. The influence of the end conditions extends only to a distance approximately equal to Rt , where R is the radius of the shell and t the thickness. The cylinders tested, however, usually have a length which is large compared to this value. Furthermore, Flugge's analysis would indicate a progressive increase of the wave amplitude until plastic deformation occurs, whereas the experimental evidence indicates that the failure of cylindrical shells under compression is not progressive but very rapid.