This slide shows the behavior of a very thin cylindrical shell under uniform axial compression or a very thin spherical shell under uniform external pressure. In these cases there is a dramatic effect of an initial imperfection on the load-carrying capacity of the cylindrical or spherical shell.

The solid curve with the small circular points on it is Fig. 4.4 from Chapter 1 of David Bushnell’s unpublished ph.d dissertatopm, Dept. of Aero- and Astronautics, Stanford University, 1965: “Some problems in the theory of thin shells”, Stanford University Library Stack No. 3781.S78B, 1965

[rho(crit) is the buckling load factor of the perfect shell with no pre-buckling edge effect, that is, no local prebuckling bending in the neighborhood of the edge or edges.

Full citation and abstract of Chapter 1:

David Bushnell, “Axisymmetric deformation of a shallow spherical cap clamped at the edge and submitted to uniform external pressure”, Chapter 1 in Some Problems in the Theory of Thin Shells, Ph.D dissertation, Aeronautics and Astronautics, Stanford University, Stanford, California, February 1965

ABSTRACT: In Chapter 1 a shallow spherical cap clamped at the edge and subjected to uniform external pressure is studied. This non-linear problem is approached from the equilibrium and the potential energy points of view. In both approaches the dependent variables, which are the displacement components of points on the middle surface, are expanded in trigonometric series each of whose terms satisfies the boundary and symmetry conditions. Undetermined coefficients of the terms in the series are evaluated by solving the set of non-linear algebraic equations which is generated in the equilibrium approach by direct substitution and in the potential energy approach by minimization of the total potential. Load-deflection curves are established for various cap geometries, and the stability curve is obtained by plotting the values of the pressure for which the load-deflection curves have maximua versus a parameter LAMBDA-squared which describes the geometry of the cap. The early post-buckled behavior is explored and the effect of deviations from sphericity on the critical pressure determined. A limited investigation is made of the effect of small non-linear terms in the meridional strain-displacement expression on the stability and early post-buckling characteristics of shallow clamped caps.

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