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load-deflection curves: externally pressurized spherical caps

This slide demonstrates the transition in behavior from that of a flat plate under uniform pressure to that of a deep spherical cap or a complete spherical shell under uniform external pressure.

The flat plate (a) exhibits increasing stiffness as the pressure is increased and membrane tension develops as the flat plate bends downward under the pressure.

A very slightly curved plate (b) initially softens, then stiffens as the external pressure is increased, but there is no local maximum load-carrying capacity.

The somewhat more curved plate (c) exhibits the type of nonlinear buckling called “snap-through”: The plate softens until it has zero stiffness, then “snaps” into an inverted position, after which it stiffens with further increase in pressure.

A plate with more curvature yet (d) exhibits non-axisymmetric bifurcation buckling (black points) before axisymmetric “snap-through”. The bifurcation buckling load from linear theory is somewhat higher than that from nonlinear theory in this example.

Deeper (or thinner) plates yet (e, f) exhibit the same type of behavior as (d), with the characteristic equilibrium paths having an increasing degree of the “doubling back” feature typical of shells the behaviors of which are extremely sensitive to initial imperfections.

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