Pressure stiffening and softening

A string has no bending stiffness at all, yet a tight string resists a sideways push. Pull it with a tension N between two posts a distance L apart and push its middle with a force F. The string takes a shallow V shape, and the tension in each half now has a small sideways part that balances the push.

The stiffness comes entirely from the tension: double the tension and the string deflects half as much. Release the tension and the stiffness is gone. Nothing about the material changed; the stress and the tilt of the geometry did the work.

Every structure has the same effect on top of its ordinary, material stiffness. The total stiffness is the material part plus a geometric part that is proportional to the stress already in the structure. Tension makes the geometric part positive, compression makes it negative.

For one shape of deformation, the result is a straight line. The stiffness falls in proportion to the compressive load and reaches zero at the buckling load P_cr; in tension it rises the same way. That is what buckling is: the load at which compression has used up all the stiffness against one shape.

Internal pressure puts a vessel wall in tension both round the hoop and along the axis. Push the wall inward at a nozzle or a clip and, like the string, the stretched wall tilts and pushes back. The shell deflects less under the same radial load, the dent is shallower, and the local bending stress round the attachment is lower than a model without the effect would give.

• The effect grows with the membrane stress P·R/t and is largest on thin shells, where the material bending stiffness is small to begin with.
• Pressure also rounds out a shell that is slightly oval, which lowers the bending that the out-of-roundness causes.
• Internal pressure raises the axial buckling load of a thin cylinder, because it holds the wall round and smooths out the small dents that make axial buckling so sensitive.

Outside pressure and vacuum put the wall in hoop compression, and axial load or wind bending put part of it in axial compression. Then the geometric part takes stiffness away. Near the buckling load a small sideways load, or a small dent, produces a deflection far larger than a linear calculation would give, and the bending stress grows with it.

The same vessel can sit on both sides of the line. A column designed for internal pressure may see vacuum during steam-out or after a failed vent, and a tall shell is in axial compression on one side under wind whatever its pressure. A check that relied on pressure stiffening for one case has to be repeated without it, and with softening, for the other.

A linear, small-displacement analysis uses the material stiffness only and keeps it the same at every load. It cannot see stiffening or softening. Under internal pressure with nozzle loads this usually errs on the safe side: the model is too flexible, so its local bending stresses come out somewhat high. Under compression it errs the other way: it misses the amplification, and the stresses and deflections it gives can be too low.

In a finite element model the effect is included by a geometrically nonlinear analysis, which updates the stiffness with the stress and the deformed shape as the load rises. It is worth the effort on thin shells where the conservatism of the linear answer under internal pressure is costly, and it is needed wherever part of the structure is in compression at a large fraction of its buckling load.

An eigenvalue buckling analysis uses the same geometric stiffness in its simplest form. It computes K_G for the applied load and finds the multiplier λ at which K₀ + λ·K_G loses all stiffness against some shape. Stiffening and buckling are the same physics seen from two sides.