Load-controlled and displacement-controlled loads

Pressure, the weight of the vessel and its contents, wind, earthquake, snow on a platform and the pull of a crane on a lifting lug all have one thing in common: they act however the part deforms. If the wall stretches a little under pressure, the pressure is still there. If a lug bends under the lift, the vessel still weighs the same. The structure must carry each of these loads all the way to its supports.

These are load-controlled loads. The stresses they cause are fixed by equilibrium: for a given load and shape there is only one set of internal forces that balances it. If the material cannot provide those forces, the part keeps deforming.

Other actions impose a movement rather than a force. A bar heated while its ends are held wants to grow; a support that settles drags the structure down with it; a flange pulled into line with bolts is forced into a new position. In each case a shape change is set from outside, and the structure develops whatever stress is needed to accommodate it.

The forces on the two walls are equal and opposite, and nothing outside has to hold them up. These are displacement-controlled loads. For a bar held fully at both ends and heated evenly by ΔT, the strain it is prevented from taking is α·ΔT, and while the metal stays elastic the stress is:

Take a displacement-controlled stress past yield and the metal gives a little. The imposed movement is taken up partly by plastic strain, so the stress goes no higher. In the heated bar, once the stress reaches yield, further heating adds plastic strain, not stress. The movement is finite, so the plastic strain is finite too.

Do the same with a load-controlled stress and nothing is taken up. The load is still there, the stress must still balance it, and yielding only lets the part deform further. Under a fixed load the strain keeps running once the section has yielded through; under a fixed movement it stops where the movement sets it.

Steel makes the point quickly, because its stiffness is high. A fully held steel bar picks up about 2.4 MPa of stress for every degree of temperature rise. A modest temperature difference is therefore enough to reach yield on paper, which is why restrained thermal growth so often produces large elastic stresses in a calculation.

Thermal stress is used here only as the clearest example of an imposed movement. The same reasoning applies to a settling support or a forced fit-up: the stress is capped by yield and the plastic strain by the size of the movement. When the bar cools again it is left with a locked-in stress of the opposite sign, which is where the questions of shakedown and ratcheting begin.

The clean split has an exception. A movement can still act like a load when a small region yields while a long, springy system around it stays elastic. The springy part stores energy as the movement is applied; when the small region starts to yield, the springy part keeps pushing, and the strain gathers in the small region instead of being shared.

A long, slender bar held between two fixed ends, with one short, weaker length, is the usual picture. Heat the whole bar and the short length yields first, but the rest of the bar keeps pushing, so the stress there relaxes less than a pure imposed movement would suggest and the plastic strain concentrates in it. The more springy the surrounding system and the smaller the yielding region, the closer the behaviour gets to load control. Codes and guidance documents treat strong elastic follow-up cautiously, in some cases by classing the stress as primary.

A force or moment given at an attachment can contain both kinds, and its origin is not always stated. If it is not known which part is which, treating the whole of it as load-controlled is the cautious choice. The governing code says how mixed cases are to be classed.