Equilibrium paths and limit points

The path

Plot load against displacement for a nonlinear structure and you get its equilibrium path — every state at which it is in equilibrium. For a hardening structure the path rises forever. For a yielding one it flattens. For a shallow arch it rises, turns over, descends, and rises again.

That turning over is the whole subject of this lesson.

Limit points

A limit point is where the tangent to the path is horizontal: the structure has reached the most load it can carry in that configuration.

Past it, the path descends: the structure continues to deflect while carrying less load. It is still in equilibrium at every point along that branch — it is a perfectly real set of states — but it is unstable, in the sense that a load-controlled system cannot sit there.

Why not? Because load control asks 'at load P, where is the structure?' and on the descending branch there are three answers for some values of P, or none for others. The solver is looking for a root that either does not exist nearby or is not the one it is walking towards.

Snap-through

Past the limit point, a real structure under a real load does not follow the descending branch. It jumps — dynamically, suddenly — to the far branch where the path has risen again to meet the applied load. That is snap-through, and it is what a shallow arch or a dome does when it inverts.

In an analysis, this shows up in one of two ways, and both are worth recognising:

• the solver fails to converge, which at least announces itself;
• the solver converges — onto the far branch, at a displacement far beyond where it was, with no indication that it jumped.

The second is more dangerous. A converged result on the far branch is a legitimate equilibrium state and the output will not say that the structure got there by inverting.

Bifurcation is different

A limit point is where the path turns over. A bifurcation is where two paths cross: the structure has a choice, and it takes the one with lower energy.

A perfectly straight strut under axial load is the classic case. Up to the critical load it shortens; at the critical load a second path — bending — becomes available, and the structure takes it. A model built perfectly straight has no reason to prefer either branch, which is why real analyses introduce an imperfection to break the symmetry and tell the structure which way to go.

A limit point needs no imperfection to find. A bifurcation usually does, and a perfectly symmetric model can sail past one entirely.

Displacement control

Instead of prescribing the load and finding the displacement, prescribe the displacement and find the load.

This works on the descending branch, because the question 'at this displacement, what load holds the structure?' has exactly one answer everywhere on the path. Nothing about the structure has changed; the question has been rephrased into one that always has a unique answer.

Arc-length control

Displacement control fails in its turn where the path turns back on itself in displacement — snap-back, seen in some composite and delamination problems.

Arc-length control prescribes neither. It advances a fixed distance along the path, letting both the load and the displacement be found. Geometrically it is walking along the curve at constant speed rather than stepping in x or in y. It follows anything the other two can and some things they cannot.

The cost is that the load at the end of an increment is not something you chose, so 'analyse at the factored design load' becomes 'analyse until you reach it' — which needs a stopping criterion and some care about what to report.

Dynamic relaxation

A different approach entirely: pretend the static problem is a dynamic one, give the structure a fictitious mass, apply the load, and damp the resulting motion until it stops. Where it stops is the static equilibrium.

There is no stiffness matrix and no factorisation, which makes it attractive for very large problems and for form-finding, where the target is an equilibrium shape and the route there is irrelevant.

Two things must be said about it every time:

The mass and damping are fictitious. They are chosen for convergence speed, not from the structure. Nothing about the path is physical.

Only the converged state means anything. The intermediate states overshoot, oscillate and settle. Reading a displacement off an intermediate step is reading a number the method invented.

It is also conditionally stable: too large a pseudo-time step and it diverges rather than converging, in exactly the way an explicit dynamic analysis does.

Explicit solvers

An explicit solver marches forward in time, computing each step directly from the last with no equations to solve. That makes each step extremely cheap and extremely numerous.

The stable time step is the time a stress wave takes to cross the smallest element:

Δt < h / c, where c = √(E/ρ) is the material wave speed

For steel, c is about 5 170 m/s. A 50 mm element gives a critical step of about 10 microseconds; a practical run uses perhaps 90 % of that.

The consequence is the one that decides whether an explicit analysis is feasible: the step is set by the smallest element in the model, not the average. One sliver element among a million multiplies the run time by however many times smaller it is. Explicit analysis rewards mesh discipline more than any other analysis type, and a mesh that would merely be inelegant elsewhere is here a schedule risk.

Explicit solvers earn their place on problems implicit methods handle badly: impact, blast, fragmentation, contact between many bodies, progressive collapse. They are not a general alternative to a static solver, and the course does not pretend that watching an animation teaches you to run one.