The roof that jumps

The stability failures in this collection have all had the same shape. A column carries load, stays straight, and at some critical value discovers that a bent shape is available at the same load; a plate ripples; a beam moves sideways. In every case two equilibrium paths meet, and the structure takes the new one.

A shallow frame does not do that. It stays symmetric from the first increment of load to the last, and what runs out is not its resistance to an alternative shape but the geometry it was using to carry load at all.

Nothing in that figure branches. There is one equilibrium path, the structure is on it throughout, and the failure is a maximum on it — a limit point. That is the second of the two ways a structure can run out of stability, and it behaves so differently from the first that the two barely deserve the same word.

Why the load falls while the movement grows #

The mechanism is entirely geometrical and can be read off the drawing. The bars carry the apex load by the vertical component of their axial force, which is $2N \sin\theta$ where $\theta$ is the angle they make with the horizontal. As the apex descends, two things happen at once:

• the bars shorten, so the compression $N$ rises;
• the angle $\theta$ falls, so the fraction of that force pointing upward falls.

Early on the first wins and the frame gets stronger. Later the second wins, and it wins decisively because $\sin\theta$ heads for zero while $N$ grows only linearly. The maximum is where the two effects balance, and it arrives at a movement of 42.5% of the rise — not at the flat position, but well before it.

The exact statement needs no approximation at all:

$$P(w) = -\frac{2N(h-w)}{L}, \qquad N = \frac{EA(L - L_0)}{L_0}, \qquad L = \sqrt{a^2 + (h-w)^2}$$

Three lines, no series expansion, and the curve above is that expression evaluated. A structure can be perfectly linear in its material and violently nonlinear in its behaviour, because the nonlinearity is in the geometry rather than in the stress–strain curve.

The cube of the rise #

The frames that snap are the shallow ones, and the penalty for shallowness is severe.

The cube can be read out of the shallow-arch expression $P_{\max} = 2EA(h/a)^3/3\sqrt3$, and it is worth pausing on because it is unusually strong. Halving the rise of a shallow frame divides its capacity by eight, which is the same brutal arithmetic a truss meets in its depth with the exponent raised by one. Nothing else in this collection is that sensitive to a proportion — span enters deflection as a fourth power but only for deflection, and the second moment of area’s cubic dependence on depth is the closest analogue.

The practical consequence is that shallow structures do not have a small stability problem. They have either an enormous one or none, and the boundary is not far from the rises architects like.

What the textbook approximation costs #

The formula quoted above is the one every treatment gives, and it is derived by assuming the frame is nearly flat — so that $L \approx a$ and the bar strain is quadratic in the rise. Since the frames that snap are nearly flat, that ought to be harmless.

That is an unusually well-behaved approximation and the figure is here to say so rather than to complain. The reason it works is that its error and its subject are governed by the same parameter: the shallower the frame, the more the approximation is entitled to its assumption and the more likely snap-through is to be the governing failure. An approximation whose error shrinks in the case it is used for is the best kind there is, and it is worth checking whether one has that property rather than assuming it.

Which failure arrives first #

The frame above reaches 133.4 kN at its limit point, and to get there its bars have to carry 776 kN of compression. On 500 mm² of steel that is 1,552 MPa, which is four times the yield stress of anything ordinary. The frame drawn in the hero figure would never snap through, because it would squash first.

That is not a flaw in the calculation; it is the calculation being asked the right question. The bar strain at the limit point is exactly $-(h/a)^2/3$, so the stress is

$$\sigma_{\text{limit}} = \frac{E}{3}\left(\frac{h}{a}\right)^2$$

and setting that equal to the yield stress gives the rise at which the two failures cross:

$$\frac{h}{a} = \sqrt{\frac{3f_y}{E}}$$

which for steel at 355 MPa is 7.1%, and contains nothing but two material properties. Below that ratio the geometry runs out first and the frame snaps; above it the material runs out first and the frame squashes. The same arithmetic for aluminium gives 8.9%, for timber about 10%, and for a glass-fibre rod nearly 20% — so which failure a shallow structure has is a property of what it is made of, even though snap-through itself contains no material property whatever.

Bifurcation and limit point, side by side #

The distinction matters because the two failures need different calculations and give different warnings.

A bifurcation is found by an eigenvalue calculation on the undeformed structure, and it needs no idea of how much the structure has deflected. A limit point cannot be found that way at all: it is a maximum on a path, so it requires tracing the path, which means a geometrically nonlinear analysis with the equilibrium equations written on the deformed shape.

That is the practical difference. An eigenvalue solver will not report a snap-through load, because there is no eigenvalue to find. A structure whose failure is a limit point and whose analysis was a buckling eigenvalue has been checked against a mode it does not have, and the check will report a comfortable factor of safety.

The structures this happens to #

Shallow arches and shallow domes, which is the obvious case and includes the lattice domes that made it famous — a reticulated shell is a set of shallow frames sharing nodes, and the relevant instability is often the snapping of a single node rather than the buckling of the whole shell.

The list past that is longer than it looks: a bowed cover plate popping the other way, a canopy over a walkway, a metal roof deck between purlins, the lid of an oil can, a bistable switch contact, a jumping toy. Everything with a slightly curved shell that can be pushed the other way is the same calculation, and in the small cases it is a feature.

It is also the mechanism of one of the most instructive structural failures of the twentieth century. The Hartford Civic Center roof came down in 1978 under snow, and the space frame’s individual compression members failed at loads well below expectation because the geometry was not what the analysis assumed — a case where the interaction between member buckling and overall geometry change was the whole story and neither calculation alone would have found it.

The path past the minimum is real #

The falling branch of the load–deflection curve, and the region where the load is negative, are not artefacts. They are equilibrium states, and the frame will sit in them — if it is held.

Under a dead load it will not be: at the limit point the applied load exceeds what the frame can carry at any nearby position, so the apex accelerates and the frame jumps to the far branch, arriving with kinetic energy that has to go somewhere. That is the dynamic snap, and it is why the phenomenon has an audible name.

Under a prescribed displacement — a jack, a screw, a testing machine with stiff control — the whole path is traversable, including the part where the frame is pulling back on whatever is pushing it. That is how the curve is measured experimentally, and it is why a testing machine’s stiffness decides whether a specimen’s post-peak behaviour can be seen at all. The same distinction decides whether a structure past first yield is observed to redistribute or observed to collapse.

The distinction is the same one an unstable post-buckling path raises: whether a structure past its maximum is dangerous depends on what is loading it, and a load that is a weight and a load that is a displacement are different boundary conditions with different answers.

The energy reading, which says why it is loud #

There is a second way to arrive at everything above, and it is the one that explains the noise.

Plot the potential energy of the frame against the apex position at a fixed load: the strain energy in the bars, which is a quartic in the movement, minus the work done by the load, which is linear. Equilibrium states are the stationary points of that function, stable ones are its minima, and the whole load–deflection path is the locus of stationary points as the load is varied.

Under a small load the function has one minimum, near the undeformed position. As the load rises a second minimum appears — the inverted state — separated from the first by a maximum, which is the unstable middle branch. At the limit point the first minimum and the maximum merge and annihilate, leaving only the far one, and there is nowhere for the frame to be except on the other side.

The energy released in getting there is the area between the constant-load line and the equilibrium path, and it goes into motion. A snap-through is not a failure that happens slowly and then stops — it is a structure falling from one equilibrium to another with the difference in energy carried as kinetic, which it then has to dissipate by vibrating, by yielding, or by breaking something.

Two states separated by an energy barrier is also the general description of bistability, and it links this essay to a much wider family: the buckled strip that clicks, the popper toy, the transition in a phase diagram. What structural engineering contributes to that family is the observation that the barrier can be crossed by a load rather than by a push, and that the load which does it is computable in closed form.

What the picture cannot show #

The bars are pin-ended and cannot buckle. A real strut at 776 kN of compression would be checked for Euler buckling long before it squashed, and the interaction between member buckling and frame snap-through is a genuinely coupled problem that neither figure here addresses.

The frame is perfectly symmetric. An asymmetric imperfection introduces a bifurcation into what was a pure limit-point problem, and the two can interact so that the actual failure load is below both — the coupled-instability case, which is the worst-behaved thing in this whole field.

The bars are elastic all the way to the limit point. For the 6% frame that is true; for anything deeper it is not, and a frame whose bars have started to yield has a lower and much less predictable maximum, because the falling stiffness of the material and the falling efficiency of the geometry are then working together. The honest version of the crossover in the section above is therefore a band rather than a line, and the interaction between two limits is the general shape of it.

Nothing here has a time in it. The jump is drawn as an instantaneous horizontal line, and the real event is a dynamic one in which the frame overshoots the far branch, oscillates and settles. The energy released is the area between the load line and the path, and it is what makes snapping loud.

Where the ladder goes #

The first rung is the imperfect case, which is where the practical numbers live: a frame with an initial asymmetry has a lower maximum than a perfect one and the deficit follows a power law with an exponent below one — the same imperfection sensitivity that makes shells the least predictable structures there are.

The second is the shell itself. A reticulated dome has thousands of nodes, each of which is a shallow frame, and its failure is a competition between one node snapping, a patch of nodes snapping together, and the whole dome buckling. Which one governs is decided by proportions, and the answer moves with the loading pattern rather than its magnitude.

The third is the useful direction. Bistability is a property to design for as often as against: a structure with two stable states and a controlled path between them is a switch, a clip, a deployable, or a shelter that erects itself. The arithmetic is identical and only the sign of the intention is different.