The load that makes itself worse

Every equilibrium equation on this site has been written on the undeformed shape. The loads were applied to a structure drawn in its original position, and the fact that the structure moves under them was ignored.

That is a first-order analysis, and it is an approximation whose error is one-directional: it always underestimates. Once a structure has moved, its own weight is acting off the line it was acting on, and the extra moment bends it further.

The loop, and the series that describes it #

The mechanism is a feedback loop with a finite gain, and following it once gives the whole answer.

A frame is pushed sideways by wind and deflects $\Delta_1$ — a first-order deflection, computed on the original geometry. Its vertical load $P$ is now acting at an eccentricity of $\Delta_1$, so it applies an extra moment $P\Delta_1$. That extra moment causes a further deflection, proportional to it — call the proportion $r$, which is the ratio of the applied load to the load at which the frame would buckle.

So the additional deflection is $r\Delta_1$, which itself produces a further $r^2\Delta_1$, and so on. The total is a geometric series:

$$\Delta = \Delta_1\left(1 + r + r^2 + \cdots\right) = \frac{\Delta_1}{1 - r}.$$

That is the amplification factor, and the whole of second-order behaviour is contained in it.

The series converges when $r < 1$ and does not when $r \geq 1$. At $r = 1$ the structure is at its buckling load, and the mathematics says the deflection is infinite — which is the same statement as the buckling load being the point where stiffness runs out, reached from a completely different direction.

The numbers, and where the trouble starts #

The factor rises slowly and then quickly.

At a tenth of the buckling load the correction is 11 percent and can be argued away. At a third it is 43 percent and cannot. Most codes draw the line at a ratio of 0.1 — an elastic critical load factor of 10 — and require an explicit second-order analysis below it.

That threshold is worth reading the other way round. A frame with a critical load factor of 10 is nowhere near buckling; nobody would call it unstable. And it already needs its first-order answers increased by a tenth. Second-order effects arrive long before instability does, which is the part that surprises people.

Two different P-deltas #

The literature distinguishes two effects with confusingly similar names, and the distinction is real.

P-Δ, “big delta”, is the storey effect. The whole floor above a column has moved sideways relative to the floor below, and the vertical load in the column now acts across that offset. It concerns the relative sway of the two ends of a member and it is a frame-level phenomenon.

P-δ, “little delta”, is the member effect. A column bows between its own two ends, and its axial load acts at that bow. It concerns the deflection of a member relative to the straight line joining its ends.

They are independent and they add. A braced frame has essentially no P-Δ, because the bracing stops the sway, and it still has P-δ in every compression member. A sway frame has both.

The remedy differs too. P-Δ is fixed by stiffening the frame — bracing, cores, moment connections. P-δ is fixed by making the member less slender, which usually means restraining it. Effective length is the packaging of exactly that.

Imperfections, which are a load #

A perfectly straight, perfectly plumb frame under vertical load only has no sideways deflection at all, so first-order analysis gives zero sway and the amplification of zero is zero. That is a correct answer to a problem nobody has.

Real frames are out of plumb — a few millimetres per storey — and real members are bowed. Those imperfections mean the vertical load is already producing sway before anything horizontal is applied.

Design handles this by converting the geometric imperfection into an equivalent horizontal force. A frame out of plumb by an angle $\phi$ carrying vertical load $P$ behaves like a plumb frame with a horizontal load $P\phi$ applied at each floor. That is the notional horizontal load, typically a few tenths of a percent of the vertical load, and it is applied in every direction that matters.

The device is neat because it converts a geometry problem into a load case, which the analysis already knows how to handle. It is also the reason a structure with no wind on it still has horizontal loads in its calculations.

Where it has mattered #

Second-order behaviour is the mechanism behind a specific and recognisable class of collapse: sudden, at a load that the strength calculations said was acceptable, with large movement immediately beforehand.

Scaffolding and falsework are the classic setting, and both are frames with barely enough restraint to count as structures at all. Both are slender, both are lightly braced, both carry large vertical loads, and both are erected with real out-of-plumb. A falsework collapse is usually a stability failure rather than a strength one, and it usually happens during a concrete pour when the load is at its highest and the bracing is at its least complete.

The other setting is the tall, slender frame. As buildings got taller and framing got lighter through the twentieth century, the elastic critical load factor of a typical frame fell, and the point at which second-order effects became routine rather than exceptional passed somewhere in the 1960s. Modern codes require the check for every frame because the answer is no longer obvious by inspection.

Where the first-order answer came from #

Every diagram elsewhere on this site was computed on the undeformed shape, and it is worth being explicit about which ones that matters for.

A pure beam has no second-order problem: the diagrams are integrals of the load and nothing amplifies them. The amplification needs an axial force and a displacement perpendicular to it, which is why columns and frames have the problem and simply supported beams do not.

Length enters twice — once by lowering the capacity and once by raising the amplification of whatever moment is present — which is why slender compression members are the ones that fail suddenly.

Where the model stops #

Elastic behaviour. The amplification factor above assumes the structure remains elastic, so the redistribution a ductile frame relies on is outside it. Once yielding begins, the stiffness falls, which lowers the buckling load, which raises the amplification — a second and nastier loop that the linear factor does not contain.

One buckling mode. The factor uses a single critical load, implying a single mode. A frame with several modes at similar loads is not described by one number.

Small rotations. Even the second-order analysis usually keeps the small-angle approximation. Genuinely large displacements need a fully nonlinear formulation.

Static loads. Nothing here concerns dynamics, and everything here concerns structures that are supposed not to move, and a structure whose stiffness has been eroded by second-order effects also has a lower natural frequency.

The figure at the top has a limitation worth naming: it is drawn as a smooth curve running to an asymptote, which suggests a structure approaching its buckling load gradually and visibly. Real structures do not get there. They yield somewhere, or a connection fails, or a brace buckles, well before the asymptote — and the collapse is sudden rather than the graceful runaway the curve depicts. The amplification factor describes the approach to a limit that is almost never reached in the manner drawn.

The ladder from here #

Later rungs: the geometric series derived properly. P-Δ and P-δ separated. The elastic critical load factor and how it is computed. Notional horizontal loads and equivalent imperfections. Amplified sway methods. Full second-order analysis, and what a geometric stiffness matrix is. Inelastic second-order behaviour. Falsework and temporary works stability. And the general theory of elastic stability, where the amplification factor turns out to be a special case of something much larger.

Engineers analysed frames on the undeformed geometry for a century and a half, and the practice was safe because structures were stocky. It stopped being safe as a matter of degree rather than of principle, and the codes caught up in the 1970s.