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.

The storey effect has a property the member effect does not, and it is the reason it is so often missed.

The load being amplified is not the load doing the amplifying. A column with a pinned base and pinned top contributes nothing to the storey’s critical load and contributes its whole weight to the numerator of the ratio, so it makes every other column in the storey worse without any check on it failing. That is a frame-level statement with no member-level version, and it is why a gravity-only column is drawn on the frame model rather than designed on its own.

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.

The free body, drawn where it now is #

Everything above is a consequence of one decision about a free body, and the decision is the only place in the subject where this site’s usual instruction has to be reversed.

Take a single column of height $h$, fixed at its base, carrying a vertical load $P$ at the top and pushed sideways by a horizontal force $H$. Cut it at the base and keep the whole column. Three things act on the piece: $H$ at the top, $P$ at the top, and at the cut a shear, an axial force and a moment.

Take moments about the cut, and everything hangs on where the top of the column is drawn. On the undeformed geometry, $P$ passes directly over the cut, has no lever arm, and contributes nothing:

$$M_{\text{base}} = Hh.$$

On the deformed geometry, the top has moved sideways by $\Delta$, so $P$ has a lever arm and contributes:

$$M_{\text{base}} = Hh + P\Delta.$$

Two free bodies, identical in every respect except which picture they were drawn on, and one has a term the other does not. The second-order effect is not an additional physical phenomenon; it is the term that appears when the sums are taken on the shape the structure actually has.

The closed form follows immediately. If the column’s lateral stiffness is $k$, then $k\Delta = H + P\Delta/h$, which rearranges to

$$\Delta = \frac{H/k}{1 - P/(kh)} = \frac{\Delta_1}{1 - P/P_{cr}},$$

with $P_{cr} = kh$ the load at which the column has no lateral stiffness left. The geometric series in the earlier section and this single equilibrium equation are the same result reached from two directions: one by following the loop round, the other by never letting it start.

Which raises the obvious question about every other figure on this site. All of them sum forces on the undeformed shape, and all of them are therefore first-order. For a beam that is exact, because there is no axial force to ride on the deflection. For anything carrying compression it is an approximation whose error is always in the unsafe direction, and the amplification factor is the correction.

What the honest analysis costs #

Doing it properly is not merely more arithmetic. It removes a property of the analysis that a great deal of engineering practice quietly depends on.

Superposition is gone. A first-order analysis is linear, so the results for dead load, wind and imposed load can be computed separately and added in whatever proportions the load combinations require. A second-order analysis is not linear in the applied load — the denominator contains $P$ — so a case computed at working load cannot be scaled up to factored load, and two cases computed separately cannot be added. Every combination has to be analysed from the beginning, at its factored values, and a structure checked against thirty combinations needs thirty analyses rather than four and some arithmetic.

The factors have to be applied first. This inverts the habit of a century. In an elastic first-order world it makes no difference whether the loads are factored before the analysis or the results factored after. In a second-order world the amplification at factored load is larger than the amplification at working load, and applying the factor afterwards understates the answer — silently, and by more the closer the frame is to its critical load.

The result is an iteration, not a formula. Practical second-order analysis either iterates the geometry until the displacements stop changing, or adds a geometric stiffness matrix that depends on the axial forces, which are themselves an output. Either way the answer arrives by convergence, and a frame near its critical load converges slowly — the numerical difficulty and the physical sensitivity are the same fact appearing twice.

Nothing is conservative by inspection. In first-order analysis a stiffer member attracts more load and a designer can reason about the direction of an error. Under amplification, stiffening one part changes the critical load of the whole, which changes the amplification everywhere, and the sign of the change in a particular member’s moment is genuinely not obvious in advance.

The compensation is that the check afterwards becomes simpler. Once the second-order moments are in hand, the member is verified against its cross-section capacity directly, and the whole apparatus of effective lengths — which existed to smuggle stability into a first-order calculation — can be dispensed with. The profession has been slowly moving from the first arrangement to the second for forty years, and the reason it took forty years is the four costs above.

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 oldest well-documented instance is also the clearest, because the warning it gave was the deflection itself. The Quebec Bridge, under construction in 1907, was to be the longest cantilever span in the world. Its lower compression chords were built up from plates and angles laced together, and the lacing — the light diagonal work holding the components of the chord to act as one member — had been designed on assumptions nobody had tested at that scale.

During August the chords near the south anchor pier were found to be bowed. The measurement was taken, repeated, and found to have grown: a chord out of line by about ten millimetres a few weeks earlier was out by nearly sixty. Work continued while the correspondence went back and forth to the consulting engineer’s office in New York. On 29 August the span collapsed in fifteen seconds, killing seventy-five men.

The bow is the whole essay in one observation. A compression member with an initial imperfection carries its load off the line of the member, which bends it further, which increases the eccentricity — and the increment is not constant but proportional to how much bow there already is. A deflection that grows at an increasing rate under a load that is not increasing is the signature of an amplification approaching unity, and it is the one symptom stability failures reliably give. It is also, characteristically, a symptom that arrives in a form that reads as a fabrication tolerance problem rather than as an approach to collapse.

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.

There is a version of the loop in which nothing is added and the answer changes anyway, because the denominator moves on its own.

A creep coefficient is not usually thought of as a stability parameter, and on that pair of figures it is the only thing that differs. It decides the long-term critical load, the amplification a working column settles at, and — for a column above the threshold — the date. None of that is available to a calculation performed on the day.

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.

Only the sway part gets amplified #

There is a step in applying the factor that is easy to get wrong in both directions, and it follows from what the loop actually is.

The amplification comes from a lateral displacement carrying a vertical load off its line. So the moments it magnifies are the ones associated with that displacement — the sway moments — and the moments a frame carries with no sway at all are not part of the loop.

So a frame’s moments are split before the factor is applied:

$$M = M_{\text{non-sway}} + \frac{M_{\text{sway}}}{1 - 1/\alpha_{cr}}$$

Take a frame whose gravity analysis with the sway prevented gives 100 kNm at a joint, whose sway analysis gives a further 40, and whose elastic critical load factor is 5. The amplifier is $1/(1-0.2) = 1.25$, and the answer is $100 + 40 \times 1.25 = 150$.

Amplify everything and the answer is $140 \times 1.25 = 175$ — seventeen per cent too high, and a member sized on it. Amplify nothing and it is 140, seven per cent too low. Neither is a small error and the second is on the wrong side.

Which is why the analysis is run twice — once with the frame held against sway and once for the sway alone — on a structure that is going to be analysed by computer anyway. The two runs are not a refinement; they are the split the factor requires, and a single run producing a single set of moments cannot be corrected afterwards because the two parts are no longer separable.

The same split explains a feature of the codes that looks arbitrary: the amplifier is written against a sway critical load factor, computed from a sway mode of the whole frame, and not against the buckling load of any individual member. Those are different numbers about different modes, and a frame can be comfortable on one and marginal on the other.

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.