Counted, not checked

A load that makes itself worse does so through one number: the ratio of the load applied to the load at which the structure buckles.

Everything in that curve is about the structure’s buckling load, and the question this page is about is whose load appears in the numerator.

The column that cannot buckle and still matters #

Consider a storey with one braced column and several that are pinned top and bottom, carrying gravity load and nothing else.

A pinned-pinned column with no lateral restraint has, on its own, no way of resisting sway at all. It is not a stable structure; it is a mechanism, and the question “at what load does it buckle” has no answer for it. Nothing about it can be checked in isolation.

It nevertheless carries load, and when the storey sways it goes with it — and its weight, displaced sideways, produces an overturning moment that something else has to resist.

The load doing the amplifying is the storey’s, and the load being amplified is the braced column’s. Those are different quantities, they are computed by different people at different times, and the second-order check has to bring them together.

Which free body produced the number #

The free body is the whole storey, cut horizontally just below the floor above and just above the floor below.

Crossing the top cut are the storey shear and the gravity loads of everything above; crossing the bottom cut are the reactions of every column. Displace the free body sideways by $\Delta$ and take moments about the base.

Every gravity load $P_i$ contributes $P_i\Delta$ to the overturning, whatever column it stands on and whether that column has any stiffness or not. The restoring moment comes from the lateral stiffness $K$ of the storey, which is the sum of the stiffnesses of the columns that have any — and a pinned-pinned column contributes exactly zero to it.

So the storey’s equilibrium is

$$K\Delta = H h + \left(\sum P_i\right)\Delta$$

and the critical condition is $\sum P_i = Kh$, with the sum over every column. The free body is a storey and not a column, and that is the whole content of the subject: a stability check on a member is a check on something that is not the structure.

That also explains why the amplifier is $1/(1 - \sum P/P_{cr})$ with a summed numerator and a fixed denominator. The denominator belongs to the bracing; the numerator belongs to the floor plan.

Two amplifications, one name #

The literature uses one symbol for two effects that live at different scales, and separating them is worth doing once.

P-Δ is the storey effect just described: the whole floor moves sideways relative to the one below, and every gravity load on it acts through that displacement. It is a global amplification, its denominator is a storey property, and it affects the frame’s moments and the bracing’s forces.

P-δ is the member effect: a column bows between its own ends, and its own axial load acts through that bow. It is a local amplification, its denominator is the member’s own Euler load, and it affects the moment in the middle of that member only.

The two are independent and they multiply rather than adding. A column in a swaying frame has its end moments amplified by the storey’s P-Δ and its mid-height moment amplified by its own P-δ, and a member can be comfortable on one and governed by the other.

A leaning column has no P-Δ of its own and a full P-δ. It contributes to the storey’s amplification without experiencing it in any way its own analysis would show — and its own bow is amplified by its own axial load in the ordinary way.

The effective length that is off the chart #

The same fact expressed as an effective length is the form most designers meet, and it produces numbers that look like mistakes.

$K$ rises as the square root of the load ratio, exactly, because the critical load falls in proportion to the load being shared and $K \propto 1/\sqrt{P_{cr}}$.

Two readings of that figure are worth carrying.

The braced column’s effective length depends on how much load other columns carry. It is not a property of the column, of its section, or of its own end conditions — which is what an alignment chart implicitly assumes and is why the answer leaves the chart.

And the leaning columns are at 2.60 rather than 1.0. A designer taking them at $K = 1$ for pinned ends is under-estimating their own slenderness by a factor of 2.6, on a member they believed needed no stability check at all. The column that leans on its neighbours is being carried in a sense that is quantitative rather than metaphorical.

How large the correction is, in a real storey #

It is worth putting the effect on a floor plan, because its size depends on the arrangement rather than on the loads.

A typical braced steel frame has bracing in two bays each way and columns everywhere else. On a 6 × 6 grid of 36 columns, perhaps 8 are in braced bays and 28 are not. If the load is roughly uniform, 78 per cent of the storey’s gravity load stands on columns with no lateral stiffness whatever.

The storey’s critical load is set by the 8, and the demand is all 36. So the ratio $\sum P/P_{cr}$ is 4.5 times what a calculation using only the braced columns would give, and the amplifier goes from a comfortable 1.05 to something between 1.2 and 1.6 depending on how the bracing was sized.

That is why the rule of thumb is expressed as a threshold on the ratio rather than on the amplifier. Below about 0.1 — an amplifier of 1.1 — the effect can be neglected; above it, it cannot. The whole design question is which side of 0.1 the storey is on, and a calculation that leaves out three-quarters of the gravity load will answer it wrongly with great confidence.

Why it is counted rather than checked #

The practical resolution is the one in this page’s title, and it is worth stating as a procedure.

A gravity-only column is put on the frame model so that its load is included in the storey’s $\sum P$, and it is not given a stability check of its own. Its section is chosen for its axial force and its own P-δ, and its lateral behaviour is inherited from the storey.

That works because it is the sum that matters. The storey’s critical load is a property of the bracing; the demand on it is the total gravity load; and where that load happens to stand is irrelevant to the storey’s equilibrium, though not to the leaner’s own bow.

Getting this wrong has a characteristic signature. A frame analysed with only its braced bays present, and its gravity columns designed separately by hand, will report a storey amplifier computed from a fraction of the real load — 30 per cent of it in the case above — and an amplifier of 1.13 where the truth is 1.57. Every gate, every check and every calculation is then internally consistent and about a different building.

What the bracing is actually being asked for #

Turning the storey’s equilibrium round gives the requirement on the bracing, and it is a stiffness rather than a strength — which is the same shape as a brace that need not be strong applied at the scale of a floor.

The condition $\sum P < Kh$ is a demand on $K$: the storey needs lateral stiffness proportional to the gravity load it carries, whatever horizontal load it is designed for. A storey with no wind on it at all still needs bracing, and the amount is set by the weight standing on it.

That is not how bracing is usually sized. It is sized for wind, or for a notional load, and its stiffness is whatever falls out. The stability requirement is then checked afterwards as an amplifier, which works, and hides the fact that the two demands are unrelated: a heavy, sheltered building can be governed by stability and a light, exposed one by wind, and neither calculation mentions the other’s driver.

There is a corollary about what counts as bracing. Anything with lateral stiffness contributes to $K$ — a shear wall, a braced bay, a moment frame, a stiff cladding system, an infill panel. Anything without contributes nothing, however substantial it looks. A heavy concrete column pinned top and bottom is a leaner, and the reason is a detail at each end rather than anything about the member.

Where a lean comes from when nothing is leaning #

There is a companion effect that is often confused with this one, and separating them keeps both honest.

A notional horizontal load is a lean rather than a wind: every frame is built slightly out of plumb, so its gravity load has a horizontal component from the day it is erected. That is an initial imperfection, it is proportional to the gravity load, and it produces a first-order sway.

The amplification on this page then acts on that sway. So the two compound: an out-of-plumb frame has a first-order lean of $\phi \sum P$, and the second-order effect multiplies it by $1/(1-\sum P/P_{cr})$.

Both terms contain $\sum P$, which is why a heavily loaded, lightly braced storey is in trouble twice over — and why the notional load and the amplifier are usually specified together, as a pair, rather than as separate provisions.

The one that arrives late #

Everything so far is instantaneous. A concrete column adds a clock.

That is the least comfortable figure in this subject. A column can pass every check on the day it is loaded and fail eleven weeks later, with nothing added and nothing having moved but its own modulus.

The mechanism is entirely in the denominator: $P_{cr}$ contains $E$, creep reduces the effective $E$ by a factor of $1+\varphi$, and a structure whose demand was a safe fraction of its capacity finds the capacity has come down to meet it. The load ratio that is safe on the day is $1/(1+\varphi)$ times the ratio that is safe in the long term, and $\varphi$ is 2.5 for an ordinary concrete.

A check that takes one line #

Given how easy the omission is, the useful defence is a single arithmetic check that can be done on any storey without an analysis.

Compute the storey’s total gravity load at the ultimate limit state, $\sum P$. Compute the storey’s lateral stiffness $K$ from the bracing alone. Form $\sum P h / K h^2$ — which is $\sum P / P_{cr}$ — and read it against 0.1.

Two things make that check worth doing by hand even when a program has done it.

A program computes $P_{cr}$ from the model it was given. If the gravity columns were left out of the model, or were modelled as pinned struts with their loads applied to the bracing instead of to their own bases, the number is right about the wrong structure.

And the answer is sensitive to a quantity nobody validates. $K$ is a series combination of several stiffnesses, and the one that usually dominates is a connection rather than a member — a gusset, a bolt group in slip, a base plate. A bracing system is only as stiff as the joints in it, and a stiffness overestimated by two puts the storey on the wrong side of 0.1.

There is a reason the square root appears rather than a linear law, and it is worth a sentence because it decides how bad the omission can get. The critical load falls in inverse proportion to the load ratio, and $K \propto 1/\sqrt{P_{cr}}$ — so quadrupling the leaning load doubles the effective length rather than quadrupling it. The penalty is real and it is bounded, which is why a storey with a great deal of leaning load is uncomfortable rather than catastrophic, and why the effect went unnoticed for as long as it did.

What to carry away #

The critical load belongs to the storey and the demand is the whole floor’s gravity load. A member with no lateral stiffness contributes to the demand and nothing to the capacity.

Count a gravity-only column; do not check it. Its load must be in $\sum P$, and its own stability is a question the storey answers.

Separate P-Δ from P-δ. One is a storey effect on the sway moments, the other a member effect on the mid-height moment, and they multiply.

And in concrete, check the long term. The buckling load falls with the modulus, the amplifier grows with nothing added, and the failure has a date rather than a load.

The same argument in a different structure #

The pattern — a member that carries load, supplies no stiffness, and lowers everyone else’s capacity — is not peculiar to building frames.

A falsework tower carrying a concrete pour has legs that are braced and legs that are not, and the unbraced ones are leaners. Falsework collapses are disproportionately stability failures, and the reason is usually that the bracing was sized for the wind and the gravity load was not counted into the stability check.

A row of props under a slab is the same arrangement with no bracing at all: each prop is pinned at both ends, the whole set is a mechanism, and what actually stops it moving is friction at the heads and the slab’s own stiffness — neither of which appears anywhere.

And a guyed mast’s own shaft carries the vertical component of every guy’s tension while the guys supply the stiffness. The shaft is the leaner and the guys are the bracing, and a mast held by something that goes soft is a storey whose $K$ falls as the load rises.

The common structure is that stiffness and load are supplied by different members, and every check that treats a member in isolation implicitly assumes they are supplied by the same one.

Where the model stops #

The storey is treated as rigid in its floor plane. A flexible diaphragm does not deliver every column’s sway to the bracing, and the columns furthest from a braced bay move more than the calculation says.

One mode is assumed. A multi-storey frame has a critical mode that may not be a single-storey sway, and a storey-by-storey check can miss it — which is the storey’s version of the same trap this page is about, one level up.

The bracing’s stiffness is taken as a number. It is the series combination of diagonals, gussets, floor connections and foundations, and the smallest of those usually decides it.

Everything is elastic. A frame at its design load has yielded somewhere, its stiffness has fallen, and the critical load fell with it — the amplification and the plasticity arrive together.

And the imperfection is treated as a lean. A real frame is out of plumb by different amounts in different storeys and in different directions, and the pattern that is worst for the sway is not the one the notional load describes.

Second-order effects arrive through three different doors and only one of them is a sway. A column leaning on its neighbours is the storey version; a base that is not as pinned as the model says changes the amplification without changing any load; and a load that cannot be buckled under is the case where the whole amplification argument does not apply at all.

The ladder from here #

Later rungs on this anchor: the geometric series derived properly, and where the amplifier’s form comes from. The elastic critical load factor computed rather than assumed, and what a geometric stiffness matrix is. Amplified sway methods against direct second-order analysis, and the range over which they agree. Multi-storey critical modes, and why a storey-by-storey check can miss them. Inelastic second-order behaviour, where the stiffness falls during the event that is loading it. Falsework and temporary works, where the bracing is provisional and the gravity load is not. And the general theory of elastic stability, in which the amplification factor is a special case of something much larger.

The leaning column is not an obscure case. It is most of the columns in most buildings: a braced frame has a few bays of bracing and dozens of gravity columns, and the whole of the storey’s stability rests on the few while the demand comes from all of them. That the members needing no check are the ones setting the check is the sort of inversion that only shows up when the free body is drawn round the right thing.