The load that is really a lean

A frame is drawn plumb and built leaning. The lean is small — a three-hundredth of the height is a typical erection tolerance, which on a 3.6 m storey is twelve millimetres — and it is not an error, because there is no such thing as a plumb building. Steel arrives with a mill tolerance on its straightness, columns are set with a plumb line against a rule, welds shrink, foundations settle differentially, and the finished frame is out of upright by an amount somebody wrote into a specification as acceptable.

The structural consequence of that lean is a horizontal force, and the size of it is fixed by nothing more than a free body.

Which free body produced the number #

One storey, cut top and bottom. Its columns carry a vertical load $N$ down to the storey below, and they are out of plumb by an angle $\varphi$. Take moments about the base of the storey.

The vertical load’s line of action arrives at the top of the storey displaced sideways by $\varphi h$ relative to where it leaves at the bottom. So it delivers a moment $N \varphi h$ about the storey base, which nothing in a plumb analysis contains. And $N\varphi h$ is exactly the moment of a horizontal force $\varphi N$ applied at the top of the storey.

That is the whole derivation. It is a statics identity between a geometry and a force, and it holds regardless of what the frame is made of, how stiff it is, or whether it is braced. Two free bodies produce the same moment, so a designer may draw whichever is easier — and the force is easier, because every analysis program in existence accepts a horizontal load and none of them accepts “the building is not quite straight”.

The identity is the reason for the name. The force is notional: nothing pushes on the building. It is the leaning of the building, written down.

The size of it, which is proportional to gravity #

Because the substitution is $\varphi N$, the notional force is proportional to the vertical load, and a column’s vertical load is a running total down the height.

Take a twelve-storey frame whose columns each carry 42 m² of floor at 7 kN/m². Each column collects 294 kN a floor and 3,528 kN at the base; with forty such columns the building weighs about 141,000 kN. At an out-of-plumb of 1/200 the equivalent horizontal force is 706 kN, applied as 59 kN at each floor.

That is a real load applied to a real bracing system, and it is worth comparing with the wind on the same building. A 48 m by 42 m elevation at a net 1.17 kN/m² gives about 2,400 kN — three times as much. Every instinct then says the wind governs and the lean is a detail.

The combination the lean governs, which contains nothing else #

The instinct is wrong for a reason that has nothing to do with the size of the two numbers.

Wind appears in some load combinations. The notional force appears in all of them, because it is proportional to the gravity load and the gravity load is always there. In particular it appears in the combination where the imposed load is at its maximum and the wind is absent — the one every project treats as the vertical case, and the one in which a bracing system is not usually thought about at all.

In that combination the horizontal load is 706 kN and nothing else. The braced bay carries it, its diagonals carry it, its connections carry it, and the foundations under it carry an uplift they were not asked about. None of that appears in a model to which no horizontal load was applied.

This is the same structural mistake as designing for the wind and forgetting the storey is free to sway, and it has the same tell: a set of results in which everything is checked and one load case was never run.

Amplification, which applies to it like anything else #

A notional force is a horizontal load, so a second-order analysis amplifies it exactly as it amplifies the wind — and the amplification is not small in the frames where the notional force matters, because those are the flexible ones.

An imperfect column has no critical load to reach at all. Its deflection grows from the first increment, is amplified by that curve, and runs away as the load approaches a value it therefore never attains — a bifurcation replaced by a divergence.

The clean way to see why is the column that was never straight. A perfect column sits on the load axis with no deflection until it bifurcates; a bowed one has a deflection from the first increment, amplified by $1/(1 - P/P_{cr})$, and never reaches the Euler load at all. A leaning frame is that column one scale up: the lean is the initial imperfection, the sway is the deflection, and the elastic critical load ratio $\alpha_{cr}$ plays the part of $P/P_{cr}$.

At $\alpha_{cr} = 5$ — a perfectly ordinary value for a braced multi-storey frame — the amplifier is 1.25, so the 706 kN is really 883. At $\alpha_{cr} = 3$, which is where most standards insist on a full second-order analysis, it is 1.5.

Note what has and has not happened here. The amplification is a second-order effect and the notional force is a first-order substitution for a geometry. They are different things and they compound: the imperfection creates a sway, and the vertical load acting on that sway makes it larger.

The same substitution, one scale down #

A frame is out of plumb; a member is out of straight. The two imperfections are the same idea at different scales, and they get combined rather than added, because a whole frame is unlikely to lean and have every member bowed the same way at once.

The member version is starker than the frame version because the ratio is so large. For a rectangular section, bending stress over axial stress is exactly $6y/d$, so a bow of one section depth costs six times the axial stress. Compression members are slender and their bows are measured in fractions of a depth, but the multiplier is what makes an initial bow of L/1000 into a design case rather than a curiosity — and it is why the shape of the buckling curve is a statement about tolerances rather than about steel.

The two imperfections also do different jobs in a check. A sway imperfection loads the bracing system; a bow imperfection loads the member. A frame can be perfectly braced and contain a column that fails on its own bow.

Where the force is applied, and in which direction #

Two decisions remain, and neither of them is a calculation.

Direction. The lean has no natural sign, so the force is applied in whichever direction is worst, in each orthogonal plan direction independently, and with the sign that adds to the wind rather than relieving it. That is a search over cases and not a load case, which is the second time this argument produces one.

Where it acts. The force is $\varphi$ times the vertical load carried by that storey, applied at that storey’s floor and taken out at the one below. It is self-equilibrating over the whole building only in the sense that the ground eventually resists it — the base shear it produces is $\varphi$ times the total weight, and it is a real base shear that a foundation resists like any other.

That last observation is a small piece of good news. Because the notional force is proportional to the weight and overturning is resisted by the weight, the stability ratio against a notional force alone is $1/(2\varphi \cdot H/B)$ — a number with no load in it, and one that a squat building passes without noticing.

Twelve storeys do not lean twelve times #

The arithmetic above applied one out-of-plumb to the whole building, and it is worth doing properly, because the correction is a factor of two and it is the difference between a load case that governs and one that does not.

Treating $\varphi$ as uniform says every storey leans the same way by the same amount, which is a building bent into a single straight stick. That is not how a frame is erected. Each storey is plumbed against the one below it, so an error at one level is corrected at the next, and the leans alternate in sign as often as not. The same argument applies across the plan: forty columns do not all lean the same way, and the horizontal force delivered to the bracing is the sum of forty errors of random sign rather than forty times one of them.

Both corrections are statistical and both are written into the standards as geometry. The height factor is $\alpha_h = 2/\sqrt{h}$, bounded below at two-thirds; the column factor is $\alpha_m = \sqrt{0.5(1 + 1/m)}$, which tends to $1/\sqrt2$ for many columns. Neither is derived from anything mechanical — they are fits to the observation that errors of independent sign accumulate as a square root rather than linearly.

On the twelve-storey frame above, $h = 43.2$ m gives $\alpha_h = 0.304$, clamped to 0.667, and forty columns give $\alpha_m = 0.716$. So

$$\varphi = \frac{1}{200} \times 0.667 \times 0.716 = \frac{1}{419}$$

and the equivalent horizontal force falls from 706 kN to 337 kN — 14% of the wind rather than 29%.

That does not make the load case go away, and it is worth being clear about why. The reduction applies to the global force delivered to the bracing system, which is the sum of many independent errors. It does not apply to a single column being checked for its own lean, nor to a small frame with two columns and one storey, where $\alpha_h = 1$ and $\alpha_m = 0.866$ and the full value very nearly stands. The buildings the reduction helps are the large ones, and the buildings the notional force governs are the small heavy ones — so the correction and the cases it matters in are almost disjoint.

Which frames it actually decides #

The notional force governs a design when three things line up, and the combination is common enough to be worth naming.

A heavy structure with a small elevation. Racking, water tanks, plant platforms, the internal frames of a car park: high gravity load per square metre of façade, so $\varphi N$ is large and the wind is small.

A sheltered or internal frame. A braced core inside a building whose façade delivers its wind to a different system carries almost no wind and all of the gravity, so its horizontal demand is entirely notional.

A very flexible frame. Low $\alpha_{cr}$ amplifies the notional force more than it amplifies anything else, because the notional force is itself proportional to the vertical load that is doing the amplifying.

The third of those is worth dwelling on because it is a feedback rather than a coincidence. A frame’s $\alpha_{cr}$ falls as its vertical load rises. The notional force rises with the same vertical load. So the two effects multiply, and the horizontal demand on a heavily loaded flexible frame grows faster than its load does.

What it looks like when it has been missed #

The failure this argument protects against is quiet, and it has a recognisable shape in a set of results.

A bracing system sized for wind is usually generous, so the diagonals themselves are rarely the problem. What is not generous is everything the wind combination happened to make easy. A holding-down bolt group is sized on the worst combination that was run; if the gravity-only case was run with no horizontal load, the tension in the windward foundation of the braced bay is a number nobody computed. A transfer beam carrying a braced bay across an opening is checked for the horizontal shear it was given. A connection between the bracing and the floor plate is sized on the load path that was modelled.

None of those is wrong in the model. Each of them is wrong about the building, and the error is invisible because every check that was made passes. That is the characteristic signature of a missing load case rather than a mistaken one — nothing to find by reading the output, because the output is internally consistent and complete.

There is a second, subtler version on a structure with no bracing system at all. A frame designed as a set of gravity columns and simply supported beams, with the stability provided by a shear wall elsewhere, is drawn with pinned connections and it has none. The columns lean, the leaning delivers a horizontal force into the floor plate, and the floor plate delivers it to the wall — a diaphragm action that only exists if somebody drew it. Where the floor is precast planks with no topping, or a steel deck with a movement joint through it, that path is absent and the notional force has nowhere to go except into the connections that were called pins.

The check is one line long and worth making on any project: apply a horizontal force of a two-hundredth of the total weight in each direction, with no wind, and see what changes.

Where the model stops #

The out-of-plumb is a specification, not a measurement. Nothing in this calculation knows how far the building actually leans. It uses the value the erection tolerance permits, which is a bound on a quantity nobody surveyed, and a building erected badly is outside the calculation entirely rather than merely closer to its limit.

Leans do not add up the way the arithmetic assumes. Applying $\varphi$ to every storey in the same direction treats the whole building as one straight leaning stick. Real out-of-plumb alternates: a storey out one way, the next out the other, because each is corrected against the one below. Standards handle this with a reduction factor that falls with the number of storeys and the number of columns, which is a statistical statement dressed as a geometric one.

A settlement is not an imperfection. A frame tilted by differential foundation movement has a lean that arrived after erection, grows with time, and is not covered by an erection tolerance. It produces exactly the same $\varphi N$ and is checked by nobody, because the settlement that matters is a difference and the tilt of a whole building is not usually measured at all.

What the picture cannot show #

The amplification curve at the top of this essay is drawn for a structure with one deflected shape and one critical load. A real building has many, and the imperfection has to be applied in the shape of the mode it is amplifying for the substitution to be exact. Applying a uniform lean to a building whose critical mode is a soft storey near the base under-represents the imperfection where it matters and over-represents it everywhere else.

Nor does the picture show the thing the whole substitution is for. There is no drawing anywhere in a project of the building leaning, because the drawings are of the building as designed and the lean is a property of the building as built. The notional force exists precisely so that a quantity nobody will ever draw can be carried into an analysis that only accepts loads.

The generalisation #

The habit worth carrying is about the difference between a load and a load case.

Everything else in this field is a force somebody can point at. A notional horizontal force is a model of a geometry, converted into a force because the analysis will only take forces — and once it is in the model it is indistinguishable from a real one, which is exactly the property that makes it useful and exactly the property that makes it easy to forget.

The move generalises, and it generalises to the thing that resists the lean as well as to the lean itself. What restrains an imperfection is a stiffness rather than a strength, and a restraint’s stiffness reaches a knee past which more of it buys nothing at all. A brace that need not be strong is doing the same job: what it restrains is not a load but a tendency to move, and the force in it is whatever the tendency produces. Prestress is the same conversion in reverse — a state of the structure, entered into the analysis as a set of applied forces. In every case the honest description is that a fact about the structure has been rewritten as a force, and that the arithmetic is exact while the fact is an assumption.

Which leaves one thing to carry into any check. When a horizontal load appears in a model, ask what it is a substitution for. If the answer is “the wind”, it belongs in the wind combination. If the answer is “the building is not straight”, it belongs in every one of them — and most importantly in the one that has no other horizontal load in it at all.