The frame is a girder stood on end

A Vierendeel girder carries its shear round the corners rather than across them, and the rung below this one establishes what that costs. This one is about where else the same object is.

The hero is a six-panel girder, 21 m long and 6,000 mm deep, under 300 kN. Each panel’s 150 kN of shear is carried as chord bending; the local moment is 131.3 kN·m, the chord’s axial force at the same point is 263 kN, and the girder deflects 109.09 mm against 6.53 for the same members triangulated.

Now stand it on end. The span becomes the height of a six-storey building, the depth becomes a 6 m bay, the chords become columns, the verticals become beams, and the 300 kN becomes wind. Nothing in the arithmetic changes — it is the same rigid-jointed rectangular grid with no diagonal in it, and this is what an unbraced building frame is.

The mapping, term by term #

The identification is exact for the panel mechanics and inexact for the boundary conditions, and it is worth separating the two before using it.

What carries across exactly. A panel’s shear divides between the two chords in proportion to their stiffnesses. Each chord bends in double curvature across the panel with a point of contraflexure inside it. The local moment is the chord’s share of the panel shear times half the panel length, giving $V\ell/4$ when the two chords are equal. The chords also carry an axial force from the global moment, which is a couple over the depth. All four of those are storey-shear, column, storey-height and overturning statements in a building, word for word.

What does not carry across. A girder is supported at both ends and a frame is a cantilever off its foundations, so the shear profile is different — constant along a girder under a point load, and increasing downwards in a building. And a frame’s base is a fixity condition with no counterpart in a simply supported girder.

So the mapping transfers panel behaviour and not global behaviour, which is precisely the division that matters, because a sway frame’s deflection is almost entirely accumulated panel shear deformation rather than global bending.

The panel length is the variable and it enters twice. The moment goes as $\ell$, and the panel’s own shear deformation goes as $\ell^3/EI$ — so halving the panel length halves the moment and reduces its contribution to the deflection eightfold, against only twice as many panels to accumulate.

Which is the frame result stated properly: a building with more, narrower bays sways less than one with fewer wide ones, at the same total column steel. That is the opposite of the trussed case, where more panels means more joints and no benefit at all.

Which free body produced the panel moment #

The $V\ell/4$ is worth deriving once, because it is the sentence the whole mapping rests on and it is three lines.

Cut the girder vertically through the middle of one panel — through both chords, between two verticals. Crossing the two cuts is the panel shear $V$, divided between the chords in proportion to their stiffnesses, and a bending moment in each chord. Cut at the panel’s middle and those moments are zero, because that is where the double curvature reverses.

Now take the piece of one chord between that cut and the next joint, half a panel away. It carries its share of the shear, $V/2$ for equal chords, over a length $\ell/2$, with no moment at the cut end. The moment at the joint is therefore $(V/2)(\ell/2) = V\ell/4$.

The depth is not in that derivation anywhere, which is the whole of the previous section stated as an absence. Neither is the span, nor the number of panels except through $\ell$, nor the chord’s own properties.

The same three lines in a building read: the storey shear divides between the columns, each column bends in double curvature over the storey height with its zero at mid-height, and the column moment at each floor is the column’s share of the storey shear times half the storey height. That is the portal method in one sentence, and it is not an approximation of anything — it is exact statics on a structure whose points of contraflexure have been assumed.

Which is why the assumption is the only thing worth arguing about, and why it gets a figure of its own further down. Everything else in the method is equilibrium, and equilibrium does not have errors.

Depth is the wrong lever, and it is worse than useless #

That figure is the one that reads as a mistake and is not.

In a truss, depth is everything: the chord force is $M/d$, so a deeper truss has smaller forces and smaller extensions, and the deflection falls as $1/d$. In a Vierendeel, the axial part of the deflection does exactly that — and it is two per cent of the total.

The other ninety-eight per cent is chord bending, which depends on the panel length, the chord’s second moment of area and the panel shear. The depth appears in none of them. What the depth does do is lengthen the verticals, which makes them more flexible, which lets the panel rack more.

Read as a building, that says something that sounds wrong and is right: making the building wider does not stiffen its unbraced frames. The overturning is shared over a longer lever arm, so the column axial forces fall — but the sway is a storey-by-storey racking quantity, and racking does not care how wide the building is. A twelve-metre-deep frame and a thirty-metre-deep one of the same columns and the same storey height sway by the same amount.

The two flexibilities, and why one of them wins #

The deflection of a Vierendeel girder is the sum of two terms, and the ratio between them is the number every figure on this page prints.

The axial term is the chords stretching and shortening under the global moment: $\sum F f L / EA$, the ordinary truss deflection, and it is what the triangulated comparison in each caption is almost entirely made of.

The bending term is the panels racking: each panel deforms as a portal, by an amount proportional to $V\ell^3/EI$ summed over the chords.

On the hero the first is about 2 mm and the second about 107. The structure has a shear flexibility fifty times its bending flexibility, which is the same series-of-two-flexibilities arithmetic a built-up column has with the proportions inverted: there the shear term was a ten per cent correction, and here it is the answer.

Two consequences follow and both are practical.

A Vierendeel behaves like a shear beam, not a bending beam. Its deflected shape is straight-sided rather than curved, its slope is largest where the shear is largest rather than where the moment is, and adding depth does nothing — all of which are properties of a shear-dominated structure and none of which a designer expects from something drawn as a girder.

And that is exactly what a frame’s drift profile looks like. A moment frame’s storey drift is roughly constant up a uniform building rather than tapering the way a cantilever’s would, and buildings that combine a frame with a core show the characteristic S-shaped interaction because the two have deflected shapes of opposite curvature. The frame’s shape is a shear beam’s because it is a Vierendeel, and this essay’s mapping is where that comes from.

The currency is the second moment #

That is the practical instruction, and it is the reverse of a truss’s.

A truss is sized by area, because its members are axial and its deflection is $\sum F f L / EA$. A Vierendeel is sized by second moment, because its members bend. The two properties are not proportional across a section table — a deep light section has a large $I$ and a small $A$, and it is exactly the wrong member for a truss and exactly the right one for a frame.

So the section a designer reaches for changes with the system, and the reason is visible in the two comparison numbers on every figure here. The Vierendeel deflects 109 mm where the truss deflects 6.5. A factor of seventeen is not a correction to be absorbed by making members heavier; it is a statement that the load is being carried by a different property.

What the identification is worth #

An identification between two structures is only useful if results travel along it, so here is what does.

Everything about the panel travels. The moment $V\ell/4$, the contraflexure at the panel centre, the shear-dominated deflection, the insensitivity to depth, and the sensitivity to panel length cubed. A result established on either object holds on the other.

Everything about the ends does not. The girder’s supports and the frame’s foundations are different boundary conditions, and the girder’s point load and the building’s distributed wind are different shear profiles.

And the sizing rules travel with a warning. A frame is sized for strength by the column moment $V\ell/4$ plus the axial force, and for stiffness by the drift — and the two want different things from a section. The strength check is nearly indifferent to the section, since the moment is a statics quantity; the drift check wants $I$ and only $I$. A frame is almost always drift-governed, which is another way of saying the Vierendeel penalty of seventeen is not something a strength check can see.

There is one more thing the identification buys, and it is the reason the rung below this one is worth having read. A Vierendeel girder is a thing a person can draw the whole of, put numbers on, and check by hand. An eight-storey frame is not. The girder is the frame with the confusing parts removed, and every result on this page was easier to obtain on it and is equally true of the building.

That is the ordinary use of an idealisation, and it is worth naming because it runs the opposite way to most: usually a model is simpler than the thing it models. Here the two objects are equally complicated and one of them is merely easier to look at.

Where the contraflexure actually is #

The whole of the panel arithmetic rests on one assumption: the chord’s moment passes through zero at the middle of its panel. It is exact for a symmetric girder, and a building frame is not symmetric top to bottom.

Right on average and wrong in every storey is the honest description, and it is why the hand methods that assume the answer survive: the assumption is exactly the sort of error that averages out over a frame and does not average out in the storey a designer is checking.

The two ends are where it breaks, and both are visible in the mapping. The base is the frame’s boundary condition — the thing the girder analogy does not carry across — and the roof is the free end where a column has a joint above it that is doing nothing. Those are precisely the panels of a Vierendeel girder nearest its supports and nearest its point load, which is where the girder’s own panel arithmetic is least accurate too.

Why anybody builds one anyway #

Everything above is a list of penalties, and unbraced frames are the commonest structural system there is, so the argument is obviously incomplete.

The panel is empty and that is the product. A diagonal across a bay is a wall the building cannot use — no door, no corridor, no window, no plant route. A Vierendeel panel is clear, and in a building the clear rectangle is what is being sold.

The penalty is on stiffness, not on strength. The girder that deflects seventeen times as far has chord forces of the same order as the truss’s, and its strength check is not seventeen times worse. What it fails is a drift limit, and a drift limit is a serviceability criterion whose number is a judgement — stiffness is not strength is the whole of why the system is viable.

And the frame does not have to act alone. A core, a braced bay in one direction, or a shear wall at a lift shaft takes the drift, and the frame carries gravity and whatever lateral load its own stiffness attracts. How a tall building stands still is a description of that division, and the frame’s contribution to it is governed by exactly the panel arithmetic above.

So the design position is narrow and defensible. Use the frame where the opening is worth the drift, size it by second moment rather than by area, and make the bays as narrow as the plan allows — which is a set of instructions that follows from a picture of a girder that nobody would think to consult.

What to carry away #

A sway frame is a Vierendeel girder on end, and the identification is exact for panel behaviour: storey shear, column double curvature, contraflexure inside the panel, and an axial couple from the overturning.

More panels is the lever that works. Four to ten takes the deflection from 166 mm to 66, because the panel length enters the moment linearly and the deflection cubically.

Depth does almost nothing and can do harm. The chord axial force falls with it and the sway does not, because 98 per cent of the movement is bending that the depth does not appear in.

And the currency is $I$, not $A$. Doubling one halves the answer; the other is holding up two per cent of it.

Where the model stops #

The joints are rigid points. A real frame’s panel zone is a piece of column web that shears, and on a deep-beam frame it can be a third of the storey drift — an entire flexibility this model puts at zero. And a connection that is neither pinned nor rigid puts another one in series with it, which the girder’s rigid corners contain none of.

The chords are prismatic and continuous. A building’s columns are spliced, change section every two storeys, and have their axial force varying up the height, none of which the girder’s uniform chord contains — and an average stiffness is not a safe stiffness for a member that changes along its length.

Nothing here is second-order. A frame that sways carries its gravity load sideways, and the moment that produces is proportional to the sway this essay has just computed — so every deflection here is the first term of a series.

And the mapping’s boundary conditions are genuinely different. A cantilever frame’s shear rises towards the base while the girder’s is constant, so the storey that governs in a building is the bottom one and the panel that governs in the girder is any of them.

The ladder from here #

Later rungs on this anchor: the classical hand method — cut at the contraflexure points, solve each panel, march along — which is one of the last hand methods for an indeterminate frame a person can carry out. Vierendeel girders with unequal chords, where the shear does not split in half and the zeros move off the panel centres. Panel-zone flexibility, and how much of a deep-chord girder’s movement it is. The Vierendeel as a stability problem, since its chords are beam-columns with moment reversal along them. And the cellular beam, which is a rolled section cut and rewelded into a Vierendeel girder, where the panel is a web post and the chord is a tee.

The girder is named for Arthur Vierendeel, who built the first ones in Belgium in the 1890s in preference to trusses because the open rectangular panel was easier to fabricate in wrought iron and easier to look at. The frames of every unbraced steel building since have been the same object, analysed by the same panel arithmetic, and almost never called by the same name — which is the ordinary fate of a structural form that turns out to be general: it stops being a type of structure and becomes a way of reading one.