The heavy chord the posts will not let work

A Vierendeel girder is a truss with the diagonals left out. Its panels are rectangles with rigid corners, and the shear that a diagonal would have carried as a single axial force has to cross each panel as bending in the two chords: each chord bends in double curvature between the posts, with a point of contraflexure near the middle of the panel, and the shear in the panel is the sum of the two chords’ shears. With both chords the same section the arithmetic is symmetric — each carries half the panel shear, and the moment at each end of each chord is that half times half the panel length. It is one of the last hand methods that assume part of the answer for an indeterminate frame that anybody still uses, and read as a sway frame stood on end it is the portal method by another name.

Real Vierendeel girders seldom have equal chords. A transfer girder under a building has a floor slab acting with its top chord. A footbridge’s bottom chord is a deck beam and its top chord a handrail member. A storey-deep truss in a façade has a heavy floor beam at one level and a light roof member at the other. And the moment the chords differ, the half-and-half split has to be replaced by something. The obvious replacement is the rule that runs through the whole of structural analysis: the stiffer path takes the load. Two members bending between the same two posts, through the same sway, should share in proportion to their stiffnesses, which for members of one length is their second moments.

That rule is right for a reason that is easy to state and easy to forget, and the girder below shows what happens when the reason is missing.

A girder with one heavy chord #

The girder is the one the first essay on this structure used: six 2 m panels over a 12 m span, 1.5 m deep, carrying 100 kN at mid-span on its top chord. The bottom chord and the posts are a section of second moment 11,700 cm⁴. The top chord is made four times as stiff in bending — a heavier section, or the same one with a slab acting on it.

The rule says 0.80. The girder, solved as the rigid-jointed frame it is, says 0.56 in the end panel and 0.59 in the next. The heavy chord, four times as stiff, takes barely more than half the shear — closer to the equal split the rule was meant to replace than to the rule itself. Only in the panels beside the load does it rise to 0.70, and there for a separate reason that comes later.

Make the posts a hundred times stiffer and the shares climb to 0.76, 0.78 and 0.80. The rule is the rigid-post answer. With posts of the size a designer would actually choose — the same section as the light chord, which is what a fabricator would usually want — it is wrong by a factor that a designer sizing the light chord would notice: that chord, sized for a fifth of the shear, is carrying 44 per cent of it.

The joint that turns #

The reason is at the joints. A chord bending in double curvature has a moment at each end, and at a joint those end moments have nowhere to go but into the post. In a girder with equal chords, the top chord’s end moments at a joint and the bottom chord’s are carried by the post together, which bends in double curvature too, its contraflexure near its own middle. Make one chord heavy and its end moments are larger, so the post at the heavy chord’s end has to carry more — and the only way a post can carry more moment is to bend more, which means the joint it is attached to turns.

A joint that turns relieves the chords meeting at it. A chord whose two end joints turn through the same angle as the chord’s own sway is not bending at all; it is a straight bar rotating as a whole, and it carries no shear. The question is how far each chord’s joints turn, and that is a small problem with a closed answer.

Take one panel in the part of a long girder where the shear is uniform. Every joint along the heavy chord turns through the same angle $\theta_t$, every joint along the light chord through $\theta_b$, and each chord’s line rotates through the sway angle $\psi$ — the relative movement of its two ends over the panel length. The slope-deflection equations — the same relations moment distribution passes round a frame by hand — give each chord-end moment as $k(6\theta - 6\psi)$ for a chord with equal end rotations, where $k = EI/s$ is its stiffness over the panel length $s$, and each post-end moment as $k_p(4\theta_{\text{near}} + 2\theta_{\text{far}})$ with $k_p = EI_p/h$. A joint is in equilibrium when the two chord ends and the post end meeting at it add to nothing:

$$12k_t(\theta_t - \psi) + k_p(4\theta_t + 2\theta_b) = 0, \qquad 12k_b(\theta_b - \psi) + k_p(4\theta_b + 2\theta_t) = 0.$$

The shear in each chord is proportional to $k(\psi - \theta)$ — its stiffness times how much of the sway is left over after its joints have turned.

For the girder drawn, with $k_t = 4k_b$ and $k_p = \tfrac43 k_b$ (a post 1.5 m long against a chord panel of 2 m), the two equations give $\theta_t = 0.87\psi$ and $\theta_b = 0.56\psi$. The heavy chord’s joints have turned almost all the way with it. What a chord carries is its stiffness times the bending it is allowed to do, and the heavy chord is four times as stiff and allowed 0.13 of the sway, against the light chord’s 0.44: four times 0.13 against one times 0.44 is a share of 0.54.

The drawing shows what the arithmetic says. The light chord is the one doing the Vierendeel’s work — bending in an S between joints that the posts hold nearly still — while the heavy chord, whose joints the posts cannot hold, tilts as a whole. The posts are bent in the opposite sense at their two ends, as they should be, and the heavier curvature is at the top, where the larger moment has had to go.

A ceiling set by the posts #

The one-panel model makes a stronger prediction than the girder showed. As the heavy chord gets heavier its joints turn closer and closer to the sway, and its share does not go to one.

With posts as stiff as the light chord, the share saturates at 0.55. Doubling the heavy chord from four to eight times the light one moves it in the third decimal place; making it thirty times stiffer moves it by one hundredth. The limit has a simple reading. As $k_t$ grows without bound, $\theta_t$ approaches $\psi$ and the heavy chord’s shear approaches $k_p(4 + 2\theta_b/\psi)/12$ times $\psi$ — a quantity with no $k_t$ in it at all. In the limit the heavy chord’s shear is set by how much moment the posts can feed into it, and the posts, not the chord, decide the ceiling.

The dots are a long girder — twenty 2 m panels — solved in full and read at its sixth panel, far from both the support and the load, where the shear is uniform enough for the one-panel model to describe it. With ordinary posts the girder gives 0.50, 0.53, 0.56 and 0.60 at chord ratios of 1, 2, 4 and 10 — the model’s curve to within a few hundredths — and then climbs past it, to 0.65 at 20 and 0.69 at 30. That departure is a second mechanism arriving rather than the first one failing. A chord thirty times as stiff as the light one has a second moment that is no longer small beside the girder’s own: the chords’ axial forces at 1.5 m apart are worth $A h^2/2$ of bending stiffness, and the heavy chord’s own $I$ is by then 31 per cent of the total. It starts to carry the girder’s global moment by bending as a beam in its own right, and the shear that goes with that moment’s gradient is in it whatever the posts do. Up to a ratio of about ten, the girder is a Vierendeel and the posts cap the heavy chord; beyond it, the heavy chord is becoming a continuous beam with a light tie hung beneath it.

How stiff a post has to be #

If the posts set the ceiling, the useful question is how stiff they must be for the rule to come true.

The curve runs from one limit to the other. With posts much softer than either chord, both chords’ joints turn nearly freely, both chords are nearly unrestrained, and they share about equally — 0.50 at a tenth — because each is then carrying its shear as a member whose stiffness hardly matters. With rigid posts, the chords are fixed-ended between the same two displacements, and they share by second moment. The transition between the two is slow: a factor of ten in post stiffness moves the share from 0.54 to 0.67, and the rule is not within a twentieth until the posts are 158 times as stiff as the light chord, or forty times as stiff as the heavy one.

Nobody builds posts like that. A post that stiff is a wall. And that is the point of the figure: the second-moment rule describes a girder with rigid posts, and a Vierendeel girder with rigid posts is a plate with holes in it. The rule is exact for an opening in a deep web, where the “posts” either side of the hole are metres of web and the tees above and below share the shear by their stiffnesses as it says. It was carried across to the girder, whose posts are members of the same size as its chords, and it does not survive the journey.

What the heavy chord does instead #

The heavy chord does not carry the shear its stiffness suggests, but it is not idle either.

The bottom chord is relieved slowly — from 33 kN·m with equal chords to 24 at four times and 17 at twenty — which is what a share capped near 0.55 means: the light chord keeps nearly half the shear and so nearly half its moment. The top chord’s largest moment, meanwhile, doubles at four times and quadruples at twenty, and almost none of that growth is in the panels the rule was about. It is in the two panels beside the load.

There the load is applied to the top chord itself, at a joint, and a heavy chord carries a load applied to it as a beam would — spanning a short way to the posts either side and passing the load down them, rather than sending it round the panel as a Vierendeel shear. The heavier the chord, the more of the load it keeps, which is why its share in the panel beside the load reaches 0.70 in the first figure and its moment there grows faster than its stiffness. The posts are correspondingly relieved — at twenty times the largest moment in any of them has fallen from 46 to 30 kN·m — because load they used to carry round the panel corners is going along the chord instead.

So the heavy chord’s stiffness is spent in two places, neither of which is the Vierendeel shear: locally, carrying the load where it is applied, and — once it is heavy enough — globally, as a beam across the whole span. The middle of the girder, where a designer applying the rule would put the heavy chord’s work, is where it does least.

Splitting a fixed amount of steel #

The same argument answers a design question directly. Suppose a girder needs a certain bending stiffness in its chords in total — twice the section drawn, say — and the designer may split it between top and bottom as convenient.

If the posts were rigid, the two chords would act in parallel and only their sum would matter: shifting stiffness from one to the other would change who carried the shear and nothing else. The stiff-post curve is very nearly that — flat to within 1.5 per cent across the whole range. With ordinary posts, equal chords are the stiffest arrangement and every departure costs: four fifths in one chord adds a tenth to the deflection, and 95 per cent adds two fifths. Stiffness put into a chord the posts cannot exploit is stiffness that does not reach the girder, and the cheapest way to make a Vierendeel girder stiff is to keep its chords alike and spend any extra on the posts.

That is a sharper statement than it sounds, because the usual motive for unequal chords is not stiffness but convenience — the slab is there anyway, or the bottom chord has to be a deck beam. The calculation does not say those girders are wrong. It says the stiffness they appear to have been given is partly fictitious, and the light chord’s moments are larger than the rule says by the ratio 0.44 to 0.20 — more than double.

The panel, by hand #

The one-panel model is small enough to solve on the back of a drawing, and it is worth doing once to see where each number comes from.

Write every stiffness as a multiple of the light chord’s $k_b = EI_b/s$. The heavy chord is $k_t = 4$; the posts are $k_p = (EI_b/h)/(EI_b/s) = s/h = 2000/1500 = 1.33$. Divide each joint equation by $\psi$, so the unknowns are the rotations as fractions of the sway:

$$(48 + 5.33)\,\theta_t + 2.67\,\theta_b = 48, \qquad 2.67\,\theta_t + (12 + 5.33)\,\theta_b = 12.$$

From the second, $\theta_b = (12 - 2.67\theta_t)/17.33$; substituting into the first gives $52.9\,\theta_t = 46.2$, so $\theta_t = 0.872$ and $\theta_b = 0.558$. The shears are proportional to $4 \times (1 - 0.872) = 0.51$ and $1 \times (1 - 0.558) = 0.44$, and the heavy chord’s share is $0.51/0.95 = 0.54$. Set $k_p$ to infinity instead and both rotations vanish: the shares are 4 and 1, and the rule’s 0.80 comes back. Set $k_t$ to infinity and $\theta_t \to 1$; the heavy chord’s shear tends to $k_p(4 + 2\theta_b)/12$ with $\theta_b = (12 - 2k_p)/(12 + 4k_p) = 0.54$, so its share tends to 0.564 against the light chord’s 0.462, or 0.55.

Every number in the argument is in those few lines, and none of them is the second-moment ratio.

Rigid joints, uniform shear, and posts that do not sway #

The model and the girder rest on assumptions worth naming.

The joints are rigid. Every corner is a full-strength, full-stiffness connection, which is what makes a Vierendeel possible at all — a pinned corner would leave a mechanism. A semi-rigid corner — a bolted end plate, say — is a spring in series with the post, and it makes the post effectively softer still; everything above then moves further from the rule, not closer.

The one-panel model assumes uniform shear. It describes a panel whose neighbours are doing the same thing, so that every joint along a chord turns alike. Near a support, where the girder starts, and near a load, where the chord carries something of its own, the joints differ, and the whole frame has to be solved. That is why the figures put the model beside full solutions and say where they part.

The posts carry no shear of their own. In the model a post only bends, from the chords’ end moments; it does not sway. In a Vierendeel girder that is right, because the girder’s shear crosses each panel in the chords and the posts carry only the change of chord force from one panel to the next. In a sway frame read the other way up — the frame that is a girder stood on end — the roles swap, and so does the conclusion: a heavy beam on light columns is a frame whose beams cannot hold its columns’ joints still, which is the same statement as a portal’s columns attracting their stiffness share only when the beam is rigid.

Everything is elastic and first-order. Under a load large enough to yield the light chord, which the argument shows is carrying more than the rule supposed, the light chord softens and sheds shear to the heavy one, and a plastic analysis — four hinges at a panel’s corners, as at a web opening — shares the panel shear by the chords’ plastic moments rather than by any stiffness. The elastic shares here are what decides the deflection and the service stresses; they are not what decides the collapse load.

The posts’ joints, and a floor on the chord #

The figures cannot show the connections the posts need. A post carrying the heavy chord’s end moments down to the light chord’s joint has to have a joint at each end that can deliver them, and the moments drawn here — 44 kN·m in a post of the six-panel girder — are of the same size as the chords’. A Vierendeel girder’s posts are often detailed as secondary members because their axial force is small, and that is exactly the habit this argument says is dangerous: the post’s moment, not its force, is what sets how the chords share.

They also cannot show a slab. The commonest way for a top chord to become four times stiffer is for a floor to act with it, and a floor acts with it only as far as its connection allows. A composite top chord in a Vierendeel girder is a chord whose stiffness is itself uncertain, under a sharing rule that the posts have already made unreliable. Neither uncertainty is resolved by the other.

Still open: a heavy chord, light posts, and a load that moves #

The girder drawn carries one load at mid-span. A Vierendeel bridge carries a vehicle along its deck chord, and the panel in which the vehicle stands is always, briefly, the panel beside the load — where the heavy chord carries the load as a beam and its share and moment jump. As the vehicle crosses, every panel takes its turn at being that panel. The question a moving load puts is whether the deck chord’s local beam action, which relieves the posts beside the load, governs the deck chord’s design everywhere along the span once the load has visited everywhere — and whether a girder whose chords are equal under a static load is still the stiffest arrangement for a load that never stays still.