The load that visits every panel

The heavy chord the posts will not let work loaded a Vierendeel girder at mid-span and asked how its two chords share the shear when one is stiffer than the other. That is how a Vierendeel is usually checked: one load, at one position, chosen because it gives the largest bending moment in the girder as a whole. A Vierendeel bridge carries a vehicle along its deck chord, and the vehicle stands everywhere in turn. The essay ended on what that changes: every panel takes its turn as the panel beside the load, and the panel beside the load is the one where the deck chord bends as a beam between its posts as well as a Vierendeel chord.

So the question is no longer how the girder carries one load. It is what each member carries at its own worst moment, which is a different moment for each, and whether the arrangements that are best for one central load are still best for a load that never stays still.

The girder, solved at every point of the deck #

The girder is the one the earlier essays used: six panels of 2 m, 1.5 m deep, its chords and posts all the same section. The load is 100 kN, travelling along the top chord, which is the deck. To let it stand anywhere — not only at the posts, where a hand method puts loads on a Vierendeel — the deck chord is cut into four members a panel, and the frame is solved with the load at every one of those points by the stiffness method. The frame is linear, so the response to any group of loads is the sum of the responses to each, and a vehicle of two axles is solved by adding two of these solutions at the right spacing.

What is recorded at each position is each member’s largest bending moment — every chord panel and every post — and the deflection under the load. Over the whole crossing, the largest of each is its envelope: the moment that member has to be designed for.

What the crossing asks, member by member #

The central load is the worst position for the girder’s deflection and for nothing else. Its shear is half the load in every panel, so the chords’ moments, which a Vierendeel makes from the shear, are moderate everywhere: 26 kN·m in the end panels of the deck chord, 35 in the middle panels, where the load’s own bending adds to them. A load standing at the first post sends five sixths of itself through the end panel’s shear, and the end panels’ chords reach 44 kN·m — 1.7 times what the central load gave them.

The posts change most. A Vierendeel’s post carries the difference between the chord moments either side of its joint, and under a central load the middle post has equal chords either side carrying equal and opposite moments: it carries nothing. Under a crossing load it carries 30 kN·m whenever the load is beside it. The posts next to the end posts carry the most of any member in the girder, 59 kN·m against the chords’ 44 — they were already the girder’s heaviest-loaded members under a central load, at 46, and the crossing raises them by more than a quarter.

So a girder checked for a load at mid-span is checked for its deflection and for the middle panels of its chords. Its end panels and its posts are checked for a load that is not where the critical vehicle will stand.

Every member’s worst position #

The figure is three influence lines — each member’s moment against the load’s position — and they make the point that a single check cannot. Each member has its own worst position, and none of them is mid-span. The end panel’s deck chord is worst with the load at the first post, where the shear through the end panel is greatest and the chord is still carrying the load directly. The third panel’s deck chord is worst with the load inside its own panel, where the chord’s bending between its posts adds to its share of the shear. The second post is worst with the load in the panel beyond it, where the difference between the chord moments either side of its joint is largest.

A Vierendeel has no diagonals to carry its shear directly, so every member’s moment is a function of the shear where it is, and the shear where it is depends on where the load is. An influence line is how a bridge designer reads a member’s worst case from a moving load, and a Vierendeel needs one for every member because its members have no single governing position between them.

The deck chord as a beam #

A hand analysis of a Vierendeel puts its loads at the panel points, where the posts are, because that is where a cross-girder or a floor beam would deliver them. A vehicle on a deck chord is not at the panel points; between them it bends the deck chord as a beam spanning between two posts, and that bending adds to the chord’s share of the Vierendeel shear.

How much it adds depends on how large the share is. With equal chords the end panels’ shear is large and the deck chord’s share of it — half — is large too; the local bending is real but smaller than the shear’s contribution, and the envelope is the same whether the load stops between posts or not. In the middle panels the shear is small, the share is small, and the local bending is a large part of what is left: it adds 14 per cent.

With a light deck chord, carrying an eighth of the girder’s chord stiffness, its share of the shear is small in every panel, and its own bending between posts governs everywhere: 107 per cent added in the middle panels, 47 in the end ones. A deck chord stiff enough to carry a large share of the girder’s shear is designed by that share; a light one is designed as a beam between posts, which is the answer to the earlier essay’s question about whether local beam action governs the deck chord everywhere. It does, if the deck chord was made light.

A vehicle, not a wheel #

A real vehicle spreads its weight over axles, and their spacing against the panel length decides how much of the deck chord’s local bending survives. With the axles a panel apart, one axle is always near a post when the other is mid-panel, and the vehicle can never put its whole weight in the middle of one panel: the middle panels’ deck chord falls from 40 kN·m to 27, a third less, while the end panels and the posts, which were governed by shear rather than by local bending, fall by a tenth or less.

The girder’s other members do not get the same relief. The posts and the end panels are governed by shear, and two axles 2 m apart put nearly as much shear through a panel as one axle carrying both their weights, because the shear in a panel is the sum of everything on one side of it. For those members a vehicle is very nearly as bad as a single load of its whole weight, and a train of vehicles can be worse, as a train is worse than its heaviest axle for any member whose influence line is long.

This is the reason a Vierendeel bridge’s deck is usually carried on cross-girders at the posts and stringers between them, rather than directly on the chord. Cross-girders turn a vehicle’s axles, wherever they are, into loads at the panel points, and the deck chord then carries only its share of the shear. A deck laid directly on the chord makes the chord a beam between its posts as well, and the figure says how much that costs: from nothing in the end panels to 40 per cent in the middle ones, for a single heavy axle.

Equal chords, still #

The earlier essay found that for a central load, equal chords make the stiffest Vierendeel of a given total stiffness: splitting it unequally moves load into the posts’ joints and relieves the heavy chord less than its stiffness suggests. A crossing load could have changed that, since its worst positions for most members are not mid-span. It does not change it for the deflection, because the deflection is worst with the load at mid-span, which is exactly the static case: 6.20 mm with equal chords, 6.44 at a 30/70 split, 7.18 at 15/85.

For strength the picture is asymmetric. Each chord’s moment divided by its share of the stiffness — its stress, for sections of the same depth — is nearly balanced with equal chords, 44 and 45. Make the deck chord lighter and its stress rises steeply, because it loses both its share of the shear and the stiffness it needs for its own beam action between posts. Make the other chord lighter and its stress rises too, but only by its share. Equal chords are the stiffest arrangement for a crossing load, and nearly the most balanced; a lighter deck chord is punished twice.

The middle post’s range #

The envelope is a design moment; a member under traffic also needs its range, because a load that never comes near failing anything can still crack it by repetition. For most members here the two are the same: the second post’s moment at its top end runs from nothing to 59 kN·m as a vehicle passes and never changes sign, so its range is its envelope.

The middle post is different. Its moment reverses. With the vehicle in the panel to its left the post bends one way, with it in the panel to its right the other, and its top end’s moment runs from −30 to +30 kN·m. Its range is 59 kN·m — the same as the heaviest post’s, although its envelope is half as large and a central load leaves it with nothing at all. A fatigue check made on the central-load moments would pass the middle post without looking at it; a check made on the envelopes would see a lightly loaded post; only a check made on the range sees a post working as hard as any in the girder.

Openings in a web are the same girder #

A Vierendeel is not only a bridge. A beam with a row of large openings in its web — a cellular beam carrying services through a floor, or a castellated crane-runway girder — is a Vierendeel in all but name: the steel above and below each opening are its chords, and the web-posts between openings are its posts. The hole that costs nothing, and everything found that what decides such an opening is the Vierendeel moment in the tees above and below it, and that the worst opening is the one with the most shear through it. A crane runway girder carries a moving wheel load on its top tee, and everything here applies to it: the worst opening moves with the wheel, the top tee bends between web-posts under a wheel standing over an opening, and the web-post that a central-load check leaves idle has a full reversing range under a passing crane.

What a hand check has to change #

None of this needs more than the hand method a Vierendeel is usually checked by, applied more times. The method cuts each chord at mid-panel, gives each cut its share of the panel shear, and reads the chord and post moments from the shears. A designer who applies it to a central load has checked one position; the changes the crossing asks for are three.

Load each panel point in turn, not only the middle one. The shear through a panel is the load on one side of it, and it is largest for the end panels when the load is at the first post. Six positions for six panels give every chord its worst shear and every post its worst unbalance.

Add the deck chord’s own bending where the load stands between posts. For a deck chord that carries a large share of the shear, this changes little except in the middle panels; for a light deck chord, or a deck laid straight on it, it is the larger term in every panel, and a fixed-ended span under the wheel is a fair estimate of it.

Check the posts for their range, not their envelope. A post between two panels that the load visits in turn reverses, and its range is twice its envelope. The middle post of an even-panelled girder is the clearest case, and whether its cycles count depends on that range, not on the moment a central load gives it.

The end panel and the deck chord, by hand #

The end panel’s shear with the load at the first post is five sixths of the load, 83 kN, against half the load, 50 kN, with it at mid-span. A Vierendeel’s chords share the panel shear and bend about a point of contraflexure near mid-panel, so each chord’s end moment is roughly its share of the shear times half the panel: with equal chords, $83/2 \times 1 = 42$ kN·m against $50/2 \times 1 = 25$. The frame gives 44 and 26; the ratio, 1.7, is the ratio of the shears.

The deck chord’s own bending under a load between two posts is at most that of a 2 m span fixed at both ends under a central load, $PL/8$ = 25 kN·m, and the posts are not fully fixed, so it is a little more where the chord is light. With equal chords in the middle panels it adds about 5 kN·m to the 35 the shear gives, close to the 14 per cent the frame finds.

Where the frame stops #

The load is a point on the chord’s centre line. A wheel spreads its load over a contact patch and a deck spreads it further; a real deck chord’s local bending is somewhat less than a point load’s.

The joints are rigid and the members elastic. A Vierendeel’s joints are its whole mechanism, and a real joint’s flexibility lowers every chord moment and raises the deflection; the comparisons between positions and arrangements are less affected than the moments themselves.

The load is static at each position. A vehicle crossing at speed adds dynamic amplification, which for a short girder of this stiffness is modest but not nothing, and which affects every member’s envelope by the same factor rather than changing which position governs.

Fatigue, and the vehicle off the centre line #

They cannot show fatigue, which for a bridge is often what a moving load decides: every member’s moment ranges from nothing to its envelope once per vehicle, and the posts’ joints, where the moments change direction across a welded corner, are where a Vierendeel’s fatigue cracks start. They cannot show a vehicle that is not centred over the girder, which loads one girder of a pair more than the other. And they cannot show the deck’s own stiffness, which on a real bridge carries part of the vehicle’s weight past the deck chord directly to the posts.

Still open: the posts’ joints under a moving load #

The posts carry the girder’s largest moments, and they carry them at their ends, where they meet the chords. Each joint sees a moment that changes sign as a vehicle passes from one side of it to the other, and the joint’s corner — a welded or haunched connection between two members bending in opposite senses — is a stress concentration whose range, not its peak, decides its life. Whether a haunched corner reduces that range in proportion to its stiffening, or moves the concentration to the haunch’s toe, and how many vehicles a joint of each kind survives, is the question a crossing load leaves at the corners that make a Vierendeel a Vierendeel.