The corner that counts its crossings

A Vierendeel girder has no diagonals, so the shear a diagonal would carry travels instead as bending in the chords and the posts, and the joints between them must be rigid. Every one of those joints is a corner where two members meet bending in opposite senses, and every corner is welded or bolted, with a flange turning through ninety degrees. The load that visits every panel followed a load across a six-panel girder and found that the posts, not the chords, carry its largest moments as it crosses: 59 kNm in the post beside each end, against 46 for a load parked at mid-span.

That essay asked about peaks, which is what a strength check reads. A bridge girder is not loaded once. Every vehicle that crosses it takes each corner through a cycle, and a welded corner’s life is decided by the range of its stress, not its peak. The question here is what the range is at each of a Vierendeel’s corners, how it differs from the peak, and what the usual stiffening — a haunch at the corner — does to it.

The moment at a corner, as the load goes by #

Take the girder from the crossing-load essay: six 2 m panels, 1.5 m deep, every chord and post the same section, with a second moment of 117 × 10⁶ mm⁴ and an elastic modulus, for a member 300 mm deep, of 780 × 10³ mm³. A 100 kN load crosses the deck chord, and at each position the frame is solved.

At the top corner of the first interior post, 2 m from the left support, the moment is small and negative while the load is on the short side of the post, rises steeply as the load passes the post, peaks at 58.8 kNm with the load a metre and a half beyond it, and falls away as the load runs on to the far support. Its swing is lopsided: from −2.3 to +58.8. At the second post, 4 m along, it is less lopsided: −13.8 to +45.6. At the middle post it is symmetric: −29.6 to +29.6.

The reason is the shear that each corner carries. A Vierendeel post’s moment at a joint is made from the chords’ end moments either side of it, and a chord’s end moment is its share of the panel shear times half the panel length. With the load to the right of a post, the panels on its left carry the left reaction as shear; with the load to its left, the panels on its right carry the right reaction, the other way. A post near the left support sees a large left reaction for most of the crossing and a small right reaction for the short part when the load is behind it, so it swings a long way one side and hardly at all the other. The middle post sees both reactions reach half the load, symmetrically.

Peaks differ; ranges do not #

Drawn post by post, the two measures tell different stories.

The peak corner moment, either sign, is 58.8 kNm at the first interior posts, 45.6 at the second, 29.6 at the middle post and 40.0 at the end posts over the supports. A strength check reading peaks makes the posts beside the supports the heaviest and the middle post half as heavy, and the crossing-load essay found exactly that.

The range is nearly flat. At the first interior post it is 61.1 kNm, at the second 59.4, at the middle 59.3. The range a corner sweeps is roughly the sum of the largest left reaction and the largest right reaction that can act on it, and for any interior post that sum is close to the whole load: the further a post is from one support, the more of its range comes from one side and the less from the other, and the total barely moves. Only the end posts, over the supports, are different — they never see a reversal, so their range is their peak, 38.9.

So fatigue reads a Vierendeel differently from strength. Every interior corner of this girder sees the same stress range from every vehicle, to within three per cent. If the corners are detailed alike, they will crack alike, and the middle post’s corner — the one a strength check would have made lightest — has the same fatigue life as the first post’s. Making the middle post lighter on the strength check’s advice would make it the girder’s weakest joint.

Why the sign of the moment does not matter #

It is tempting to treat the middle post’s fully reversed cycle, −29.6 to +29.6, as harsher than the first post’s pulsating one, −2.3 to +58.8, because the first never puts the corner’s far flange into much tension. For a welded corner that distinction does not survive. A weld leaves residual tensile stress near its toe at about the yield stress, so the metal where a fatigue crack starts is already in tension whatever the applied moment’s sign, and the applied cycle moves it down from yield and back. What it sees is the range. That is why the S–N curves for welded details are written in stress range alone, with no correction for the mean, and why a crack grows on the loops of the range regardless of where the loops sit.

So the two corners are equivalent in fatigue, cycle for cycle, and the lopsided swing near the supports is no gentler than the symmetric one in the middle. The same equations govern a building frame, which is a Vierendeel girder stood on end, and its beam-to-column corners see reversing moments from every gust of wind. The difference is the count. A bridge girder sees its heavy vehicles hundreds or thousands of times a day; a building’s corners see their design wind a few times in a lifetime, and their smaller cycles mostly fall below the limit at which they do damage. A Vierendeel used as a bridge is the case where every corner is a fatigue detail.

A haunch at the corner #

The usual way to stiffen a rigid corner is a haunch: the member is deepened towards the joint, by a welded plate or a cut-and-turned section, so that the joint’s section has a larger modulus where its moment is largest. In a portal frame a haunch is a strength device, and the section that governs moves inside the haunch, to its tip, where the moment is still large and the section has returned to the plain member’s. A fatigue check asks the same question about the range.

The figure follows the stress range up the first interior post, from the bottom chord to the top. Plain, it is largest at the corners — 78.3 MPa at the top — and falls linearly to a minimum near mid-height, because the post bends in double curvature and its moment passes through zero between its ends. With 250 mm haunches twice the member’s depth at both corners, the range at the top corner itself falls to 43.3 MPa: the section there is twice as deep, so its modulus is twice as large, and the moment is shared more evenly. But at the haunch’s toe, 250 mm down the post, the section drops back to the plain member’s, and the range jumps to 61.2 — the largest anywhere in the post, and above the constant-amplitude limit of 52.3 MPa for a detail of category 71.

The worst stress range has not gone away. It has moved 250 mm along the post, to a point where the moment is about two thirds of the corner’s and the section is half what the haunch gives the corner.

Depth stops paying #

It would be natural to suppose that a deeper haunch, which makes the corner stiffer, makes the joint safer in proportion.

At the corner it does. The corner’s range falls from 78.3 MPa, plain, to 56.0 at one and a half times the member’s depth, 43.3 at twice and 29.7 at three times. At the toe, it does the opposite. The toe’s range rises, from 55.4 MPa with a haunch only a tenth deeper than the member to 61.2 at twice the depth and 63.0 at three times. Past about 1.5 times the depth the toe governs, and from there every millimetre of extra haunch depth makes the joint’s worst range larger.

The rise is not a local effect. A haunched corner is a stiffer corner, and a Vierendeel girder’s moments are distributed by stiffness: the stiffer the joints, the larger the share of the panel shear that goes into the end moments of the chords and posts, and the larger the moment at every joint. A haunch twice as deep raises the first post’s corner moment range from 61.1 to 67.5 kNm — ten per cent more moment, drawn from the rest of the girder into the corners where the haunch has made room for it. The corner’s own section absorbs that increase easily. The toe, which has the plain member’s section and two thirds of the corner’s moment, does not.

Length keeps paying #

The toe’s moment is a fraction of the corner’s because the post’s moment falls linearly from its ends towards mid-height. A longer haunch puts its toe further along that line.

With the haunch twice the member’s depth, its toe’s range is 72.0 MPa at a length of 100 mm, 65.0 at 200, 61.2 at 250, 48.9 at 400 and 40.1 at 500 — the last two below the constant-amplitude limit for category 71. The corner’s range barely changes with length, rising a little from 41 to 45 MPa as the longer haunch stiffens more of each member and draws a little more moment into the joints.

So the toe’s range is governed by how far along the member it sits, not by how deep the haunch is. That is the same lesson the haunched portal rafter teaches for strength, with one difference. In a portal, lengthening the haunch stops buying anything once its tip passes the point where the rafter’s moment has fallen to what the plain section can carry. In a Vierendeel post the moment falls to zero at mid-height and the toe’s range keeps falling until the haunch meets it — but a post 1.5 m long has room for haunches of perhaps a third of its length at each end, and that is where the range falls below the limit.

How many vehicles #

A corner’s fatigue life under one kind of vehicle is the number of crossings at which its stress range reaches the detail’s S–N curve. EN 1993-1-9 gives that curve as a straight line of slope three on logarithmic axes, $N = 2 \times 10^6 (\Delta\sigma_C/\Delta\sigma)^3$, with a constant-amplitude limit at 0.737 of the detail category below which a single kind of vehicle does no damage at all. The cycles that do not count showed what that limit does to a real traffic spectrum; here there is one vehicle, crossing over and over.

At 100 kN the plain corner, at category 71, survives 1.5 million crossings. The haunched joint — twice the depth, 250 mm long — survives 3.1 million if its toe is as good a detail as the corner, because the worst range has fallen from 78 to 61 MPa and life goes as the cube. At 150 kN the three are 441 thousand, 926 thousand and 454 thousand: the cube law makes every vehicle half as heavy again about three times as damaging.

The third line is the one a designer should look at first. A haunch’s toe is where the haunch’s flange plate ends on the member’s flange, and the end of a welded plate is a classic fatigue detail, usually worse than the corner it was added to protect. With the toe two detail categories lower, at 56, the haunched joint survives 1.5 million crossings of 100 kN — the same as the plain corner. The haunch has moved the crack from the corner to its own toe, and bought nothing.

What the girder here leaves out #

The girder is one section throughout, and every corner is detailed alike. A real Vierendeel bridge girder is often deeper in its chords than in its posts, which moves the share of moment between them and changes which corners see the largest ranges; the argument that ranges are flatter than peaks along the girder survives, because it comes from the reactions, but the numbers move.

The load is a single 100 kN axle. A real vehicle has several axles, and a group of axles closer together than a panel puts several cycles into a corner on one crossing, or one larger cycle, depending on the spacing. Rainflow counting of the corner’s moment history for a multi-axle vehicle would replace “one crossing, one cycle” with a small spectrum per vehicle. And a real traffic stream is a spectrum of weights, for which the damage is summed by Miner’s rule and the constant-amplitude limit is replaced by a cut-off further down — so the haunched toe’s lower range buys less below the limit and more above it than the single-vehicle figure suggests.

The haunch is modelled as a prismatic deepening of the end segments, with an I-section’s flanges making its second moment grow as the square of its depth and its modulus as the depth. A tapered haunch varies continuously, puts its toe at the end of a slope rather than a step, and has a lower stress concentration there than a square-ended plate. The toe’s detail here is therefore on the harsh side, and the figure’s lesson about depth is the safer reading of it: depth beyond what the toe needs does nothing but draw moment.

And the stresses are nominal — moment over modulus, at the corner and at the toe. The corner itself is a stress concentration, with a flange turning through ninety degrees and a weld across the turn, and its detail category already reflects that; but the corner’s real range is larger than its nominal one, and a hot-spot analysis would sharpen the comparison between corner and toe without reversing it.

The two readings of one girder #

A Vierendeel girder read for strength and read for fatigue is two different girders. Its strength is governed by peaks, which are largest beside the supports and smallest at mid-span; its fatigue life is governed by ranges, which are nearly the same at every interior corner because each sees a reversal whose two halves add up to about the whole load. A girder proportioned by peaks has its lightest post in the middle and its weakest fatigue joint there.

At each corner, the range is where the crack will start, and a haunch does not remove it — it relocates it. The relocation is worth having only if the new place is better: further along the member, where the moment has fallen, and detailed so that the haunch’s own end is not a worse detail than the corner it protects. Depth past what the toe needs only draws moment out of the rest of the girder and into the joints, which is the opposite of what a haunch was for.

Still open: the girder whose posts are not all the same #

Every post here is the same section, so the flat range along the girder is a statement about the load, not about the posts. A designer reading peaks would make the middle posts lighter, and a lighter post is more flexible, which changes how the panel shear is shared between the chords and the posts at the joints around it. Whether a girder with posts proportioned to their peaks has its fatigue range concentrated into the lightened middle posts, whether proportioning the posts to their ranges instead — all alike — costs more steel than it saves in inspection, and whether a girder whose middle posts are deliberately stiffer than its end posts can spread its fatigue damage evenly, is what the posts’ own sizes ask of this one.