The post sized by what it must survive

A Vierendeel girder has no diagonals, so the shear a diagonal would carry passes from chord to chord as bending in the posts, and the posts’ corners carry the girder’s largest moments. The corner that counts its crossings followed a 100 kN load across a six-panel girder, every member the same section, and found two different girders in one. Read for strength, the posts beside the supports peak at twice the middle post’s moment. Read for fatigue, every interior post sweeps the same range within three per cent, because the middle post’s moment reverses fully as the load passes it and the end posts’ moments hardly reverse at all.

That girder was uniform, so its flat ranges were a statement about the load and not about the posts. A designer who reads the peaks will not leave it uniform. The middle post carries half the moment of its neighbours, so it is the obvious post to lighten, and a lighter post is a more flexible post, which changes how the frame shares its shear. The question here is what happens to the ranges once the posts are sized one by one — to their peaks, to their ranges, or to both — and what each choice costs in steel and buys in life.

A post’s range is its peak times a number the load decides #

Take the girder again: six panels of 2 m, 1.5 m deep, its chords of a section with a second moment of 117 × 10⁶ mm⁴ and an elastic modulus of 780 × 10³ mm³, and a 100 kN load crossing the deck chord. At every position of the load the frame is solved, and at every post the moment at each corner is recorded. Two numbers come out for each post: the largest moment it ever carries, of either sign, and the difference between the largest and the smallest — the range it sweeps in one crossing.

The ratio of the two is a property of where the post stands, not of how big it is. An end post, over a support, is bent one way by every position of the load; its moment rises from almost nothing to a peak and falls back, so its range is its peak, 0.97 times it here because it never quite returns to zero. The first interior post is nearly as one-sided: the load on the near side of it bends it slightly backwards and the load beyond it bends it hard forwards, so the range is 1.04 times the peak. The second post is 1.30. The middle post sees a load that is as far on one side as on the other for every position, and its moment is a mirror image either side of mid-span; it is bent backwards as hard as it is bent forwards, and its range is exactly twice its peak.

The consequence is that a post’s fatigue demand is its strength demand multiplied by this ratio, and the ratio rises exactly where the peak falls. The uniform girder’s peaks, in moment, are 40.0, 58.8, 45.6 and 29.6 kNm from the end post to the middle one. Its ranges are 38.9, 61.1, 59.4 and 59.3. The posts a strength check finds lightly loaded are the ones that reverse, and the ones that reverse are the ones a fatigue check finds fully loaded. The earlier essay on the stress range rather than the stress is the general form of this; a crossing load on a Vierendeel gives it a precise geometry.

None of this tells yet what happens when the posts are no longer alike, because a frame shares its moments by stiffness and the stiffnesses are about to change.

Three designers, three girders #

Give each post its own section, a multiple of the chords’ — heavier flanges or lighter ones on a post of the same depth, so that its second moment and its modulus scale together. Three designers size the posts three ways.

The first sizes each post to its peak, so that every corner’s largest stress equals the worst corner’s in the uniform girder: 75.4 MPa, which is the first interior post’s 58.8 kNm over the modulus. The second sizes each post to its range, so that every corner’s stress range equals the uniform girder’s worst: 78.3 MPa, the first interior post’s 61.1 kNm, which gives every corner the same fatigue life. The third sizes each post to both, taking whichever of the two sections is larger — a static check and a fatigue check both made, which is what a designer is actually required to do.

None of these can be sized in one pass. A post made lighter rotates more at its joints, attracts less moment and passes more of the panel’s shear to its neighbours, which is the rule that stiffness takes the load acting inside one frame. So each design is iterated: size the posts to their demands, re-solve the girder with the new posts under the whole crossing, size them again to the new demands, until no post changes by more than a ten-thousandth.

The three girders come out with very different shapes. Sized to its peaks, the girder keeps its heavy posts beside the supports — 1.03 of the chords at the first interior posts — and cuts everywhere else: the end posts to 0.64, the second posts to 0.70 and the middle post to 0.39, two-fifths of a chord. The posts together hold 5.14 chord sections’ worth of steel against the uniform girder’s 7, a saving of 27 per cent, and on a strength check alone the saving is entirely legitimate: every corner peaks at 75.4 MPa and none above.

Sized to its ranges, the girder looks almost like the uniform one in the middle and quite unlike it at the ends. The interior posts are 0.96 to 1.03 of the chords. The end posts, which never reverse, drop to 0.59. The total is 6.14, a saving of 12 per cent on the uniform girder — the opposite of what intuition says a fatigue design costs. The posts a fatigue check can lighten are not the middle ones but the ones over the supports, whose range is no larger than their peak and whose peak is small.

Sized to both, the girder takes the ranges design and adds a little steel to the end posts, from 0.59 to 0.64, because a post lightened to 0.59 peaks at 80.6 MPa, above the uniform girder’s 75.4. The total is 6.24: 11 per cent less than uniform, and only 1.7 per cent more than the ranges design. In this girder the two checks govern different posts and barely compete — the static check governs the ends and the fatigue check the interior.

What lightening the middle post does to it #

The peaks design is the one a strength-only check produces, and its middle post is the one to follow. In the uniform girder its top corner sweeps from −38.0 to +38.0 MPa as the load crosses. In the peaks design it is two-fifths of the section, and something has to give.

What gives is less than the designer hoped. A lighter post is more flexible, and the frame responds by putting less moment into it: the middle post’s moment range falls from 59.3 kNm in the uniform girder to 45.8 in the peaks design, to 77 per cent. The other 23 per cent is shed into the chords and the neighbouring posts. But the post’s modulus has been cut to 39 per cent, and stress is moment over modulus. The moment falls by a quarter; the modulus falls by three-fifths; the stress doubles. Its curve in the figure is the uniform curve stretched vertically, reversing at the same position of the load — when the load stands at mid-span, directly over the post — because the reversal is a matter of symmetry, and lightening the post does not change the girder’s symmetry.

The peak, at 75.4 MPa, is exactly what the peaks designer asked for. The range is 151 MPa, twice that, because this post’s range was always twice its peak. The peaks designer has sized the middle post to half its fatigue demand.

The same thing happens, more gently, to the second posts: sized to 0.70 of the chords, their range rises from 76.1 to 102 MPa. The end posts, sized to 0.64, rise from 50 to 73 MPa — which matters less than it seems, because the uniform girder’s end posts were below the detail’s constant-amplitude limit and these are only just above it.

Where the range goes #

Set the three designs side by side, post by post, and read the stress range at each corner against the line the uniform girder drew.

The uniform girder’s ranges are flat across the interior and low at the ends. The peaks design turns that profile upside down in the middle: the range rises towards mid-span as steadily as the peak fell, to nearly double the line at the middle post. The design is exactly as safe as the uniform girder on its strength check and nowhere near it on fatigue, and the post that has been made weakest for fatigue is the one a designer reading peaks would think least worth inspecting.

The design sized to both does what the other two cannot do separately. Every interior corner sits at the uniform girder’s worst range, the end posts sit at its worst peak, and nothing exceeds either. The flat line the uniform girder drew across its interior survives, because the interior posts were left at very nearly the uniform section — the uniform girder was already very close to the right fatigue design across its interior, by the accident that a crossing load gives every interior post nearly the same range. Where it was wasteful was at the ends.

There is a further point in the figure about the first interior post. In every design it is the heaviest, and in the ranges design it is slightly heavier than in the uniform girder: 1.03 of the chords, because when the end posts are lightened they shed moment to it. Its range in moment rises from 61.1 to 63.1 kNm. A post that is both peak-critical and range-critical, with a ratio near one, is the post a girder cannot economise on by either reading, and the redistribution from lightening others lands on it.

What each girder survives #

A detail category turns a stress range into a number of crossings. Take the corner as category 71 of EN 1993-1-9 — a welded cruciform joint with full penetration, typical for a Vierendeel corner — and its line of slope three: a range of 71 MPa survives two million cycles, half that range eight times as many, and below 52 MPa, the constant-amplitude limit, a range of a single repeated size does no damage at all.

The uniform girder’s shortest-lived corners are the first interior posts at 1.49 million crossings; the middle ones last a little longer, 1.63 million, and the end posts, below the limit, last indefinitely under one repeated vehicle. A girder lasts as long as its shortest-lived corner, so the uniform girder’s life is 1.49 million.

The peaks girder’s life is 209 thousand crossings — a seventh of the uniform girder’s — and it is decided by the middle post. Its second posts last 682 thousand, under half. Only its first interior posts, left at full section, last as long as the uniform girder’s. A bridge carrying a hundred heavy vehicles a day reaches 209 thousand crossings in under six years; it reaches 1.49 million in about forty.

The girder sized to both lasts 1.49 million crossings, exactly as long as the uniform girder, with every interior corner at that life and the end posts a little longer. It has eleven per cent less steel in its posts and loses nothing.

The figure also says something about inspection. In the uniform girder and in the girder sized to both, the damage is spread: every interior corner reaches its life at nearly the same moment, so an inspection programme has to look at every corner equally. In the peaks girder the damage is concentrated in one post, which would be the place to look — except that a programme drawn from the strength calculations would not look there, because the strength calculations say it is the least loaded post in the girder.

Steel for life #

The fair comparison is not against the uniform girder but against a uniform girder of the same steel. Scale every post together, lighter or heavier, and the girder traces a line of steel against shortest life. A design sized post by post is worth having only if it lands above that line.

The uniform line rises steeply at first and then flattens, because a uniformly heavier post attracts more moment as it gets stiffer, so the stress falls more slowly than the section grows. At 6.24 chord sections — the steel the design sized to both actually uses — a uniform girder would survive 1.12 million crossings. The design sized to both survives 1.49 million with the same steel, a third longer. Or, read the other way, it reaches the uniform girder’s full-section life with eleven per cent less steel.

The peaks design falls far below the line. A uniform girder with its 5.14 chord sections of steel — every post at 0.73 of the chords — would last about 700 thousand crossings, more than three times as long as the peaks design’s 209 thousand. Sizing to peaks is not merely a design that ignores fatigue: it is a worse fatigue design than spreading the same steel evenly, because it takes the steel from exactly the posts where the range is largest relative to the peak.

The general principle is the one the cycles that do not count states for a single detail: under a slope of three, a range made twice as large is eight times as damaging, so steel spent where the range is largest buys life fastest. The design sized to ranges puts its steel where the ratio of range to peak is largest, and takes it from where that ratio is smallest; the peaks design does the reverse.

What the comparison leaves out #

The posts here are scaled as a whole — second moment and modulus together, by heavier or lighter flanges on a post of fixed depth. A real designer chooses from rolled sections, which come in steps, so the design sized to both would be built a little heavier everywhere than the iterated numbers say; the comparison between designs survives the rounding, since all three round the same way.

The chords were held at one section throughout and not re-checked. A lightened post sheds moment, and in the peaks design the middle post sheds about 13 kNm of its range; that moment goes somewhere — into the neighbouring posts, whose ranges were seen to rise, and into the chords meeting them — and a full design would check the chord panels on either side of the lightened post as well as the posts. The heavy chord the posts will not let work is the essay about how far a chord can be made to carry what the posts decline.

The load is one vehicle of one weight, and the end posts’ “no limit” in the uniform girder is a statement about that vehicle alone. Under a spectrum of vehicles, heavier ones take the end posts above the constant-amplitude limit and the lighter ones below it then count too, at the lower slope, which a Miner sum over a real traffic spectrum handles and a single vehicle does not. The ranking of the three designs does not change under a spectrum, because each corner’s range scales with the vehicle’s weight; their absolute lives do.

And the stresses are nominal — moment over modulus at the corner. The corner detail’s category already reflects the stress concentration of a flange turning through ninety degrees, and a haunch would move the critical point along the post, as the earlier essay on these corners showed. A haunched girder sized post by post would need its toes sized by the same reading of ranges, and the conclusion — that the middle post must not be lightened — would move with the toe rather than disappear.

A post sized by what it must survive #

The two readings of one girder become, once the posts are sized separately, two different girders with very different futures. A post’s peak and its range differ by a factor that the load’s geometry sets: one at the supports, two at mid-span. A designer who sizes by peaks makes the posts lightest exactly where that factor is largest, and the frame cannot rescue the design by redistribution, because a post that sheds a quarter of its moment while losing three-fifths of its modulus has still doubled its stress.

Sizing by range does not mean making the posts alike, and it does not cost steel. The uniform girder was close to right across its interior and wasteful at its ends, and the design that respects both checks takes its saving from the ends — the posts that never reverse — and leaves the middle alone. The middle post of a Vierendeel girder looks idle on a strength diagram and is working as hard as any post in the girder. It should be sized for the crossings it must survive, which a frame stood on end would call its storeys’ wind reversals, and a bridge calls its traffic.

Still open: the girder whose chords are sized too #

Here only the posts changed. A girder whose posts have been sized to their ranges has a frame whose shear is shared differently between chords and posts, and the chords themselves have peaks and ranges that the crossing load sets panel by panel. Whether the chords, sized each panel to their own ranges, concentrate the moment back into the posts the posts’ design has just settled; whether the two sizings converge to one girder or chase each other; and whether a girder sized member by member for fatigue ends up much lighter than one sized for strength and then checked, is what sizing every member of the frame would ask.