Structural Engineering & Statics
where the load goes
Stability

The frames the deck pushes first

A half-through girder's U-frames are the ends of the cross-girders that carry its deck. Load the deck and every cross-girder sags, its ends rotate, and every frame's foot swings inward by millimetres before the compression chord has asked the frames for anything. It looks like a second imperfection on top of the chord's bow. It is a much weaker one: the frames all move together and the chord simply goes with them, so a 7.5 mm push adds 1.4 kN to frames that the bow loads with 13.6, and stiffer frames make it smaller, not larger. What the push leaves is mostly in the frames' own members, not in what they ask of the chord.

Assumes Held everywhere, and it forgets its length and The load that makes itself worse.

15 min read 6 figures The load must go somewhereWhich failure arrives firstScale changes everything

What the frames hold is the bow found the force a row of U-frames must carry when the compression chord they hold arrives bent. For a chord 24 m long carrying 930 kN, held sideways by frames of 1,400 N/mm every 4 m, every bow from one half-wave to twelve was tried, each with an amplitude of its own half-wavelength over 500, and the worst — four half-waves — put 13.6 kN into a frame, 1.47 per cent of the chord force. The force was the bow, grown by how near the chord works to buckling.

Every frame in that essay was a spring that did one job. Its last section pointed out that a real U-frame does two. It is the end of a cross-girder carrying the deck, and the deck’s load bends the cross-girder and rotates its ends, which rotates the verticals and pushes the chord sideways before any bow is involved. The essay asked whether that deck-induced push, amplified like any bow, adds to the frame force or dominates it.

It adds, a little. The interesting part is why it is so little.

Where the push comes from

A U-frame is a cross-girder with a vertical rising from each end to the girder’s compression flange, the chord. Load the deck and the cross-girder sags. If its joints with the verticals are rigid, the verticals rotate with its ends, and the chord at the top of each vertical is pushed toward the bridge’s centre line by the end rotation times the vertical’s height.

The size of the push follows from the same frame stiffness the chord sees. The frame’s sideways stiffness at the chord, KK, is set by how much the cross-girder bends and the verticals bend when the chord pushes on them. Say a share ψ\psi of the frame’s flexibility is the cross-girder’s bending. For a cross-girder of span BB carrying a line load qq from the deck, simply supported against the deck’s load and bent by equal end moments against the chord’s, the push at a vertical of height hh is

d=qB2ψ12 Kh.d = \frac{q B^2 \psi}{12\,K h}.

Take a footbridge 3 m wide, loaded at 7 kN/m² over the 4 m between cross-girders, q=28q = 28 kN/m; verticals 1 m high; half the frame’s flexibility in the cross-girder. With the earlier essay’s frames, d=7.5d = 7.5 mm. That is the same order as the bow the earlier essay used — the four-wave bow’s amplitude is 12 mm — and every frame is pushed by it at once.

The chord goes with the frames

The calculation is the earlier essay’s with one change. The chord is still cubic beam elements under its 930 kN, the frames still springs at their nodes; but each frame’s foot has moved by dd before the chord arrives, so the frame carries K(u−d)K(u - d) rather than KuKu, and the chord is pushed by KdKd at every frame. The bow and the push are both loads on the same linear second-order problem, so they add as two solutions.

The deck moves the frames' feet, and the chord goes with them. A chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames of 1,400 N/mm every 4 m, carrying 930 kN, in plan, sideways movements drawn 6 times the scale of the length: each frame's foot pushed 7.5 mm inward by its cross-girder's rotation under the deck (arrows), the chord's movement under that push alone (solid), and under the worst bow alone, four half-waves at 12.0 mm (dashed). The chord follows the push, and under its compression goes a little further — 9.1 mm at its middle frames against the 7.5 mm the feet moved — because a chord 24 m long is far softer sideways than the frames holding it; the frames that carry most are the end ones, beside the chord's fixed ends.
Fig. 1 The 24 m chord in plan, sideways movements drawn at six times the scale: each frame’s foot pushed 7.5 mm inward by its cross-girder (arrows); the chord under that push alone (solid); and under the worst bow alone, four half-waves (dashed). The chord follows the push, and under its compression overshoots it — 9.1 mm at its middle frames.

Under the push alone, the chord simply moves with the frames. At its middle frames it moves 9.1 mm — further than the 7.5 mm their feet moved, because the compression in it makes it bow a little more in the direction it has been pushed. Only near its ends, which are held in place by the girder’s end supports, does it lag behind the frames, turning from nothing at the end to the frames’ line within the first bay.

That is the whole explanation of what follows. A chord 24 m long is laterally far softer than five frames of 1,400 N/mm each: its stiffness against a single long half-wave between its ends is a few tens of newtons per millimetre — a point load at its middle meets 48EI/L348EI/L^3, under 9 N/mm. When every frame’s foot moves the same way by the same amount, the chord offers almost no resistance and goes along. The push is not resisted by the chord; it is carried by it.

What the frames feel

The push adds a little, mostly at the ends. For a chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames of 1,400 N/mm every 4 m, carrying 930 kN, the force in each frame, kN: from the deck's push of 7.5 mm alone, from the worst bow alone (four half-waves), and from both in the sense that makes the worst frame worse. The push alone loads the end frames 1.37 kN and the middle one −0.68 the other way — its forces add to 3.32 kN in all, the ends of the chord taking the rest. The bow alone puts 13.63 kN into its worst frame; with the push, 14.99, 10% more.
Fig. 2 The force in each frame: from the 7.5 mm push alone, from the worst bow alone, and from both together in the sense that makes the worst frame worse. The push loads the end frames 1.37 kN and the middle frame 0.68 kN the other way; the bow puts 13.63 kN into its worst frame; both together, 14.99, 10% more.

The frames feel only the part of the push the chord does not follow. The end frames, beside the chord’s fixed ends, feel most: 1.37 kN each, the force needed to bend the chord through its first bay. The middle frame, where the chord has overshot, pulls back by 0.68 kN. The others carry a little. Together they push the chord by 3.32 kN, which the girder’s end supports take.

Against the bow these are small. The worst bow alone loads its worst frame with 13.63 kN. Add the push in the direction that hurts most and the worst frame carries 14.99 — 10 per cent more. The deck’s push is real and it does add; it is a tenth of the frame’s design force, not a second imperfection of the same size.

How large a push would matter

How far the deck would have to push. For a chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames of 1,400 N/mm every 4 m, carrying 930 kN, against how far the deck pushes each frame's foot: the largest frame force from the push alone (dashed) and with the worst bow (solid), beside the bow alone (dotted), in kN. The push's frame force grows 0.18 kN for each millimetre; at 7.5 mm it is 1.37, and the worst frame carries 14.99 against the bow's 13.63. The push would have to be about 75 mm before it alone loaded a frame as much as the bow does.
Fig. 3 Against how far the deck pushes each frame’s foot: the largest frame force from the push alone, from the push with the worst bow, and from the bow alone. The push’s frame force grows by 0.18 kN a millimetre; at 7.5 mm it is 1.37; it would take about 75 mm of push to load a frame as much as the bow does.

Everything here is linear, so the push’s frame force grows in proportion to the push: 0.18 kN for each millimetre. It would take a push of about 75 mm at chord level before the push alone loaded a frame as heavily as the bow. That would mean a cross-girder end rotation of 0.075 radians on a 1 m vertical — a cross-girder sagging by about a fortieth of its span. No deck is designed to deflect like that, so on any frame that is stiff enough to hold the chord, the bow governs.

The reason is the shape, not the size. The bow is short waves, four half-waves in 24 m, and the frames sit near its crests and troughs, where the chord has to bend against them. The push is one long wave: every frame the same. A chord between fixed ends that is asked to move uniformly by the same amount at every frame needs only to bend in its end bays, and bending in two short places costs very little. The settlement that matters is the difference found the same rule for a building on settling ground: what loads a structure is not how far its supports move but how differently. A push that moves every frame alike is a uniform settlement of the chord’s supports, and it loads almost nothing.

Stiffer frames take less of it

The push's force is the chord's, not the frames'. For a chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames every 4 m, carrying 930 kN, against the frames' stiffness on a log scale: the largest frame force from a 7.5 mm push alone (dashed) and from the worst bow alone (solid), in kN. Stiffer frames hold the bow harder and the bow's force settles near its stiff limit; the push's force barely moves — 1.93 kN at 700 N/mm, 1.07 at 60,000 — because frames that all move together just carry the chord with them, and what resists is the chord's own bending between its fixed ends and the frames.
Fig. 4 Against the frames’ stiffness: the largest frame force from a 7.5 mm push alone and from the worst bow alone. The bow’s force has its minimum near the stiffness at which the chord stops using the frames; the push’s barely moves, from 1.93 kN at 700 N/mm to 1.07 at 60,000.

The earlier essay found the bow’s frame force passing through a minimum: least near the stiffness at which the chord stops buckling on the frames, larger for frames just stiff enough and settling toward a limit for very stiff ones. The push behaves differently. Its frame force falls slightly as the frames stiffen — 1.93 kN at 700 N/mm, 1.37 at 1,400, 1.07 at 60,000 — and settles at a limit that does not depend on the frames at all.

The limit is the chord’s. With rigid frames, a push of 7.5 mm at every frame is exactly a support settlement of 7.5 mm at five of the chord’s seven supports, and the forces it causes are the chord’s own reactions to being bent through its first and last bays. Frames softer than rigid give a little, and the chord bends a little less; frames stiffer cannot do better than rigid. The push’s force is a property of the chord, and adding stiffness to the frames cannot attract any more of it.

That reverses the usual intuition about restraint, in which a stiffer support attracts more force. The stiffness a row of braces shares out worked through the ordinary case, where stiffer braces take a larger share of a load that has to go somewhere. Here the load is not a force but a movement, imposed on the frames’ feet, and the chord can follow a movement for free.

Why the bow is amplified and the push is not

There is a second difference between the two, and it is the one that would matter most near the chord’s buckling load. The bow’s frame force grows steeply as the chord’s compression rises; the earlier essay found it at 0.95 per cent of the chord force at a fifth of the critical load and 2.35 per cent at eight-tenths. The load that makes itself worse is the reason: a compression acting through a deflection adds to the deflection, by a factor 1/(1−P/Pm)1/(1 - P/P_m) for a shape whose own critical load is PmP_m, and the four-wave bow is close in shape to the chord’s buckling mode on these frames.

The push is a long wave, and its critical load is far higher. A row of frames is not a foundation found that the chord’s lowest critical load with these frames is in a mode of several half-waves, because the frames prevent the long ones; a single half-wave over 24 m, held at five frames, would buckle only at many times the working force. So under 930 kN the push’s shape is hardly amplified at all. The 9.1 mm the chord moves at its middle frames against the frames’ 7.5 is most of the amplification there is, and it barely loads the frames because it is the same at all of them.

This is the result held everywhere, and it forgets its length found from the other side: a chord on a stiff enough row of restraints does not buckle in long waves, and so it is not sensitive to long imperfections. A push that moves every restraint alike is the longest imperfection there is.

A deck loaded over part of the span

A deck loaded over part of the span pushes unevenly. For a chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames of 1,400 N/mm every 4 m, carrying 930 kN, the force in each frame from a 7.5 mm push at the frames under the loaded part of the deck, kN: all frames, largest 1.37; the first three, largest 1.53; the first two, largest 2.27; the first alone, largest 2.14. An uneven push has more of the short waves the chord resists, so its largest frame force is up to 1.66 times the uniform push's — with the worst bow, 15.89 kN against the bow's 13.63.
Fig. 5 The force in each frame from a 7.5 mm push at the frames under the loaded part of the deck only: all five frames, the first three, the first two, the first alone. An uneven push has more of the short waves the chord resists; its largest frame force is up to 1.66 times the uniform push’s, 2.27 kN, and with the worst bow the worst frame carries 15.89 kN.

A deck is loaded unevenly: a crowd on one half of a footbridge, a vehicle near one end of a road bridge. Then only the cross-girders under the load rotate, and only their frames are pushed. An uneven push is more like the bow, with the chord asked to change direction between pushed and unpushed frames, and the chord resists it more. With the first two frames pushed and the rest not, the worst frame carries 2.27 kN, 1.66 times the uniform push’s largest; with the worst bow, 15.89 kN, 17 per cent above the bow alone.

That is the realistic upper end of the deck’s contribution. It is still a sixth of the frame force, and the load pattern that produces it — the deck loaded over a third of the span — is one the frames must be checked for in any case, because it is also the pattern that maximises the chord’s compression in that third.

What the push does to the chord itself

What the push does to the chord. For a chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames of 1,400 N/mm every 4 m, carrying 930 kN, the chord's own sideways bending moment along it, in kN·m: from a 7.5 mm push alone (solid) and from the worst bow alone (dashed), each under the chord force. The push bends the chord most in its end bays, 1.87 kN·m, where it has to turn from its fixed ends to follow the frames; the bow bends it 12.63 kN·m at its crests. The bow's is 6.7 times the push's.
Fig. 6 The chord’s own sideways bending moment along it: from the 7.5 mm push alone and from the worst bow alone. The push bends the chord most in its end bays, 1.87 kN·m, where it turns from its fixed ends to follow the frames; the bow bends it 12.63 kN·m at its crests, 6.8 times as much.

The chord carries the push as a bending of its own, and the bending is where the chord has to turn: in its end bays, from its fixed ends to the line the frames have moved to. The push’s largest lateral moment is 1.87 kN·m there. The bow’s is 12.63 kN·m, at its crests, 6.8 times as much. The two peaks are in different bays, so they do not even add at the same section.

For the chord’s own check — compression plus lateral bending — the push is therefore smaller still than it is for the frames. It lands in the end bays, where the compression in a half-through girder’s top flange is smallest, because the bending moment that drives it falls to nothing at the supports.

Where the deck’s load does matter

None of this makes the deck irrelevant to the U-frames. It makes it irrelevant to the force the frames must exert on the chord. The frames’ own members are another matter.

A cross-girder with rigid joints to its verticals is a portal, and under the deck load it is not simply supported at all: its ends are partly restrained by the verticals, and the verticals carry a share of the cross-girder’s end moment down to their feet and up to the chord. For the footbridge drawn, the deck puts qB2/8=31.5qB^2/8 = 31.5 kN·m into a simply supported cross-girder. If the verticals restrain its ends by even a quarter, they carry several kN·m each — against the 1.4 kN·m that 1.4 kN of chord restraint puts into a 1 m vertical. The frame’s members are designed by the deck; the frame’s restraint of the chord is designed by the bow. The two are additive in the vertical, and the deck’s share is the larger.

So the earlier essay’s caution that “the frame’s stiffness against the chord is partly spent before the chord asks for it” is half right. The stiffness is not spent: the frame is still a spring of 1,400 N/mm against further movement, whatever its foot has done. Its strength is partly spent, by a moment that has nothing to do with the chord.

What a designer checks, then

The practical consequence is a short list. The U-frames are sized for the chord’s restraint by the bow, with a margin of a tenth or so for the deck’s push, a sixth where the deck can be loaded over a third of the span. The frames’ members are sized for the deck’s own moments, which are larger than anything the chord asks of them. And the chord’s own check needs nothing new: the push’s lateral bending is a seventh of the bow’s, in a part of the chord whose compression is small.

What is not needed is a second, deck-shaped imperfection of full size added to the bow, which is the reading the earlier essay’s last paragraph invited. The brace that need not be strong found that a restraint against buckling is mostly a matter of stiffness, and that the strength it needs is small; this is the same finding for the frames’ strength against a movement the deck imposes on them. The frames need to be stiff to hold the chord’s short waves, and a movement in one long wave costs them almost nothing however large the stiffness.

There is one place where the push would matter, and it is the case the figures cannot reach: frames so soft that the chord is close to buckling on them. There the chord’s long-wave stiffness is no longer negligible beside the frames’, every frame force is amplified, and one support too many applies in reverse — the push is a settlement of supports that the chord now leans on hard. A frame stiffness half of the one drawn already takes the push’s force from 1.37 kN to 1.93, and the bow’s from 13.6 to 18.5. The answer is not to design for the push; it is to keep the frames stiff enough that the chord does not lean on them.

What the calculation takes for granted

Every push is the same and all are inward. On a straight bridge the two girders’ chords are pushed in opposite directions, toward each other, and each is calculated alone; a skew bridge or a bridge with its deck off-centre loads its two sides differently, and the frames then also carry a twist the model does not have.

The verticals are rigidly joined to the cross-girders. A pinned or semi-rigid joint lets the cross-girder rotate without turning the vertical, and there is less push. A rigid joint is the case that makes the push largest, which is the case to check.

The frames are springs, as before. A real U-frame’s stiffness changes a little with the moment its members already carry, and its connections may slip; both make the frame softer and, by the stiffness figure, make the push’s force slightly larger.

The chord force and the deck load are taken at once. The chord’s compression comes from the same deck load that pushes the frames, so in reality they rise together. The figures take the full chord force with each push; a push at partial load meets a smaller compression and is amplified less.

Still open: the frame that is also the bearing

Every frame drawn is an intermediate frame, and the chord’s ends are held rigidly. At a real girder’s ends the chord is held by an end U-frame or a bearing stiffener over the bearings, whose stiffness is larger than the intermediate frames’ but finite — and which is itself loaded by the end reaction’s eccentricity and by the bearing’s own rotation under the girder’s sag. The push’s frame forces were largest at the end frames because the chord’s ends did not move. Whether an end frame that gives, rather than a fixed end, relieves the end frames of the push or simply moves the problem to the end frame itself, and whether the girder’s own end rotation pushes the chord’s end the same way the cross-girders push its frames, is the question the girder’s supports put to the frames between them.

Where this goes next

The bow the frames have to hold. A chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames every 4 m, carrying 930 kN critical at 1548 kN, drawn with its sideways movement exaggerated; its critical load is 1.66 times the chord force. Dashed, the worst of the bows tried: four half-waves, each 12.0 mm high, a five-hundredth of its own length. Solid, the chord once the compression has grown the bow; each arrow is the force the frame there must supply, the largest 13.6 kN — 1.47 per cent of the chord force. What the frames hold is the bow A row of U-frames is sized for a force, and the rules give that force as a percentage of the compression chord's force that nobody derives. Derived, it is not a property of the frames at all. It is the chord's initial bow, grown by how near the chord works to buckling — and it is least, not greatest, near the stiffness at which the chord stops using the frames. Stiff frames tend to a limit that is π²/500 of the chord force: the familiar two per cent, arriving from a bow of one five-hundredth. The same restraint, spread and gathered. The buckled shape of a 24.0 m compression chord with the same smeared restraint, 0.35 N/mm per mm, delivered by U-frames at two spacings. With frames every 2.0 m the chord buckles in half-waves of about 5.1 m that ignore the frames — the smeared shape — at 1892 kN, matching the smeared answer. With frames every 4.0 m it buckles between them, with a node at every frame, at 1548 kN: 18 per cent below the smeared 1897 and at the Euler load of one bay. The dots are the frames. A row of frames is not a foundation The top chord of a half-through girder is held sideways by U-frames, and the standard calculation smears them into a continuous elastic foundation. That is right while the frames are close and quietly wrong once they are not. The crossover sits at seven-tenths of the buckle's own half-wavelength, whatever the frames' stiffness — and beyond it the chord buckles between frames at a load no stiffening of the frames can raise. The load that makes itself worse. The amplification of a deflection against the ratio of applied load to buckling load. A structure at half its buckling load deflects twice as far as first-order analysis predicts, and the curve runs away well before the load is reached. The load that makes itself worse A structure that has leaned carries its weight off the axis, which makes it lean further. The amplification is one over one minus the load ratio, and it runs away long before the buckling load.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

AmplificationBracingContinuous restraintCross-girderImperfectionSecond-orderSupport settlementU-frame