The stiff frame among soft ones

The deck whose frame is its crossbeams took a 60 m steel box girder, 4 m wide and 2.5 m deep, whose top flange is an orthotropic deck: a 14 mm plate on longitudinal ribs, with no continuous section frame of its own worth the name. Its only stiffness against distortion — the rectangle becoming a rhombus under an eccentric load — came from a ring frame at each crossbeam, the crossbeam welded into a stiffener on each web and one across the bottom flange. With ring frames every 4 m the bare deck’s 145 milliradians of distortion fell to about one, and the frames, though discrete, acted exactly as a continuous foundation smeared along the span while they stood within half a decay length of each other.

Real steel boxes rarely rely on ring frames alone. The common arrangement, which that essay ended on, puts a lighter cross-frame — two diagonals across the box — at every second, third or fifth crossbeam, and leaves the crossbeams between as they are, sometimes full ring frames and often only tees under the deck. The springs then come in two kinds: stiff ones at a wide spacing and soft ones at a close one. Whether the soft ones carry enough of the distortion that the stiff ones can be spaced as diaphragms are, or whether the box between the stiff ones distorts as if the soft ones were not there, is a question the two kinds of spring put to a single decay length.

Two kinds of spring #

A cross-frame’s stiffness against distortion is a matter of its diagonals. Distorting the section by a corner-angle change γ stretches one diagonal and shortens the other by $(bh/d)\gamma$, d being the diagonal’s length, so a pair of diagonals of area A resists with a generalised stiffness $K = 2EAb^2h^2/d^3$ — for two 1,900 mm² angles across this box, $7.4 \times 10^{11}$ N·mm per radian. A ring frame is a closed frame of four bending members, and its stiffness is a frame solve: $1.7 \times 10^{11}$ for the crossbeam and stiffeners of the earlier essay. An open tee crossbeam — the same crossbeam, with nothing but the box’s own plates at the webs and the bottom flange — is $1.6 \times 10^9$, a hundredth of the ring frame, because the plates it must bend at the corners are 14 and 20 mm thick.

So the cross-frame is four times as stiff as a ring frame and four hundred times as stiff as an open tee. The two arrangements are solved the same way: the crossbeams smeared into a continuous foundation of $K/s$ per metre, less the walls’ own plate stiffness the foundation already contains, and the cross-frames as discrete springs on it.

The ring frames alone hold the distortion to 1.13 milliradians at its worst, near the ends. The cross-frames alone, every 20 m with nothing between them, allow 3.22: the box sags into a rhombus between each pair, held at the frames. Both together give 0.90. Adding the cross-frames has taken a fifth off the ring frames’ distortion — less than might have been expected from springs four times as stiff, and the reason is the length over which a ring-framed box can feel a support at all.

The decay length of the soft frames #

A box on a continuous foundation against distortion is a beam on an elastic foundation, and a point support on such a beam is felt only over its decay length, $(4EI_{dw}/K_d)^{1/4}$. The bare box’s walls give it a decay length of 24 m, which is why a single diaphragm reaches a long way in a box with no frames. Ring frames every 4 m raise the foundation stiffness a hundred and seventy times and shorten it to 6.5 m.

A cross-frame 20 m from the next, in a box whose decay length is 6.5 m, is three decay lengths from its neighbour. Between them the ring frames have taken over: the distortion midway between two cross-frames is what the ring frames would have given anyway, and the cross-frame holds only its own few metres. It is the same arithmetic one diaphragm is nearly none found for a single rigid diaphragm in a long box, applied to a box whose foundation is already stiff.

A hard point in a soft foundation #

The distortional warping stress tells the other half of the story. With the ring frames alone it is a gentle wave, 4.1 N/mm² at its largest near the end diaphragms and almost nothing in the middle third of the span, where the distortion is uniform and so needs no warping to change. With the cross-frames among them, it spikes at each cross-frame to 7.1 — three-quarters more than the ring frames alone produced, at exactly the stations where the stiffness was added.

The mechanism is a beam on an elastic foundation with a stiff point support in it. The warping stress is the “bending moment” of that beam — it is $E w_c \gamma''$, the curvature of the distortion along the span — and a stiff point support on a soft foundation makes a sharp kink in the deflected shape, with a peak of curvature at the support. A box held by ring frames distorts gently and smoothly; a cross-frame among them holds one cross-section firmly while its neighbours a few metres away are still distorting, and the walls must warp sharply to accommodate the difference. The stiffer the cross-frame relative to the ring frames, the sharper the kink.

The size of the peak can be had by hand. A cross-frame holding a box on a foundation is a point reaction R on a beam on an elastic foundation, and the largest bending moment under a point load on such a beam is $R\lambda/4$, λ being the decay length — the result the beam that sits on the ground found under a column on soil. For the distorting box the “moment” is $EI_{dw}\gamma''$ and the stress is $E w_c \gamma''$, $w_c = bh/8$ being the warping function’s value at a corner, so the peak warping stress at a cross-frame is

$$\sigma_w \approx \frac{w_c\,R\,\lambda}{4 I_{dw}}.$$

At 20 m each interior cross-frame takes a distortional reaction of about $3.6 \times 10^8$ N·mm, and with λ = 6.5 m and $I_{dw} = 9.5 \times 10^{16}$ mm⁶ that is 7.7 N/mm², against the 7.1 the full solution gives where the cross-frame’s neighbours and the box’s ends also act. The formula also says why the curve flattens as the spacing grows: the reaction a cross-frame can attract is limited by the distortion the ring frames leave at its station, so once the cross-frames are several decay lengths apart each takes the same reaction and makes the same peak, however far its neighbour is.

This is not only a number. The distortional warping stress is a longitudinal stress at the web-to-flange corners, added to the girder’s bending stress, and it is a range under every eccentric lorry. The load that never came near failing anything found that a small, repeated stress at a weld is what cracks steel bridges, and a cross-frame connection is a welded detail at the corner where this stress peaks. A designer who adds cross-frames to a ring-framed box to be safe has put a fatigue concentration at each of them. And the fatigue check would see it twice over: once as the stress range, raised by the hard point, and once as the detail, because the detail decides and the steel does not — a cross-frame gusset welded to a web stiffener at a corner is one of the poorer categories a box girder contains, sitting where the range is largest.

Open tees are another box #

Open tees change everything. A crossbeam that is a tee under the deck stiffens the deck plate between the webs, but the frame’s corners are still the box’s own plates bending, and the ring it closes is only a little stiffer than the walls already were: with open tees every 4 m the box distorts 104 milliradians, against the bare box’s 145. Its decay length is 21 m. The cross-frames are now the only real frames in the box, and with them added the distortion is 3.19 milliradians, against 3.22 with the cross-frames alone. The open tees between them make no difference: the box between the cross-frames distorts as if they were not there.

The difference between the two crossbeams is a pair of stiffeners and a weld round the corner. It is small in steel and in fabrication, and it decides which of two different structures the box is. With ring frames, the crossbeams are the distortion’s foundation and the cross-frames are hard points in it. With open tees, the cross-frames are diaphragms and must be spaced as diaphragms are — close enough, as the earlier essays found, that the distortion between them stays small.

The spacing that matters, and the one that does not #

With nothing between them, the cross-frames’ spacing is the whole design: the peak warping stress is 3.8 N/mm² at 8 m, 21.8 at 20 m and 60.8 at 30 m, rising steeply once the cross-frames are more than a fraction of the bare box’s decay length apart. Open tees follow the same curve within a few per cent. With ring frames between them, the curve flattens: 3.1 at 8 m, 7.1 at 20 m, 7.3 at 30 m, settling at the ring frames’ own level plus the hard point’s peak.

Read the other way, the solid curve says when cross-frames pay. At 8 m, two ring-frame spacings, they bring the peak below the ring frames’ own 4.1 — they are close enough to work together as a stiffer foundation. From about 10 m they are above it: each is a hard point, and widening the spacing only removes some of them. In a ring-framed box, cross-frames earn their place close together or not at all. At a wide spacing they do nothing the ring frames were not doing, except concentrate the stress.

Which spring governs #

The two extremes are one box with different crossbeams, and the transition between them can be put on one axis.

When the cross-frames stand within about one decay length of each other — measured with the crossbeams’ decay length, not the bare box’s — the ratio is near one: the crossbeams barely change the answer, and the cross-frames act as diaphragms. Beyond about two decay lengths the ratio falls steeply, and the crossbeams carry the box. A crossbeam a hundredth as stiff as a ring frame — the open tee — has a decay length of 21 m, and the cross-frames would have to be spaced at 40 m before it mattered; the ring frame’s 6.5 m makes it matter at 13.

That gives a designer a single test for the arrangement on the drawing. Compute the decay length of the box with its crossbeams smeared, and compare it with the cross-frame spacing. If the spacing is shorter, the cross-frames are the diaphragms and the crossbeams are decoration as far as distortion goes. If it is longer, the crossbeams are the foundation and the cross-frames are hard points, to be justified by something else — erection, the support of a bearing, a change of section — and detailed for the stress they concentrate.

Who carries the distortion #

The forces say the same. With nothing between them, or with open tees, the interior cross-frames at 20 m take 72 per cent of the distortional load between them and the end diaphragms most of the rest. With ring frames between, the same cross-frames take 27 per cent; the ring frames carry the rest, a little at each. Each cross-frame’s diagonals are then designed for about a third of the force a designer who ignored the ring frames would give them — but the connections at its corners still see the warping peak, because that is a property of the stiffness contrast, not of the force.

The diaphragm that is a few angles found that a cross-frame of light angles is stiff enough to act as a rigid diaphragm in a box with nothing else in it. In a box with ring frames it is stiff enough to be an unwanted one.

By hand, for the decay lengths #

The walls of the box give it a distortional foundation of $K_w = 2.5 \times 10^5$ N·mm per radian per millimetre of span, and a distortional warping rigidity $EI_{dw}$ with $I_{dw} = b^2h^2(b t_f + h t_w)/96$. The decay length is $(4EI_{dw}/K_d)^{1/4}$: 24 m with the walls alone.

A ring frame of $1.7 \times 10^{11}$ every 4,000 mm adds $4.3 \times 10^7$ per millimetre, 170 times the walls; the fourth root of 171 is 3.6, and $24/3.6 = 6.6$ m. An open tee of $1.6 \times 10^9$ every 4,000 mm adds $4.0 \times 10^5$, less the walls’ $2.5 \times 10^5$ that the strip of plate inside it already counted, so $1.5 \times 10^5$ — and the fourth root of $(2.5 + 1.5)/2.5 = 1.6$ is 1.12, so $24/1.12 = 21$ m. A factor of a hundred in the crossbeam’s stiffness is a factor of three in the decay length, and the cross-frame spacing of 20 m sits between the two.

The ends are hard points too #

The same reading explains where the ring-framed box’s own warping lives. A box held by ring frames alone has its largest warping stress near its end diaphragms, 4.1 N/mm² about a decay length in from each end, and almost none in its middle. The end diaphragms are the box’s original hard points: rigid frames over the bearings, holding the section square while the ring frames a few metres away let it distort. It is the reason the section that cannot stay flat found an open section’s warping stress gathered at its restrained ends: warping stress is made where warping is prevented, and a diaphragm prevents the distortional kind. The concentration reaches about one decay length — 6.5 m here, 24 m in the bare box.

So the ring-framed box with cross-frames has two kinds of hard point and they behave alike: the end diaphragms, which it must have, and the interior cross-frames, which it may not need. A designer reading the warping stress along the span sees peaks at both, and the interior peaks are the ones that can be designed away by leaving out the cross-frames, or softened by making them slip.

What smearing the frames leaves out #

The crossbeams are smeared. The earlier essay found that discrete ring frames every 4 m behave as their smeared foundation to within 6 per cent; a fully discrete calculation of the cross-frames among ring frames gives the peak distortion within 2 per cent of the smeared one at a 20 m spacing and within 13 per cent at 12 m, and the peak warping within 7 per cent at both. Near a cross-frame the nearest ring frames are a metre or two away, and their discreteness softens the hard point’s peak a little.

The cross-frame’s diagonals stay elastic and both work. Slender diagonals that buckle in compression halve the cross-frame’s stiffness, which softens the hard point; connections with slip in their bolt holes soften it further, and a cross-frame whose bolts are in clearance does nothing at all until the distortion takes up the clearance.

The load is one eccentric line load along the whole span. A moving lorry distorts the box locally, over a decay length or so either side of it, and the stress range at a cross-frame is the difference between the lorry over it and the lorry far away — which is where the hard point’s peak matters most, and where the section that will not keep its shape found distortion to be at its least intuitive.

Still open: the box whose cross-frames slip #

Every cross-frame here is welded or bolted tight, so it is stiff from the first millimetre of distortion. Most are bolted, and the bolts are in clearance holes: the frame does nothing until the box has distorted enough to take up the clearance, and then everything. In a box with ring frames, whose distortion under a lorry is a fraction of a milliradian, a millimetre of clearance in each diagonal’s connection may never be taken up at all — the cross-frame would then be invisible, and the hard point would vanish. Whether a slipping cross-frame among ring frames is a hard point that switches on only under the heaviest lorries — a fatigue range concentrated into the rare events — or whether the clearance it never takes up makes it, in service, exactly the decoration the decay length says it is, is the question a bolt hole puts to a spring that was assumed to be one.