The frame that waits for a heavy lorry

The stiff frame among soft ones drew a steel box with two kinds of spring against distortion. Ring frames every 4 m were a soft foundation smeared along it, and cross-frames of two angle diagonals every 20 m were hard points on that foundation. Every one of those cross-frames was taken as tight, stiff from the first thousandth of a milliradian. That essay’s question was how many a box needs. This one asks whether a cross-frame is really there.

Most of them are bolted, and in a bolted site connection the holes are 2 mm larger than the bolt so that the steel can be put together. A diagonal does nothing until its bolts have come into bearing, which requires the connection to move through whatever clearance happens to have been left, and until then a frame the calculations count as a spring is a loose angle hanging in a box. Against a box 4 m wide a millimetre sounds like nothing. The right comparison is not with the box’s width. It is with how far a lorry actually distorts the box, and among ring frames that is very little, so the question arises whether the clearance is a detail or the whole of the frame’s behaviour.

How far a millimetre goes #

The box is that essay’s: 60 m long, 4 m wide and 2.5 m deep, with orthotropic walls. Its distortion is a beam on an elastic foundation, $EI_{dw}\gamma'''' + K_d\gamma = m/2$, where $\gamma$ is the change in the corner angle, the warping rigidity plays the part of the beam, and the transverse stiffness of walls and ring frames together plays the foundation. Its decay length is 6.6 m. On top of that sit discrete springs for the cross-frames at 20 m, each two diagonals of 1,900 mm².

A diagonal’s length follows from the corner angle by geometry. For a rectangle $b \times h$ with diagonal $d$, a change in the corner angle lengthens one diagonal and shortens the other by

$$\delta = \gamma\,\frac{bh}{d},$$

which is 2.12 m on this box: a thousandth of a radian moves the diagonal 2.1 mm. So a clearance $c$ in the connection is a dead band in the angle, $\gamma_g = c\,d/bh$, and each millimetre of clearance is 0.47 mrad of distortion in which the frame takes no load. Outside that band it has the stiffness it always had.

The load is a four-axle lorry of 440 kN standing with its centre line 1.5 m off the box’s, the lorry the earlier essay’s fatigue argument was about, and it is that load, moving along the span, that the frame has to answer to.

What the lorry asks #

With no cross-frame at all, the lorry’s largest corner-angle change is 0.49 mrad. That is the whole of what the frame is being asked to take up, and a millimetre of clearance is 0.47 of it. The tight frame pulls the distortion down to 0.21, a dent in the curve at its own station, exactly the point restraint the earlier essay drew. With a millimetre of clearance the curve is indistinguishable from the one with no frame: 0.48 at the station against 0.49, a dent of a hundredth of a milliradian. Among ring frames, a one-millimetre hole is nearly as large as the problem the frame was put there to solve.

That is the ring frames’ doing. They are what have brought the lorry’s distortion down to half a milliradian, and they are stiff enough that nothing here is much affected by the frame at all, as the earlier essay found: the cross-frames’ share of the distortion had already fallen to 27 per cent. Clearance removes what is left of that share.

A lorry below which the frame does not exist #

Read against the lorry’s weight, a clearance is a threshold. Distortion is proportional to the lorry, so a 1 mm band is a weight of about 425 kN, 2 mm is 850 kN, and half a millimetre is 210 kN, and under that weight the frame takes nothing whatever. Above the threshold the range rises along a line parallel to the tight frame’s. It does not rise to meet it. Once the bolts are in bearing the frame answers each further newton at its full stiffness, but the movement that went into the hole is gone, and the tight frame’s line stays above it by a fixed amount for every lorry heavier than the threshold.

The picture looks like a joint in a line of braces with slack in their bolts, where it was slack against a storey’s own sway, and the same reasoning applies: slack subtracts from a displacement and so from every force the displacement causes, and it does this by a constant. It is the opposite of the case the force a filler can promise worked through, where a gap delays a contact and the contact then arrives as an impact; in a static reading of a lorry crossing at its slow pace there is no impact to deliver. A real lorry takes the band up at a speed, and a real bolt arrives in bearing with a small knock, and neither is on this picture.

What the frame sees of one passage #

As the lorry crosses, the tight frame sees a smooth hump, with a small reversal when the lorry is a span away, on the far side of a neighbouring frame. With half a millimetre the hump’s top has been kept and its shoulders cut off: the frame carries nothing until the lorry is 8 m from it, and nothing after the lorry has gone 8 m beyond it. With a millimetre, it carries a small bump for the few metres of road nearest it, and that is all it does, under this lorry, in the life of the bridge.

The reversal is lost as well. A tight diagonal has a small compression before its tension and so a slightly larger range than its peak; a slack one has no such thing, because a small movement of the box the other way stays inside the hole. So the frame in the slack condition sees only one sign of stress for any given lorry lane, and the diagonal that was working in compression for the tight frame has its turn when a lorry goes by on the other side of the box.

The hard point, which was already smaller than it looked #

The earlier essay’s concern about cross-frames was that a stiff point in a soft foundation raises the warping stress at itself, from 4.1 to 7.1 N/mm² under a load spread along the whole span. So a frame that goes slack would seem to be a benefit, since slack removes the hard point. Under a lorry the case comes out differently. The warping range at the frame’s own station is 2.41 N/mm² with the tight frame and 4.03 with no frame. A lorry is a concentrated distortional load, with its own sharp curvature of $\gamma$ under it, and when it stands on the station the frame takes away more of that curvature than it adds. The frame’s own point curvature opposes the lorry’s.

So there are two opposed effects. A load spread over a span gives no curvature of its own, and the frame’s hard point is all there is. A lorry supplies its own, and the frame is relieving it. The warping range even falls slightly as a little clearance is introduced, to 2.02 at 0.13 mm, and rises from there to the no-frame value by about 1.1 mm. All of these are small numbers, a few N/mm², and the earlier essay’s 7.1 is a statement about a different loading. What can be said is that clearance moves warping at the station by at most a factor of two in either direction, and in neither case to anything that would govern.

The spectrum, and the rare event #

The question put to this frame was a fatigue question: whether a slipping frame is a hard point that switches on only under the heaviest lorries, concentrating the range into rare events, or whether it is exactly the decoration the decay length says it is. Both are true at different clearances, and the first of them turns out not to be a penalty. The spectrum here has five lorries, 200, 310, 390, 450 and 490 kN, in the usual proportions of 20, 5, 15, 10 and 50 per cent, with damage going as the cube of the range as the same weight dropped again and every fatigue reading on a box like this assumes. The tight frame’s damage-equivalent range is 18.9 N/mm². With half a millimetre, 10.2, and 80 per cent of lorries reach it. With a millimetre, 2.3, and 60 per cent. With a millimetre and a quarter, none of them do.

So the slack frame does concentrate its working into the heavier end of the spectrum, but since each event it sees is smaller than the tight frame’s by a constant, the result is less damage rather than more, and by a large margin: at 1 mm the cube is less than a five-hundredth of the tight frame’s. A slipping frame has a longer fatigue life than a tight one, if the only thing that is being counted is steel stress and the movement is quiet.

What a fatigue check cannot count here is the hole. A connection that opens and closes through its clearance a few hundred times a day has its faying surfaces fretting and its bolts knocking, and a joint the fatigue tables were not written for. There is good reason not to draw a static curve for the case at all. Whether a bolted connection that slips into bearing under every heavy lorry stays in that state for fifty years is a question for its own detail category, and the tables have only partial answers to it; the figures above answer the different question of what the frame carries.

Whether anything is lost #

The diagonal’s stress may go down, but the corner angle goes up. The box without the frame’s restraint distorts 0.49 mrad instead of 0.21, the plates and ring frames carry the difference in transverse bending, and a deck plate’s fatigue depends on exactly those stresses. Since the ring frames are carrying the whole of the distortion, the deck whose frame is its crossbeams is the relevant check, and the increase there is the factor between those two curves, about 2.3 at the station and less away from it. If the design relied on the frames to bring the transverse stress in the plate within its limit, then the clearance has quietly unbuilt the thing it relied on.

There are two ways to deal with it. One is to fit the connection so it has no clearance, with preloaded bolts in normal holes. The force is small: a diagonal’s 19.3 N/mm² on 1,900 mm² is 37 kN, well inside the slip resistance of two preloaded M20s. Such a frame stays tight for every lorry on the spectrum and at the design lorry too, and the wrench that reads the rust sets out how to find whether a preload is still there decades later. The other is to accept what the numbers suggest: drop the frames that do nothing, and rely on the ring frames. Only the first of those is a decision a site can make, and only a design that took the clearance into account can make the second.

Without ring frames #

All of the above depends on how little the box moves, and that is the ring frames’ doing. Take them away, leaving walls of orthotropic plate, and the lorry’s distortion is of another order: about 28 mrad with no frames at all, and 0.41 once tight frames are holding it. A 2 mm clearance is then 0.94 mrad of dead band, a thirtieth of what the lorry would do if nothing stopped it, and the frame still has to stop it. The angle at the station rises from 0.41 to 1.34 mrad, and the diagonal’s range falls only from 41.7 N/mm² to 36.2. A clearance takes a fixed movement from a frame, and only matters in proportion to how much movement there is. A box that truly depends on its cross-frames does not care about its bolt holes, and the box that has been given enough ring frames for the frames to be a small correction is the box in which the holes decide the matter.

So this is why the difference between two sorts of box is worth knowing before the fit-up specification is written. The diaphragm that is a few angles gave the stiffness a set of angles has to reach to count as a diaphragm, and found that a modest one is enough. Clearance does not lower that stiffness, and it raises the amount of movement the frame needs to reach it. In a box with few other frames, one diaphragm being nearly none makes the opposite point about their spacing, and there the clearance can be neglected.

By hand, for the threshold #

The corner-angle change the 440 kN lorry would produce if there were no frame is 0.49 mrad. A diagonal’s movement is $\gamma\,bh/d$, so the band for 1 mm of clearance is $\gamma_g = 1 \times 4{,}717/(4{,}000 \times 2{,}500) = 0.472$ mrad. The lorry that just reaches it weighs $440 \times 0.472/0.489 = 425$ kN.

The tight frame’s own effect takes one line more, since the frames are far enough apart to be treated one at a time. A 20 m spacing against a 6.6 m decay length leaves little for a neighbour to do. A point restraint on an elastic foundation has a flexibility at itself of $F = \lambda/2K_d$, with $\lambda = 1/6{,}600$ mm⁻¹ and $K_d = 4.06 \times 10^7$ N·mm/mm, so $F = 1.87 \times 10^{-12}$ per N·mm. The frame’s stiffness is $K = 2EA\,b^2h^2/d^3 = 7.4 \times 10^{11}$ N·mm, so $KF = 1.39$, and the angle under the frame falls by $1 + KF$: 0.489/2.39 = 0.205 mrad. That is the 0.21 in the first figure, from a formula with no series in it, and $KF$ is the number to look at. A frame with $KF$ much below one has nothing to give whatever its bolts do, and a frame with $KF$ far above one is holding the box and can afford its holes.

With a tight frame the angle at the station is 0.205 mrad under that lorry, and the diagonal’s strain is the movement over its length: $\gamma\,bh/d^2$, or $0.205 \times 10^{-3} \times 10^7/(2.225 \times 10^7) = 92 \times 10^{-6}$. At 205,000 N/mm² that is 18.9 N/mm², matching the peak in the passage figure.

What these springs leave out #

A symmetric, frictionless clearance. The dead band is drawn as ±1 mm with the bolt centred in the hole. A bolt is centred only by accident. Fit-up usually pushes it hard against one side, leaving no clearance in one direction and twice as much in the other, and a dead load that loads the frame presses it against that side for good. Such a frame acts tight in one sense and very slack in the other, and a lorry in one lane is then treated differently from a lorry in the other.

No friction. Even a snug bolt has some, and it holds the frame together up to its small slip force. The model passes through that as if it were not there. With full preload the frame is tight and this essay does not apply to it.

A static lorry on a straight box. The frame’s dead band is taken up at no speed, and the bolt arrives without impact. Support skew, and the distortion from any cross-fall or other eccentric loading already on the box, would use some of the band before the lorry arrives.

One lorry at a time. Two lorries side by side on opposite carriageways produce less distortion, and two in a lane more; the threshold stated as a weight applies to the sum, in a manner that depends on their spacing.

Still open: the plate that the frame was protecting #

Every number here has been about the frame. Its purpose was the box’s plates: the transverse bending in the walls and the fatigue-prone details at their stiffeners, whose stress follows the angle. A clearance of a millimetre leaves that angle 2.3 times what the tight frame gave at the station, in a band of a few metres around each frame, and much less elsewhere. Whether a plate detail near a frame loses a factor of about twelve in fatigue life to its unused hole, or whether the same detail at midway between frames never noticed there was a frame at all — so that a fitting tolerance has moved the critical location along the box — is what this frame leaves to the plate it was supposed to protect.