The braces with slack in their bolts

The braces that pull only one way took the 24 m plate girder of the bracing essays — 1,225 mm between flange centres, under a uniform load that brings its moment to 5,627 kN·m at midspan — braced it with three torsional braces at midspan and 6 m either side, and asked what happens when each brace resists the girder’s twist in one direction only. The girder buckles in whichever direction its braces leave it free to, and a row of one-way braces never brings it to its target. Every brace there was either fully slack in one direction or fully stiff in the other.

Many braces that resist both ways are neither. A cross-frame between two girders is bolted to stiffeners with bolts in clearance holes, a millimetre or two larger than the bolts, and until the girder has twisted far enough to take up that clearance the cross-frame does nothing at all. Then it resists fully, in either direction. That essay ended on whether a row of braces with a few millimetres of slack each brings a girder to the moment its stiffness promises, or whether the slack acts as an imperfection the braces have to overcome before they help, as the brace that need not be strong found an imperfection does to a brace’s force.

A girder that is not straight to begin with #

A girder that buckles needs to be told which way. A perfect one has no twist until its critical moment and then any amount, in either direction; a real one is erected with a little twist already in it — a compression flange a few millimetres out of line with its tension flange — and the load amplifies that twist from the first kilonewton. Braces exist to hold the amplification down, and the force they carry is the force of holding it. The column that was never straight found the same arithmetic for a strut and its bow.

The girder here is erected with an initial twist of 5 milliradians at midspan, in a single half-wave: its compression flange 6 mm sideways of its tension flange at the centre. The braces are fitted to it as it stands, and each is twice the stiffness that would just bring a perfect girder to 5,627 kN·m — 5,691 kN·m/rad against 2,846, the doubling that bracing rules ask for so that the twist at the design moment stays a small multiple of the girder’s own. The analysis is second-order: the twist is a sine series, the lateral movement is eliminated exactly, and each brace adds its stiffness only once the twist at its station exceeds its slack.

Twisting into the braces #

With tight braces the girder’s twist rises from its initial 5 milliradians to 11 at the target, the curve steepening towards the braced girder’s critical moment beyond. With slack braces the girder starts the same way, but on its own: the first few hundred kN·m twist it as an unbraced girder, along the flat curve that would run away at 947. Before it gets there the twist reaches the slack — at 579 kN·m for 3 milliradians of slack, at 795 for 12 — the braces bite, and the girder climbs again on a braced curve. But it climbs from where the slack left it, a few milliradians further along than the tight girder was at the same load. At the target it has twisted 18.2 milliradians with 3 of slack, 25.4 with 6 and 39.8 with 12.

So the answer to the first half of the question is yes: a row of slack braces does bring the girder to its target. The braces always engage before the unbraced moment, because an imperfect girder’s twist grows without limit as it approaches that moment and must pass any finite slack on the way. The cost is in the second half: the girder arrives at its target more twisted, and so more highly stressed in its flanges and more heavily loaded at its braces.

Slack is costlier than crookedness #

The natural guess is that slack is simply more imperfection: a girder whose braces start 3 milliradians late is a girder 3 milliradians more crooked. The figure tests it. With tight braces and the girder’s own 5 milliradians, the middle brace carries 34.1 kN·m at the target. Every milliradian of slack adds 8.0 kN·m; every milliradian added instead to the initial twist adds 6.8. A milliradian of slack costs the middle brace about a sixth more than a milliradian of crookedness.

The difference comes from when each is paid for. An initial twist is resisted by the braces from the first kilonewton, so only its amplified part loads them: of the 11 milliradians at the target, the braces resist the 6 the load has added. Slack is free until it is taken up, and the girder twists through it under an unbraced girder’s amplification, which near the unbraced moment is large; when the braces engage, the girder is already at its initial twist plus the slack plus the amplification of both, and the braced girder then amplifies that total for the rest of the load. The slack is an imperfection that the girder enlarges before the braces start, rather than one they hold from the start.

At ordinary clearances the effect is not small. A pair of bolts in 2 mm clearance holes at the top and bottom of a cross-frame over the girder’s 1.2 m depth is about 3 milliradians of slack, and that raises the middle brace’s force from 34 to 58 kN·m — by 70 per cent — and the girder’s twist at the target from 11 to 18 milliradians.

No stiffness takes it back #

The usual remedy for a brace force that is too high is a stiffer brace, which holds the twist closer to the girder’s initial shape and so asks less of itself. Slack changes what that remedy can reach.

With tight braces, stiffening them works as it should: the middle brace’s moment falls from 34.1 kN·m at twice the stiffness to 21.7 at five times and 18.2 at thirty — towards the force needed simply to hold the girder at its initial twist under the target moment, which is what a rigid brace carries. With slack braces it falls too, but towards a higher floor: 58.1, 37.3, 31.4. A rigid brace with slack still lets the girder twist through the slack before it bites, so it ends holding a girder that much more crooked, and the force of holding it does not depend on the brace’s stiffness at all.

The twist tells the same story. Tight braces at thirty times the stiffness hold the girder to 5.2 milliradians at the target, barely more than its initial 5; slack ones hold it to 8.4, the initial twist plus the slack plus what the girder did on its own before the slack was taken up. Stiffness buys back the amplification; it cannot buy back the slack. The only thing that removes slack’s effect is removing the slack: tightened connections, slotted holes set with the girder held straight, or bolts installed after the girder is plumbed.

The slack is always taken up early #

When the slack is taken up depends on how crooked the girder was. An unbraced girder’s twist grows roughly as $\theta_0 M^2/(M_0^2 - M^2)$, so it reaches a slack g at $M = M_0\sqrt{g/(\theta_0 + g)}$: the faint curves, which the full solution follows to within a few kN·m, because the middle brace sits at the crest of the half-wave the girder first twists in. A girder erected with 10 milliradians of twist takes up 3 of slack at 454 kN·m, under half its unbraced moment; one erected with 2.5 takes it up at 699. A straighter girder runs longer on its own before its braces start, and arrives at them later and closer to its unbraced moment — but never past it.

That last point is the part of the question a perfectly straight girder would answer differently. With no initial twist at all the girder has nothing to amplify until its unbraced moment, and at 947 kN·m it buckles, free, until it hits its braces. The imperfect girders draw that event out into a curve; the perfect one would make it a jolt. Real girders are imperfect enough that the jolt is a curve, but it is worth knowing that the straighter the girder was erected, the closer its braces come to being engaged by a buckle rather than by a drift.

The braces bite in turn #

The curves in the first figure have a second kink, higher up, and it is the outer braces. The middle brace sits at the crest of the girder’s half-wave and takes up its slack first; the outer braces, 6 m either side, see less of the twist and take up theirs later — with 3 milliradians of slack, at about 1,550 kN·m, more than two and a half times the moment at which the middle one bit. Between the two the girder is braced at midspan only, and the twist it gathers there is the twist the outer braces then inherit. With 12 milliradians of slack the outer braces wait until about 2,040 kN·m.

That sequence is why the slack’s cost is not a simple offset. A row of braces that engaged together would add the same imperfection everywhere; a row that engages from the middle out lets the girder take a two-wave component while it is held at one point only, which is the shape twice the moment, four times the brace found a single midspan brace leaves the girder free to adopt. The outer braces then hold a girder twisted into a shape their tight counterparts never let it reach.

A slack brace loads its neighbours #

Slack in one brace is not that brace’s problem alone. With only the middle brace slack, the girder twists further at midspan before it is held there, and the outer braces, tight, carry 22.5 kN·m each instead of 15.7 — the middle brace’s delay has handed them part of the work. The middle brace itself, once engaged, ends at 44.5, more than its tight 34.1, because it is holding a more twisted girder. With only the outer braces slack, the middle brace carries 47.7. Every arrangement with any slack loads every brace more than the tight row does, and the one with all three slack loads them most.

A designer checking braces one at a time, each against its own stiffness and the girder’s initial twist, would miss this: the force in a tight brace depends on whether its neighbours have slack. The braces that stay where the framing puts them found that braces in a row share the girder’s twist in proportion to where its mode puts it; slack moves the mode, and the shares with it.

What a brace check should assume #

Bracing rules size a torsional brace for two things: a stiffness, usually twice the ideal, and a strength, the force of holding the girder’s initial twist amplified by the load. Neither mentions slack, and the figures say how it enters: as extra initial twist, at a premium. A brace check that adds the connection’s slack to the girder’s assumed initial twist, and then a sixth to a fifth more, reproduces the middle brace’s force that the second-order analysis finds for this girder at every stiffness drawn: the premium is 1.17 at twice the ideal stiffness and 1.21 at thirty times. For the twist the premium is smaller — each milliradian of slack adds 2.4 milliradians at the target where a milliradian of initial twist adds 2.2 — because a brace’s force is set only by the twist the girder adds past its fitted position, less its slack: a milliradian of initial twist adds 1.2 milliradians of that at the middle brace, and a milliradian of slack adds 1.4.

Or the slack can be designed out. A cross-frame installed after the girders are plumbed and held, with its bolts tightened in that position, has its clearance taken up on the side that resists the initial twist; a preloaded friction connection has none until it slips; holes reamed in place have none at all. Each of those is cheaper than doubling a brace’s stiffness, and stiffness, as the figures show, cannot do the same job. The beam that fails sideways treated bracing as a matter of where and how stiff; a brace that starts late adds when, and the answer to when is decided on site by the order in which bolts are put in.

The effect also says something about erection. A girder is most crooked, and its braces least likely to have been fitted tight, during erection, before the deck is cast — exactly when its compression flange has no other restraint, and exactly the stage the load that makes itself worse warned amplifies every imperfection most. The slack matters most on the day it is easiest to ignore.

By hand, at the target #

The braced girder’s critical moment with braces at 5,691 kN·m/rad is 7,612 kN·m. At 5,627 the amplification of an initial twist is about $1/(1 - (5{,}627/7{,}612)^2) = 1/(1 - 0.546) = 2.2$, so 5 milliradians becomes 11 — the figure’s tight curve. The braces resist the added 6, and the middle brace, at 5,691 kN·m/rad and a twist of about 6 milliradians at its station, carries about 34 kN·m.

With 3 milliradians of slack, the girder takes up the slack at $947\sqrt{3/8} = 580$ kN·m by the closed form, 579 by the full solution, having twisted to 8 milliradians at midspan. From there the braced girder amplifies its new state: roughly 8.2 milliradians times 2.2 is 18, the figure’s 18.2. The middle brace has twisted 13.2 of those past its fitted position, less its 3 of slack: 10.2 milliradians at 5,691 kN·m/rad is 58.1 kN·m.

What the analysis rests on #

The slack is a clean gap. Real clearance is taken up in steps as bolts come into bearing at different times, and a slipping friction connection resists something before it slips; both make the brace softer at first rather than absent, which lies between the tight and the slack cases drawn.

The initial twist is one half-wave of 5 milliradians. A girder’s real imperfection is a mixture of shapes and its amplitude is a tolerance rather than a value. Both the tight and the slack results scale with it, and the comparison between them — slack costing a sixth more than crookedness per milliradian — holds across the 2.5 to 10 milliradians drawn.

The girder stays elastic. Its twist at the target, 11 to 18 milliradians, puts a lateral bending stress in the compression flange that adds to the stress the moment already puts there; the analysis does not reduce the girder’s stiffness as the flange yields, so near the girder’s own strength it understates the twist, for slack and tight braces alike. And the braces’ connections, which are where the slack is, are also where the brace force arrives: the bolts that do not share found that the bolts of a joint carry unequal shares even when every one is snug; with clearance as well, the bolts that close their gap first carry the brace’s force alone for a while.

Still open: the slack that comes back #

Every brace here takes up its slack once, under a load that only grows. A girder under traffic is loaded and unloaded thousands of times, and the twist it takes on loading returns most of the way when the load goes — leaving the bolts in their holes, at the other side of their clearance from where they started. A brace that took up its slack on the first lorry may have to take it up again in the opposite direction on the first gust that twists the girder the other way, and between the two the girder is, for a moment, unbraced at a moment well above zero. Whether a girder whose braces have slack in both directions settles into one side of its clearance and stays there, or rattles across it under every reversal — and whether the fatigue of a connection that is struck rather than loaded is worse than its static force suggests — is the question that a moving load puts to a brace that starts late.