Three braces where the moment is

The stiffness a row of braces shares out held a 24 m plate girder with a row of equally spaced torsional braces and found that, below the row’s ceiling, the girder responds to the braces’ total stiffness and hardly to their count. It also found the one place where count and position showed through: braces near the supports, where the twist is small, do less than braces near the middle, and a row with more of its stiffness near its ends needs a little more of it in all. That essay’s girder was in uniform moment, which is the case every bracing rule is calibrated on and almost no girder carries. A girder under its own weight, a deck or a point load has its moment at midspan and very little near its supports, and its buckled twist gathers where the moment is.

So the question that essay left is a question about where to put the braces rather than how many. If the twist is in the middle, a row that concentrates its braces there should need less stiffness; but every brace moved in lengthens the bay between the last brace and the support, and a girder held at fewer points near its ends can buckle there instead. Which of those two effects wins depends, it turns out, entirely on the shape of the moment.

The girder, the braces and a moment that varies #

The girder is the earlier essays’ plate girder over 24 m: 1,225 mm between flange centres, $I_z = 2.67 \times 10^8$ mm⁴, $J = 4.57 \times 10^6$ mm⁴, $I_w = 1.00 \times 10^{14}$ mm⁶. It is asked to reach its plastic moment, 5,627 kN·m, at the section where its moment peaks, and it is held by three torsional braces — cross-frames or diaphragms to a neighbouring girder, each a rotational spring of stiffness $\beta_T$ — one at midspan and two at a distance $s$ either side. Equal spacing is $s$ = 6 m.

The calculation behind the earlier essay used a property of uniform moment that no other diagram has: under a uniform moment, the sine terms of the twist and of the sideways movement do not couple, so the only coupling is through the braces. Under a moment that varies along the span, every term couples to every other through the moment itself, and the buckling problem has to be posed with the whole diagram in it. It is posed here as a Ritz problem — forty sine terms each for the lateral movement and the twist, the braces as springs on the twist, the moment doing work as the section moves sideways and turns — with the lateral terms eliminated exactly. Under uniform moment it gives the earlier essay’s numbers to the last digit. Unbraced, it gives the girder 837 kN·m in uniform moment, 947 under a uniform load and 1,141 under a central point load: ratios of 1.13 and 1.36, the equivalent uniform moment factors the design rules tabulate.

How much the clustering saves #

Under uniform moment the answer the earlier essay implied holds: equal spacing is nearly the best there is. Pulling the outer braces in to 5.25 m saves six per cent of the stiffness, and pulling them further costs a great deal — at 4 m the row needs 18,385 kN·m/rad, at 3 m 134,000, and a little inside that it cannot reach the moment at all.

Under a uniform load the curve is shallow and its minimum is deep. The row needs 8,537 kN·m/rad equally spaced and 6,282 with the outer braces 3.75 m from the middle one: a saving of 26 per cent for moving two braces two and a quarter metres. Under a central point load the saving is 34 per cent, the minimum is at 2.75 m, and the curve is still almost flat at the tightest spacing drawn.

Those numbers are what bracing rules miss and what bracing practice has never needed to know, because braces are placed where the framing puts them — at panel points of a truss, at cross-girders, at the stiffener spacing — and nobody designs their positions. They are worth knowing for the one case where positions are free: a girder braced during erection, by temporary cross-frames or bracing to its neighbour, whose positions are chosen on the day. And they are worth knowing as a correction to the instinct, built on the brace that need not be strong, that bracing is a matter of how stiff and how many and never of where.

The ceiling the end bays set #

The two lines of the previous figure that behave differently do so because of the ceiling: the moment the girder reaches when its braces are rigid and it buckles between them. With rigid braces the girder is a set of bays, and it buckles in whichever bay reaches its own critical moment first.

Under uniform moment every bay carries the full moment, and the longest bay governs. Equally spaced, all four bays are 6 m long and the ceiling is 9,704 kN·m. Clustered, the middle bays shorten and the end bays lengthen, so the end bays govern and the ceiling falls with every metre the braces close in — to 7,032 kN·m at 4 m and below the target at about 2.6 m. A row that approaches its ceiling needs stiffness that rises steeply toward infinity, which is the turn upward in the previous figure. Under uniform moment, the price of clustering is paid in the end bays.

Under a moment that falls to nothing at the supports, the end bays carry very little of it — the same reason a restraint can beat the gradient by being placed where the moment is rather than where it is not. An end bay 8 m long under a moment that rises from zero to two thirds of the peak is a long bay under a small and steeply varying moment, and its critical moment, expressed as a peak at midspan, is far higher than the middle bays’. So the first thing clustering does under a uniform load is to shorten the middle bays, where the moment is, without making the end bays govern: the ceiling rises, from 12,019 kN·m to 14,320 at 5 m. Only when the outer braces are very close to the middle one does the end bay start to govern and the ceiling fall back. The girder’s ceiling under a gradient is set by the bays where the moment is, and those are the bays clustering shortens.

Which brace does the work #

The reason clustering saves stiffness is visible in where the braces’ work goes. A torsional brace resists the girder’s twist at its own position, and the energy it stores is its stiffness times its twist there squared. Under a uniform load the buckled twist is concentrated in the middle of the span, so equally spaced braces see very different twists: the middle brace stands at the largest and does two thirds of the row’s work, and the outer two, 6 m from the supports, stand where the girder twists far less and do a sixth each.

Braces of equal stiffness are paid for equally. A row that puts two of its three braces where they are barely used has bought stiffness that buys nothing, and the stiffness the row needs is set by how much the braces that are being used must supply. Clustered 4 m either side of the middle, the three braces see similar twists and share the work almost equally, and the row needs a quarter less in all. The saving is not that clustered braces are better braces; it is that equally spaced ones were partly idle.

The same picture says what the alternative to clustering is. A row that cannot move its braces can give them different stiffnesses — a stiff cross-frame at midspan and lighter ones beside it — and reach the same economy by putting the stiffness where the twist is rather than the braces. That is the brace sized for its own job the single-brace essay found, applied to each brace of a row in turn.

The sharper the peak, the closer the cluster #

Between the uniform moment and the point load, the saving rises steadily, and the best spacing tightens as it rises. A diagram that is mostly uniform with a small peak rewards a small movement of the braces; one that is all peak rewards bringing them in close around it. That is the relation one would expect from the work figure: the more sharply the moment peaks, the more sharply the twist does, and the braces belong where it is.

The figure also says what clustering is not worth. For a girder in uniform moment — a continuous girder’s span between points of contraflexure is closer to that than a simple span — the saving is six per cent, which no designer would chase at the cost of irregular framing. For a simply supported girder under its own weight it is a quarter, and for a girder carrying a concentrated load, a third.

What the rules credit, and what they cannot see #

The bracing requirements in design rules — the torsional bracing expression of AISC 360’s stability appendix, and its relatives — give the stiffness a row needs as a function of the moment it must reach, the span, the girder’s section and the number of braces. A moment gradient enters through $C_b$, the factor by which a gradient raises an unbraced girder’s critical moment — the factor that reads four numbers off the diagram — squared in the denominator: a gradient that raises the critical moment by 13 per cent lowers the stiffness the row needs by about 22.

That credit is very nearly right for equally spaced braces. Under a uniform load the uniform-moment total over $C_b^2$ is 8,227 kN·m/rad against the 8,537 an equally spaced row actually needs, and under a point load 5,673 against 5,929 — within four or five per cent, and on the light side. What the rule cannot see is where the braces are. It has a count and no positions, so a row placed where the moment is gets the same requirement as one spread evenly, and the clustered rows’ real requirements, 6,282 and 3,912, sit a quarter and a third below anything the rule could give them.

That is not an error in the rule. A rule that credited positions would have to know the diagram’s shape as well as its peak and gradient, and would be harder to apply than the saving is worth on most girders. It is an observation about what the rule’s credit for a gradient is: a credit for the girder’s own resistance rising, not for the braces being used better.

The bays near the supports #

The uniform-moment case says something the earlier essays did not about braces close to a support. A brace near a support in uniform moment does little, because the girder twists little there — but moving it inward lengthens the end bay, and in uniform moment the end bay carries the full moment and sets the ceiling. So in uniform moment the brace near the support is doing a job after all: it is holding up the ceiling rather than resisting the twist. Take it away or move it in and the ceiling falls.

Under a gradient the end bay carries little moment, and the brace near the support is doing neither job. That is why clustering is nearly free under a gradient and expensive in uniform moment, and it is the answer to the question the earlier essay ended on. The brace near the support earns its place only if the moment there is large enough for the end bay to govern.

The uniform load, brace by brace #

Under a uniform load the equally spaced row needs 2,846 kN·m/rad in each of its three braces. Clustered 3.75 m either side of the middle one, each needs 2,094 — about three quarters as much, for the same three cross-frames moved closer together. The ceiling with the braces rigid is 12,019 kN·m equally spaced and 11,862 clustered: the clustered row’s ceiling has passed its peak and is coming back down, but it is still more than twice the moment asked for, so the stiffness the row needs is set by the springs and not by the bays.

The twist at each brace tells the same story from the other side. In the equally spaced row’s buckled shape the twist at the outer braces is half the twist at the middle one; in the clustered row’s, the three twists are within two per cent of one another. A spring is cheapest when it is worked hard, and a row of equal springs is cheapest when it is worked evenly. That is the whole of the economy, and it needs no more than the buckled shape to see it.

A designer who wants the economy without moving the braces has two other ways to it. One is to grade the braces, as the last section asks. The other is to accept the equal spacing and the 8,537 kN·m/rad, and to notice that the outer two braces are then doing a sixth of the work each: they could be lighter than the middle one without the girder noticing much, which is the same saving bought with steel instead of geometry.

The uncoupled limit and a 6 m bay #

Under uniform moment the Ritz solution must reduce to the uncoupled one, and it does: the earlier essay’s closed form for three equal braces of 3,510 kN·m/rad each, 5,627 kN·m, is reproduced exactly. Unbraced, the uniform-load and point-load critical moments, 947 and 1,141 kN·m against 837 uniform, give factors of 1.131 and 1.362 — the textbook values for those two diagrams are 1.13 and 1.35 to 1.36.

The ceiling with rigid braces under uniform moment, equally spaced, is the critical moment of a 6 m bay: $\frac{\pi}{L}\sqrt{EI_z(GJ + EI_w\pi^2/L^2)}$ with L = 6 m gives 9,704 kN·m, the figure’s value.

Where the Ritz solution stops #

The braces are torsional springs at the shear centre, and the load is applied there too. A load applied at the top flange destabilises the girder and moves the buckled twist toward the load; a brace connected to one flange is partly a lateral brace and partly a torsional one, as the brace on the wrong flange found. Both change the numbers, and neither changes the argument about where the twist is.

The girder is elastic and perfect. A real girder carries an initial twist, and its braces carry force as well as needing stiffness; the force a brace must resist is proportional to the twist at it, so a clustered brace at a larger twist carries more force than an equally spaced outer brace would have. The stiffness saving does not come with a strength saving for the middle three.

The diagrams are the three ideal ones. A real girder carries its self-weight’s parabola plus whatever the deck and its traffic add, and the best spacing for that is somewhere between the curves drawn; the blend figure is the map for finding it.

The connections and the girder next door #

They cannot show a cross-frame’s connections, whose flexibility adds to the brace’s and is often most of it; nor the girder’s web distorting between its flanges, which lets the compression flange twist more than the section, as a beam that fails sideways does when its web is slender. And they cannot show the neighbouring girder, which a cross-frame needs to brace against and which is buckling too.

Still open: braces of different stiffness #

A row whose positions are fixed by the framing can still put its stiffness where the twist is, with a heavy cross-frame at midspan and light ones beside it. Whether a row graded in stiffness reaches the same saving as a row graded in position — and whether the optimum, for a given total, is to put nearly all of it in one brace at the peak, which is where the single-brace essay’s ceiling then returns — is the question the equal stiffnesses here leave, and it is the one a designer who cannot move a cross-frame can still answer.