The braces that stay where the framing puts them

Three braces where the moment is moved the outer two of three torsional braces toward the middle of a 24 m plate girder and found that, under a moment that peaks at midspan, the row then needs a quarter to a third less stiffness to bring the girder to its plastic moment. It is a good result for a designer who can put a cross-frame anywhere. Most cannot. On a building the cross-frames go where the secondary beams frame in; on a bridge they go at the bays the deck’s formwork was drawn to; on a crane girder they go where the walkway’s supports are. The positions are decided before the bracing is designed, and the bracing designer’s only freedom is how stiff to make each one.

So the question this essay asks is the one a designer who cannot move a cross-frame can still answer. With the braces where the framing puts them — equally spaced, at midspan and 6 m either side — how much of the saving does a row recover by making the middle brace stiffer than the outer two? And does the obvious extreme, nearly all of the stiffness in the brace at the peak, bring back the single brace’s ceiling that a row was meant to escape?

The same girder, the braces fixed and their stiffness free #

The girder is the one the bracing essays have used since a row of braces first shared out its stiffness: 1,225 mm between flange centres, its minor-axis second moment $2.67 \times 10^8$ mm⁴, its torsion constant $4.57 \times 10^6$ mm⁴ and its warping constant $1.00 \times 10^{14}$ mm⁶, over 24 m. It is asked to reach its plastic moment, 5,627 kN·m, where its moment peaks. The three braces are rotational springs on the twist at midspan and 6 m either side, and the middle one is given a share $q$ of the row’s total stiffness, the outer two $(1-q)/2$ each. Equal stiffnesses are $q = 1/3$.

The buckling problem is posed exactly as before: forty sine terms each for the lateral movement and the twist, the moment doing work as the section moves sideways and twists, the braces adding stiffness to the twist at their own points. For each share the total that just reaches 5,627 kN·m is found by bisection, and the best share by a golden-section search on the total. Nothing about the girder or the loads is new; the one thing that changes is that the three springs are no longer the same spring.

The hero figure is the answer for a uniform load. Equal, the row needs 8,537 kN·m/rad, which is the number the equally spaced row needed when the braces were moved toward the moment. Graded so that the middle brace has 61 per cent of the total, the row needs 7,067, which is 17 per cent less. The middle brace is half as stiff again as an equal brace would be, at 4,280; the outer two are half as stiff, at 1,394. The buckled twist shows why it works: in the equal row the girder twists twice as much at the middle brace as at the outer two, and in the graded row it twists the same at all three.

A best share for each diagram #

The share that needs least depends on how sharply the moment peaks, and the next figure sweeps it for the three diagrams the bracing essays have used.

Under uniform moment the best share is 41 per cent, hardly more than a third, and grading saves 3 per cent. That is the same verdict the earlier essay reached on moving the braces under uniform moment: when the moment is the same everywhere, the girder’s twist is nearly the same at every brace already, and there is little to gain by treating them differently. The middle brace does a little more work than the outer two because the twist is a little larger away from the supports, and a share of 41 per cent is what that difference is worth.

Under a uniform load the best share is 61 per cent and the saving 17; under a central point load, 78 per cent and 27. The sharper the peak, the more of the girder’s twist is concentrated at the middle brace — the shape of the diagram and not its peak is what decides it — and the more of the row’s stiffness belongs there. The curves are flat around their minima — under a uniform load any share between 50 and 65 per cent is within 3 per cent of the best — so a designer does not need the optimum to get most of the saving. A middle brace half as stiff again as each outer one, a share of 43 per cent, already saves 11.

What the figure also shows is that every curve turns up steeply past its best share, and much more steeply than it falls on the way there. Starving the outer braces costs more than over-providing them. That asymmetry is the subject of the next two figures.

Too little in the middle, and too much #

The buckled shape of the girder changes with the share, and the change explains both the best share and the steep climb past it.

With a share of 36 per cent, close to equal, the girder buckles in one broad wave whose twist is flat across both middle bays, and the middle brace is the one it is pressing against. The outer braces sit on the wave’s flanks, twisting at about half the peak, and do less. At the best share the middle brace is stiff enough to pinch the wave at midspan, to 85 per cent of its largest, and the twist at all three braces is the same. That equality is not a coincidence of this girder; it is what the best share is, and the section on checking by hand says why.

With a share of 81 per cent, the shape is a different one altogether. The girder buckles in two opposite waves, one in each half of the span, and the twist at midspan is exactly zero. The middle brace, however stiff, is at a point that does not move; it contributes nothing to resisting this shape. Only the outer braces resist it, and they have been starved to pay for the middle one. A girder held this way is two 12 m girders, each braced at its own middle by a brace of 9.5 per cent of the total.

That is the answer to the second half of the question the earlier essay left. Putting nearly all of the stiffness in the brace at the peak does not bring back the single brace’s ceiling — it brings back something worse, because the single brace’s ceiling at least has the girder buckling in a shape the brace resists. A rigid middle brace on its own lets this girder reach 3,532 kN·m under a uniform load, well short of 5,627, and the outer braces are what take it the rest of the way.

The floor under the outer braces #

The antisymmetric shape sets a floor that the outer braces can never go below, whatever is done to the middle one.

The dashed line is the stiffness each outer brace needs when the middle one is rigid: 1,238 kN·m/rad. It is the stiffness that holds the two-wave shape, which the middle brace cannot touch, at 5,627 kN·m. Below a share of 68 per cent the outer braces need more than this, because the one-wave shape governs and they share in resisting it. Above 68 per cent they need exactly this, and the middle brace’s line has a kink: from there on, every kN·m/rad added to the middle brace is wasted, and the total is twice the floor divided by the outer braces’ share. At 81 per cent it is 2 × 1,238/0.19 = 13,026 kN·m/rad, almost twice the best row’s total.

The best share, 61 per cent, sits a little short of the kink. There the one-wave shape still governs, and the outer braces are at 1,394, about an eighth above their floor. It is the one-wave shape’s own optimum, and the two-wave shape is not yet in play; it is close enough to it, though, that pushing the share a few per cent further brings the two-wave shape in and the cost rises quickly.

Under a central point load the floor is far lower — 260 kN·m/rad, about a sixteenth of the best row’s total — because the girder’s moment near the outer braces is so small that the two-wave shape needs little to hold it. That is why a point load’s best share is as high as 78 per cent. And yet it is not 100: a rigid middle brace alone takes this girder to 4,881 kN·m under a point load, still short of 5,627, and the two light outer braces are what make up the difference. The outer braces are cheap, but they are not optional.

Moving, grading, or both #

The earlier essay saved stiffness by moving the outer braces toward midspan with all three equal. Grading saves stiffness with the braces where they are. The obvious hope is that both together save more than either alone, and the next figure tests it.

The dashed line is the earlier essay’s curve: equal braces, moved in from 8 m to 2.5 m either side of the middle one, at their least near 3.75 m. The solid line grades the braces at each spacing to their best share. Far from the best spacing grading helps a great deal — at 6 m it saves 17 per cent, at 8 m 21 — and its best share labels tell the story: with the outer braces far out the middle one needs more than half the total, and with them close in it needs less than a third. At the best spacing the two lines meet. The best share there is about a third, which is equal stiffness, and grading has nothing to add.

That is not a coincidence either. Moving the braces in and grading them are two ways of doing the same thing: making the girder twist by the same amount at every brace. Moving them puts the braces where the twist is the same; grading makes the twist the same where the braces are. Once one has done it, the other has nothing left to do. The earlier essay found that its clustered row’s three braces twisted within two per cent of one another — which is the reason grading that row saves nothing.

What the framing leaves on the table #

The last figure is the summary a designer would ask for: how much of the saving from moving the braces can be had by grading them where they stand.

Under a uniform load grading saves 17 per cent and moving 26; under a point load, 27 and 34. Grading recovers two-thirds of what moving would save under a uniform load and four-fifths of it under a point load. The remainder is what fixed positions cost: grading can make the twist the same at the three braces, but it cannot change where the braces are relative to the twist, and a brace at a point of small twist must be stiffer to do the same work as a brace at a point of large twist.

For a designer the conclusion is practical. A row of cross-frames whose positions are fixed by the framing should not be one size. Under the gravity load of a floor or a bridge deck, a middle cross-frame half as stiff again as the outer ones — or the outer ones two-thirds as stiff as the middle one — captures most of what is available. And a row whose positions are free should be moved first and graded never, since moving buys more, and grading on top of it buys nothing.

Why every brace twists alike at the best share #

The equal-twist result has a short derivation. The critical moment of the braced girder is an eigenvalue, and the rate at which it rises as one brace’s stiffness rises is proportional to the square of the twist at that brace in the buckled shape. That is the standard sensitivity of an eigenvalue to a spring: a spring does work only through the movement at its own point. To reach a given moment with the least total stiffness, the last kN·m/rad of stiffness must buy the same rise in moment wherever it is put — otherwise moving it to where it buys more would save some. So at the best share every brace that has stiffness twists by the same amount.

The calculation confirms it to three figures: at each diagram’s best share the twist at the outer braces is 1.000 times the twist at the middle one, against 0.71, 0.49 and 0.46 for equal braces under uniform moment, a uniform load and a point load. The equal row’s ratio is a quick estimate of how much grading can save: the further it is from one, the more stiffness is being spent at braces the girder hardly turns.

Two bays and a half-parabola, by hand #

With the middle brace’s share at a third, the row is the equally spaced one, and its total under a uniform load, 8,537 kN·m/rad, is the earlier essay’s value exactly. With the middle brace rigid and the outer two at nothing, the girder is two 12 m bays. Under uniform moment it then reaches 2,637 kN·m, which is exactly the critical moment of a 12 m bay, $\frac{\pi}{L}\sqrt{EI_z(GJ + EI_w\pi^2/L^2)}$ with L = 12 m. Under a uniform load it reaches 3,532, a factor of 1.34 on that: each half of the parabola rises steeply from the support and flattens toward the peak, so it sits between uniform moment’s factor of one and the 1.75 or so a straight rise from zero would earn. The formula that reads four numbers gives 12.5/(2.5 + 3 × 0.44 + 4 × 0.75 + 3 × 0.94) = 1.30 for the half-parabola, 3 per cent on the safe side of the solution. And past the kink the total must be twice the floor over the outer share, $2 \times 1{,}238/(1 - q)$, which at $q$ = 0.9 is 24,750 — the value the solution gives.

Where the springs stop being cross-frames #

The braces are ideal rotational springs at the shear centre. A cross-frame’s stiffness comes from its diagonals, its connections and the girder’s web between its flanges, and a stiffer cross-frame is not simply a heavier one: past a point the web’s distortion limits the stiffness however heavy the frame, as twice the moment, four times the brace found for a single brace with a slender web. A graded row’s middle brace is the one most likely to reach that limit.

The girder is perfect and the braces need only stiffness. A real girder’s initial twist puts force into its braces in proportion to the twist at each, and a brace’s force is its stiffness times that twist. At the best share the twists are equal, so the forces go as the stiffnesses: the middle brace, three times as stiff as each outer one under a uniform load, carries three times the force. A graded row is graded in strength as well as stiffness, and the brace that need not be strong is the arithmetic of how much force there is.

The load acts at the shear centre. A load on the top flange moves the twist toward the load and raises the middle brace’s best share further, by the effect the restraint that beats the gradient traced for a single restraint.

The girder next door and the drawing office #

They cannot show the web distorting between the flanges, which lets the compression flange twist further than the section as the beam that fails sideways does when its web is slender. They cannot show the neighbouring girder, which a cross-frame reacts against and which is buckling at the same time, so that a row of cross-frames is a system of girders and not a girder on springs. They cannot show a deck that, once cast, braces the top flange continuously and makes the cross-frames’ job temporary. And they cannot show what grading costs to detail: three cross-frames of one size are one drawing and one jig, and the saving of a sixth of the stiffness has to be weighed against two.

Still open: braces that only act one way #

Every brace here is a spring that resists twist in both directions equally. Many braces do not. A brace made of a single tie, or a cross-frame with tension-only diagonals, resists the girder twisting one way and not the other, and a row of them is stiff against the one-wave shape and possibly slack against the two-wave shape’s opposite half. Whether a row of one-way braces needs its outer braces oriented in opposition to hold the two-wave shape — and whether the floor this essay found then doubles — is the question the symmetric springs leave, and the brace on the wrong flange is where a brace’s direction first mattered.