The braces that pull only one way

The braces that stay where the framing puts them graded three torsional braces on a 24 m plate girder, giving the middle one more of the row’s stiffness, and found the saving: a sixth of the total under a uniform load, most of what moving the braces would buy. It found too that the outer braces have a floor they can never go below, set by the two-wave shape the middle brace cannot touch. Every brace in that essay was a spring that resisted twist equally in both directions.

Many braces are not. A brace made of a single tie, a cross-frame whose slender diagonals buckle as soon as they are pushed, a strap from a girder’s top flange to a neighbouring one: each resists the girder twisting one way and goes slack when it twists the other. That essay ended on whether a row of such braces needs its outer braces oriented in opposition to hold the two-wave shape, and whether the floor then doubles.

A girder has no preferred direction #

The girder is the one the bracing essays have used throughout: 24 m between supports, 1,225 mm between flange centres, under a uniform load that brings its moment to a peak of 5,627 kN·m at midspan, which is the moment it must reach. Three torsional braces stand at midspan and 6 m either side. Two-way braces of 2,846 kN·m/rad each take it there; unbraced, it buckles at 947.

A perfect girder’s buckling is a bifurcation: at the critical moment it can move sideways and twist in a shape, and it can do so in that shape or in its mirror image, twisting the other way. Which one it takes is decided by an imperfection too small to draw — a sweep of a millimetre, a load a little off the shear centre, a gust. A two-way brace resists either. A one-way brace resists one of them and offers nothing against the other.

So the buckling problem is no longer a single eigenvalue. Each brace is either engaged — the twist pulls it and it resists — or slack, and the girder can buckle in any shape for which the braces it engages and the braces it slackens are consistent with the way each faces. The answer is solved by trying every set of engaged braces, finding the girder’s lowest modes with only those braces as springs, and admitting a mode only if, in one sign or the other, it pulls every engaged brace and slackens every other. The critical moment is the lowest admissible one: the girder buckles in whichever direction its braces leave it free to.

The bottom row of the figure is the whole of the problem in its plainest form. Three one-way braces facing the same way leave the girder exactly as weak as if they were not there, because the girder simply buckles the other way: the same one-wave shape, mirrored, slackening all three. Any stiffness, any number of braces facing one way, gives the same 947.

Facing alternately #

Alternating the braces — the outer two resisting one direction, the middle one the other — closes that door. Whichever way the girder now buckles in its one-wave shape, some brace is pulled. If it twists the way the outer braces resist, they engage and the middle one goes slack; if it twists the other way, the middle one engages and the outer two go slack. The girder takes the weaker of the two.

At the stiffness that brings two-way braces to the target, the alternating row reaches 3,087 kN·m — 55 per cent of what it needs. Made stiffer it rises, but towards a ceiling of 4,063, well short. The girder, offered the choice, buckles the way that slackens the middle brace, and the two outer braces at 6 m and 18 m are then all that hold it; a girder braced rigidly at those two points reaches 4,063 and no more.

The gap widens as the braces stiffen. At 500 kN·m/rad each, a sixth of the design stiffness, the alternating row reaches 1,859 kN·m against the two-way row’s 2,596, 72 per cent of it; at the design stiffness, 55 per cent; at 10,000 kN·m/rad, 3,711 against 9,573, 39 per cent. Soft braces barely engage in either direction and the girder is nearly unbraced either way, so a one-way row loses little; stiff braces are where a two-way row climbs towards its plateau and a one-way row stalls at the shape it can always escape into. The braces a designer adds to make a girder stronger are exactly the ones a one-way detail wastes.

The figure’s other lines are worse. Two braces facing one way and the third the other leave the girder free to buckle against the pair, held only by the third brace at one side: 1,734 kN·m at the design stiffness, 1,891 rigid. And all three facing the same way are a flat line at 947. No stiffness and no orientation of three one-way braces brings this girder to its target. The deficit is not something stiffer braces can buy back, because it is not about stiffness; it is about which braces a mirror-image buckle can avoid.

Rigid, a third of a two-way row #

The ceilings tell the same story under every moment diagram. Rigid two-way braces force the girder into four half-waves between them and lift it to 12,019 kN·m under a uniform load; the best orientation of rigid one-way braces reaches a third of that. The peakier the diagram, the higher every ceiling, but the ratio between the alternating row and the two-way row stays near a third. A one-way row is not a weaker version of a two-way row. It is a different structure, with half its braces missing in whichever direction the girder chooses.

Opposing the outer pair is not enough #

The earlier essay’s question was specific. The two-wave shape — the girder twisting one way in each half of the span, with no twist at midspan — is the one that sets the outer braces’ floor, because the middle brace sits where the shape does not twist. It twists the two outer brace points in opposite directions. Surely, then, outer braces oriented in opposition would hold it.

They do the opposite. In the two-wave shape, if the left brace point twists one way, the right one twists the other; opposed braces at those points resist those opposite twists only if the shape happens to have the sign that pulls both. The girder buckles in the mirror image, which slackens both. The two-wave shape, with outer braces opposed, is held by nothing at all.

With the middle brace two-way and the outer pair opposed and rigid, the girder reaches 3,532 kN·m — exactly the moment at which it buckles in two waves with only its midspan held, the number the earlier essay found for a rigid middle brace alone. The opposed outer braces have contributed nothing. And the figure’s other bars say something stronger. With the outer braces two-way and only the middle one a single tie, the girder reaches 4,063; with one outer brace a single tie, 4,578. One one-way brace anywhere in the row is enough to cap it below its target. The girder finds the direction that slackens that one brace and buckles in a shape the remaining braces cannot hold.

So the answer to the question is not that the floor doubles. It is that a one-way brace has no floor to double: the row reaches its target only if every brace in it resists both ways.

No spacing rescues the row #

Moving the braces does not help enough. The alternating row’s ceiling is set by the weaker of its two directions — the outer pair alone, or the middle brace alone — and the outer pair alone is best when the pair is close to midspan, where the moment is. Clustered 2 m either side, the alternating row reaches 5,093 kN·m, closer, still short. The two-way row clustered the same way reaches 7,972. The geometry that helped two-way braces in three braces where the moment is helps one-way braces too, but it cannot make up for the braces the girder is free to ignore.

What a two-way brace made of ties costs #

A brace that resists both ways can be a single member that works in tension and compression — a cross-frame with stocky diagonals — or two one-way members facing opposite ways: a pair of crossed ties, of which whichever is pulled does the work while the other goes slack. The second is the cross-frame with slender, tension-only diagonals, and it is the commonest light bracing there is.

A pair of ties is a two-way brace whose stiffness, in either direction, is one tie’s. To give the girder 2,846 kN·m/rad at each brace point, each tie must have that stiffness by itself, and the row has six of them: 17,074 kN·m/rad of ties, against 8,537 for three members that work both ways. Made of ties, a two-way brace costs twice the steel, which is the arithmetic two diagonals, one of which is absent found for a truss panel whose counter cannot take compression, met here in the bracing of a girder.

The row by hand #

Two numbers in the figures can be checked without the unilateral search. With every brace facing the same way the girder buckles away from all of them, so its critical moment is its unbraced one: the textbook $(\pi/L)\sqrt{EI_z(GJ + EI_w\pi^2/L^2)}$ with the uniform load’s factor, 947 kN·m. And with the outer pair opposed and the middle two-way, the two-wave shape slackens both outer braces and does not twist the middle one, so the girder buckles in two waves as if braced at midspan only: the earlier essay’s 3,532.

The alternating row’s ceiling is the girder braced rigidly at 6 m and 18 m only, its middle free to twist the way the middle brace does not resist: 4,063 kN·m. The other direction would look weaker. Engaging only the middle brace, the girder held at midspan alone buckles at 3,532 in two waves — but those two waves twist the outer brace points in opposite directions, and the alternating row’s outer braces both face the same way, so one of them is pulled and the shape is not admissible. The girder cannot use its weakest shape, and settles for 4,063. That is the kind of result the unilateral search exists to find: the lowest mode with a given set of braces is not always one the braces allow.

What the stiffness rules do not say #

The design rules for torsional bracing were built on the smeared stiffness a row of braces shares out and on the strength a brace that need not be strong has to have; they give a brace a stiffness and a force and say nothing about which way it must resist them, because they are written for braces that resist both ways. A designer who satisfies them with single diagonals has satisfied a rule about a structure that is not the one being built. The rule’s number for each brace is right; the row reaches it only if each brace delivers that number in whichever direction the girder chooses.

The same omission runs through every place a brace’s direction matters. The brace on the wrong flange found that where a brace attaches decides whether it restrains the twist at all; twice the moment, four times the brace found that a slender web lets a brace’s stiffness leak away before it reaches the flange. A brace is a stiffness at a point, in a direction, and a rule that gives only the first of those three leaves the other two to the drawing.

Why the girder’s choice is the designer’s problem #

A bifurcation is an idealisation: a real girder is never perfect, and it begins to twist under load in the direction its imperfections give it, long before the critical moment. If that direction were known, one-way braces facing against it would do — they would be engaged from the start and stay engaged, and the girder would never meet the mirror image. The difficulty is that it is not known. The column that was never straight found that a member’s crookedness is a few millimetres in an unknown direction, and a girder’s sweep, its load eccentricity and the twist a wind puts into it during erection are the same: small, and unsigned.

There is one common case where the direction is known. An outer girder of a bridge carries the formwork for the deck’s overhang on brackets hung from its outer side, and the overhang’s weight twists it outward every time, before any buckling begins. One-way braces facing against that twist are engaged by it from the start, and as long as the twist stays the same way they behave as two-way braces would. The moment the overhang formwork is struck, or the load is applied in a different order, the direction is no longer guaranteed. A design that relies on it has to say so, and has to know which loads can reverse it.

Ideal one-way springs #

The braces are ideal one-way springs. A real tie has slack before it engages — a bolt in a clearance hole, a strap with a little looseness — and engages gradually; a real tension-only diagonal resists a little in compression before it buckles. Both make the slack direction slightly stiffer than nothing, and the slack makes the engaged direction softer at first. Neither changes the conclusion that the girder takes the direction its braces resist least.

Each brace acts at a point and alone. A tension-only cross-frame between two girders has its diagonal attached at a top corner of one girder and a bottom corner of the other, so it couples the two girders’ twists and the girders’ lateral movements as well; whether it is pulled depends on how both girders move, not one. Treating it as a one-way spring on one girder’s twist is the simplest reading of a detail whose real behaviour is a system of two buckling members and a strut that may or may not be in tension.

The girder is perfect and the analysis is a bifurcation. The unilateral search finds the moment at which the girder can first move in an admissible shape. A real girder with an imperfection in the slack direction moves from the first load, its braces never engaged; with an imperfection in the engaged direction it behaves as if braced until the imperfection’s growth reverses it, which it cannot.

The braces are at the shear centre. A tie attached to one flange restrains that flange’s movement as well as the twist, and the brace on the wrong flange found that where it attaches decides whether it braces at all — another way a brace’s direction matters.

The girder next door #

They cannot show the neighbouring girder a cross-frame reacts against, which is buckling at the same time and may choose the same direction or the opposite one, so that a tie between them is pulled by both or by neither. They cannot show a deck cast on the girders, which braces the top flange continuously and in both directions and makes every cross-frame’s job temporary. And they cannot show how often light bracing is drawn as a single diagonal to save a member, on the reasoning that a girder can only buckle one way at a time — which is true, and is exactly why the girder chooses the way the single diagonal does not resist.

Still open: the brace with slack in it #

Every one-way brace here is fully slack in one direction and fully stiff in the other. Many two-way braces are neither: a cross-frame connected by bolts in clearance holes resists nothing until the girder has twisted far enough to take up the clearance, and then resists fully, in either direction. Such a brace is two-way, but with a gap at the start, and a girder with an imperfection smaller than the gap meets a row of braces that are, for the first part of its twist, not there. 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 — is the question a real connection puts to the springs these essays have used.