Twice the moment, four times the brace

A brace on the wrong flange ends by naming a second kind of restraint and declining to compute it. A lateral brace holds a point in the section against sideways movement, and everything about it depends on which point: on the compression flange it works, at the shear centre it costs three times as much, and on the tension flange it does not work at any stiffness whatever. A torsional brace holds the section against rotation instead — a cross-frame between two girders, a secondary beam with a moment connection into the web, a stiffener tied into a deck — and it does not care where it is attached, because it is resisting the twist rather than a displacement the twist produces.

That is a sentence, and this is the arithmetic behind it. The rotational spring goes into the same energy determinant the first rung used, in a different entry, and three things come out that the sentence does not contain.

The shape of that curve is the same shape the first rung drew, and the sameness is worth a sentence before the differences. There is a plateau, it sits at the two-half-wave critical moment, and past it a stiffer brace buys nothing at all. The reason is unchanged: the braced beam is using the second buckling mode, and a restraint at midspan is at the node of that mode. Whether the restraint holds a displacement or a rotation, the mode it is trying to obstruct has neither of them at the point where it sits.

The height that is not in the equation #

The first difference is the one the previous essay stated. A lateral spring at height $a$ above the shear centre resists the movement of a point that goes as $u + a\varphi$, so the spring’s energy contains $u^2$, $a^2\varphi^2$ and a cross term $2au\varphi$ — three entries of the matrix, and the cross term carries the sign of $a$. That sign is the whole of the wrong-flange result: on the compression flange it subtracts from the destabilising coupling and on the tension flange it adds to it.

A rotational spring’s energy is $\tfrac12\beta_T\varphi^2$ and nothing else. It enters one entry, it has no $a$ in it, and there is no arrangement of the section for which it is on the wrong side of anything.

That is why a torsional brace is the answer wherever the compression flange cannot be reached — the bottom flange of a continuous beam over its support, the inside of a box, a girder whose deck sits on the tension side during a construction stage. The restraint that is indifferent to height is the one available when height is not a choice.

It is also the reason a torsional brace can be a pair of members doing something neither of them looks like it is doing. A cross-frame between two girders restrains each girder’s rotation by pushing the other one, and the pair rotates as a unit only if the two are twisting in opposite directions. Two girders twisting the same way have nothing between them, which is one of the ways a bracing system passes every member check and restrains nothing.

The demand is a quadratic, and its root is the beam #

The second difference is the one that decides how big the brace has to be, and it comes out of the determinant as a formula rather than as a curve.

With the lateral spring set to zero the critical condition on the first half-wave rearranges to

$$\beta_T = \frac{(M\,c_{u\varphi}s)^2}{c_{uu}s} - c_{\varphi\varphi}s$$

which is $\alpha M^2 - \beta$: a quadratic in the moment with a negative intercept. Both terms are worth reading.

The $M^2$ says that the stiffness demanded rises as the square of what is being held. A lateral brace’s requirement, by contrast, is very nearly linear in the moment, because it is written against a force — the notional restraining force is a fixed fraction of the compression flange force, and the flange force is proportional to the moment. The torsional brace is holding a rotation against a destabilising term that is itself proportional to $M$ times the rotation, and the two multiply.

The intercept says the beam braces itself up to a point. Set $\beta_T$ to zero and the equation returns $M = M_{cr}$, the unbraced critical moment, which is the number the curve crosses the axis at: 143 kNm on the beam drawn. That is not a coincidence and it is not fitted. It is the same determinant evaluated with no spring in it, so the demand curve’s root is the answer to the question the brace was introduced to improve on, and any brace at all is being asked only for the difference.

The two terms together make the practical behaviour steeper than the square law suggests. Doubling the moment from 201 to 403 kNm — 45 to 90 per cent of the plateau, which is the range a design occupies — multiplies the demand by 7.1 rather than by 4, because the subtracted term is a fixed quantity being outgrown. The design rules quote $\alpha M^2$ alone, which is conservative everywhere and increasingly so at the low end, where it demands a brace for a beam that needs none.

Which free body produced the number #

There is no free body here in the sense the rest of this collection uses, and saying so precisely is more useful than pretending otherwise.

The quantity being computed is an eigenvalue, and the object it belongs to is the whole beam between its end supports together with everything attached to it. The calculation is an energy balance over that object: the strain energy stored in minor-axis bending, in St Venant torsion, in warping and in the spring, set against the work the applied moment does as the section twists and the compression flange goes round the corner.

Two half-waves in each of the lateral displacement $u$ and the twist $\varphi$ give a four-by-four matrix, and the critical moment is the smallest $M$ at which it stops being positive definite — the load at which a shape exists that costs no energy to adopt. The brace enters as one added term on the diagonal.

What that means for the reader is a caution rather than a technique. Nothing in this figure is a force, and nothing in it can be converted into one. The buckling problem returns a stiffness requirement and a mode shape whose amplitude is undetermined, which is why the force a brace has to carry is a separate calculation with an assumed imperfection in it — the same division of labour a column brace has, and the reason the code rules on bracing always come in pairs.

The girder, where the numbers get large #

A rolled beam is the easy case. The member that actually gets a cross-frame is a plate girder, and the same three quantities move by two orders of magnitude.

Nine thousand kilonewton-metres per radian is a large number to have to deliver through a joint, and the first useful thing to say about it is that no design ever asks for it.

The brace is not asked for the plateau #

The plateau is the beam braced into two beams and reaching the elastic critical moment of the half-span. A design does not want that. It wants the beam to reach the strength its cross-section has, and once it does, more restraint is buying capacity the section cannot use.

So the requirement is not “reach the plateau” but “reach 5,627 kNm”, and the demand curve answers that directly: 2,647 kNm/rad, less than a third of the ideal stiffness and a quarter of what the square law alone would ask for.

That is the whole of the design calculation, and it is worth noticing what it did not need. No effective length was chosen, no factor was looked up, and the brace’s own section never appeared. The question asked was how stiff the restraint has to be for the beam to reach a stated moment, and the determinant answered it.

What the brace is actually as stiff as #

Which leaves the question the curves cannot answer, and on a deep girder it is the one that decides the answer.

A torsional restraint is applied to the section by something outside it, and it reaches the two flanges through the web. A web is a plate in transverse bending, its stiffness goes as $t_w^3$, and it is in series with the brace. Two springs in series are always softer than the softer of them, so a cross-frame stiff enough to be drawn as rigid delivers whatever the web will pass on.

Three readings, and the third is the one worth carrying off the page.

A cross-frame on an unstiffened web is very nearly not a brace. It raises the girder’s capacity by six per cent, and the six per cent is indifferent to how large the cross-frame is. Every member in it passes every check; the girder is braced on the drawing and not braced in the structure.

The stiffener is the bracing member. Its size is decided here and nowhere else on the drawing: 12 mm is not enough and 20 mm is, for a member carrying no load, framing into nothing, and appearing on the schedule as a plate. That is the same shape of finding as a rib that is a boundary condition — a component whose job is to change what another component is allowed to do.

And the web thickness is a bracing variable, at the cube. The sweep runs from 4 mm to 24 and the bare web goes from 5.5 kNm/rad to 1,198: a factor of two hundred and sixteen for a factor of six of steel, in a dimension nobody chooses for this reason.

The rolled case is the surprising one, because a rolled beam’s web looks stocky and the effect is assumed to belong to plate girders. It does not. A 400 mm section with an 8 mm web is a 50-to-1 plate, and a torsional brace on it without a stiffener is delivering a fifth of what it is credited with.

The mode the section is left with #

There is a way of reading the series calculation that makes it something other than a loss, and it is the honest one.

When the web bends, the section distorts: the compression flange moves sideways, the web bows, and the tension flange stays where the brace put it. That is not the mode the critical moment was computed for. It is a different buckling mode with its own shape, and the beam has found it because the restraint closed off the other one.

So the series stiffness is not a deduction applied to a good answer. It is the shortest way of writing down that a braced beam with a flexible web has two ways to buckle and the brace only obstructs one of them — which is the same distinction that separates local and distortional buckling in a cold-formed section, at the scale of a girder rather than a lip.

Read that way, the stiffener is not compensating for a deficiency in the brace. It removes a mode, by joining the two flanges with something that is rigid in the plane of the section, and once that mode is gone the brace does what the determinant said it would.

Where the model stops #

Four boundaries, and the first two are about the mode.

One brace at midspan, under uniform moment. With several braces the plateau becomes the $n$-th half-wave and the ideal stiffness rises with the number of braces rather than falling — a result that surprises people who expect the work to be shared. With a moment gradient the whole picture moves, because the gradient factor changes what the beam was going to do unbraced.

The brace is at the node of the second mode. Move it away from midspan and it obstructs the second mode as well, the plateau disappears, and the curve rises without limit toward the mode the brace cannot reach. That is a better-behaved problem and a less useful one, because it has no ideal stiffness to design to.

The rotational spring is linear and its connection is not. A cross-frame’s stiffness is dominated by the flexibility of its end connections, which is the argument that a joint is neither pinned nor rigid applied to the brace, and the numbers here are what has to arrive at the section rather than what the members would offer if they were welded.

And nothing here yields. The web stiffness used is elastic plate bending, which is the right model up to the point where the flange-to-web region goes plastic. In a beam being taken to its plastic moment on purpose — which is exactly the case the 5,627 kNm figure describes — the last part of the rotation is being delivered by a web that is no longer elastic, and the bracing stiffness at that moment is smaller than the one computed.

What the pictures cannot show #

The curves are all of critical moment against stiffness, and the two things they leave out are the two things that go wrong on site.

Where the brace is anchored. A cross-frame between two girders restrains each against the other, and the pair is only restrained if something eventually reacts the couple — a plan bracing system, a deck, a pair of end frames. A row of cross-frames between two girders that both twist the same way is a mechanism with a full set of members in it, and no figure of a single beam can contain that.

And whether the brace is there yet. Every number here is the finished structure. During erection the deck is absent, the cross-frames may be fitted after the girders are set, and the girder is a beam with no torsional restraint anywhere along a span it was never designed for — which is the most dangerous day and is the reason a lifted girder is checked as a member with a completely different unbraced length.

The assumption the figure rests on #

That the section is rigid in its own plane everywhere except at the brace point.

The series calculation admits web flexibility at the braced section and nowhere else, so the rest of the beam is still assumed to hold its shape while it twists. That is what allows a single warping constant to describe the whole member. On a girder slender enough for the brace-point distortion to matter, the same distortion is available along the span under any transverse load, and the honest version is a finite-strip analysis in which the cross-section is free to change shape everywhere — which is what the signature curve is a picture of, and which returns a lower answer than this one.

The history, which is a laboratory rather than a theory #

The lateral bracing rules go back to Winter’s work of the late 1950s, which is where the ideal stiffness and the small brace force came from. The torsional half arrived thirty years later, from Joe Yura’s tests at Texas through the 1980s and early 1990s, and it arrived in the form it did because of what the tests found rather than what the theory predicted.

The theory says a rotational spring at midspan doubles the buckling capacity, and the tests said it did not. What was missing was the web: the specimens were braced through a web that bent, and the measured stiffnesses were the series values rather than the brace values. The $\beta_{sec}$ expression that carries the web and the stiffener into every modern bracing rule is a fitted plate-bending calculation added to make the arithmetic agree with the beams — and the reason it is remembered as an experimental result is that no one had thought to look for it in the algebra.

That order of events is common in stability and worth expecting: the eigenvalue is the easy half, and what a real structure actually does with a restraint is decided by a flexibility somebody drew as a line.

The ladder from here #

Later rungs on this anchor: multiple torsional braces along a span, where the plateau is the $n$-th half-wave and the demand rises with the count. The brace force from an assumed imperfection, and where the two per cent rule comes from. Load height combined with torsional restraint, since a load applied above the shear centre is destabilising and the brace has to hold that too. Continuous torsional restraint from a deck, where $\beta_T$ becomes a distributed rotational stiffness and the beam forgets its length in the same way a continuously restrained one does. The cross-frame as a structure in its own right, with its own diagonal forces and its own connections. And the case where the beam braces the brace — a row of girders whose cross-frames have nowhere to react against, which is a plan bracing problem wearing a member’s clothes.