The force the brace leaves behind

An inverted V of two braces runs from the beam above down to the two column bases. Push the frame sideways and one brace goes into tension and the other into compression, equal and opposite, and their vertical components at the apex point in opposite directions and cancel.

That is the picture that gets a chevron brace drawn, and the beam above is usually sized for gravity alone because of it. The picture is correct for exactly as long as both braces are elastic.

Which free body produced the number #

The apex node. Cut both braces just below it and cut the beam either side, and four forces act.

The tension brace pulls down and outward along its own line. The compression brace pushes up and outward along its own. While the two forces are equal in magnitude, the vertical components are equal and opposite and the beam sees nothing but the horizontal residue, which is the storey shear it was going to see anyway.

Now buckle the compression brace. A slender member that has buckled does not carry its critical load any more; it carries a residual somewhere between a fifth and a half of it, falling as the member is pushed further, because a buckled strut is a mechanism with a plastic hinge at midlength and its capacity falls as the hinge rotates.

The tension brace does not care. It goes on to yield at $A f_y$, which for a section chosen to buckle at 445 kN is 1,065 kN.

Vertical equilibrium of the node now gives

$$P_{unb} = (T_y - \beta P_{cr})\sin\alpha$$

and with $\beta = 0.3$ that is 659 kN, applied downward at the middle of the beam.

Why the number is so large #

The size of the residue is not a detail of the buckling model; it is a consequence of two independent choices that nobody makes together.

A brace is sized for the larger of its tension and compression demands, and the compression demand is the one that governs, because a slender member’s compression capacity is a fraction of its yield. So the section is chosen so that $P_{cr}$ is adequate, which means $T_y$ is much more than adequate — a factor of two or three is entirely ordinary.

Then the buckled brace retains only a fraction of even the smaller of the two.

Multiply the two effects together and the unbalanced force is around 60% of the tension brace’s full yield capacity. It is not a correction to the design; it is comparable with the whole brace force.

The moment it puts into the beam is $P_{unb}L/4$, which on an 8 m bay is 1,318 kNm — against the 200 kNm the 25 kN/m gravity load asks for. Six and a half times, and it arrives as a point load at the one section of the beam where a point load does the most.

Why the elastic check passes #

This is the part worth dwelling on, because it is a general lesson about what a check is a check on.

Run a linear analysis of the frame at its design lateral load. Both braces are elastic, their vertical components cancel exactly, the beam carries gravity, and every member is inside its capacity. The check passes, and it is arithmetically correct.

It is a check on a state the frame passes through. The state it will be in when the demand actually arrives — a wind event large enough to matter, or an earthquake — is the one where the compression brace has buckled, and in that state the beam is carrying six times what the analysis gave it. Every quantity in the elastic check is right and the check is about the wrong structure.

The panel that carries more after it fails is the same shape of argument with a happier ending: a plate girder web buckles and then carries more, because a new mechanism appears. Here a brace buckles and a new demand appears, on a member that was not part of the mechanism at all.

There is a further reason the elastic check is so persuasive, and it is worth naming because it applies far beyond chevrons. The elastic state is the one the structure spends its entire life in. It is the state on the day of the opening, the state under every wind the building will ever feel except one, and the state in which every measurement anybody ever takes of it is made. The post-buckling state exists for a few seconds, once, possibly never — and it is the only one in which the beam’s size matters.

Designing for a state that may not occur, against an analysis of the state that always does, is an uncomfortable position to argue from, and it is the position capacity design is always argued from. Two ways of being wrong is about bounding a collapse load. This is about bounding the demand one member can place on another, which is the same kind of statement and has the same property: it does not depend on knowing what the load will be.

The design response, which is capacity design #

The fix is not to make the braces stronger. Making the tension brace stronger makes the residue larger, because the residue is the tension brace’s yield.

The fix is to decide, in advance, which member is allowed to fail, and then to make sure everything else can survive that member doing so. That is capacity design, and it is the same discipline as the part that is meant to be weak and made weaker on purpose.

Here it means the beam above a chevron has to be designed for the unbalanced force computed from the braces’ expected strengths — not their design strengths, because a brace made from steel that came out 15% stronger than its certificate delivers 15% more unbalanced force. Overstrength is a hazard rather than a bonus in a system where the demand on one member is the capacity of another.

The consequence is a beam much heavier than a gravity calculation gives, often two or three times, and frequently deeper than the floor zone allows. That is the real reason chevrons are unpopular in seismic regions and the reason the alternatives exist.

Three ways out are used and each is a different structure.

Make the beam strong enough. Honest, expensive, and the default.

Use a two-storey X. Run the braces across two storeys so that the unbalanced vertical force at one floor is balanced by the brace continuing above, and the residue goes into an axial force in the intermediate beam instead of a moment.

Use a buckling-restrained brace. A brace that cannot buckle has $\beta = 1$ and its compression capacity equals its tension capacity, so the residue is zero by construction. That is the cleanest answer available and it costs a proprietary component.

A fourth is used more often than any of them and is not really a solution: use a single diagonal instead. One brace per bay has no apex and therefore no unbalanced force, and the price is that the bay is stiff in one direction of loading and soft in the other, so the frame is asymmetric and the tie that spends an afternoon as a strut becomes the whole design. Two single diagonals in opposite bays restores the symmetry at the cost of two bays, which is usually why the chevron was chosen in the first place.

Where the force actually goes #

A point load at midspan of the beam is where the argument usually stops, and it is worth following one step further, because the load does not stay there.

The beam delivers it to the two columns as end reactions, half each, downward. Those reactions are new axial forces on the columns, they are large — 330 kN each on the frame drawn — and they are in addition to everything the gravity analysis gave them. On a multi-storey braced frame the effect accumulates: every braced bay above contributes its own residue, and the column at the base of a braced bay carries the sum.

That is the same argument as the beam’s, applied to the members the beam leans on, and it is the reason capacity design is described as a hierarchy rather than as a check. Each member’s demand is the previous member’s capacity, so the chain has to be followed all the way to the foundation — and it ends at a holding-down bolt whose demand is a column’s capacity rather than a load.

What the frame does next #

There is a second consequence of the unbalanced force, and it is about stiffness rather than strength.

Once the beam is carrying a large midspan point load it deflects, and the apex of the chevron moves down. Both braces are attached to that apex, so both shorten geometrically — which relieves the tension brace and unloads the frame. The braced bay’s lateral stiffness falls because its beam is bending, and the softer the beam, the more it falls.

So the beam is not merely a member that has to survive the unbalanced force. It is part of the lateral load path, and a beam that is only just strong enough is a beam that is far too flexible: the frame’s storey drift under the post-buckling state can be several times what the elastic model predicted, at a load the elastic model said was fine.

That coupling is why the requirement on the beam is usually written as a strength requirement and a stiffness one, and why the two together are what make the member large.

It is also a case of a general effect that is easy to miss in a lateral system: held and not held is about a bracing system whose stiffness is not what the model says, and a chevron whose beam has started to bend is exactly that. The bay has not lost its braces — they are both still there, both still attached, and one of them is still elastic — and it has lost most of the lateral stiffness it was drawn to provide, through a member the lateral model never treated as flexible.

The same shape of problem elsewhere #

A chevron is the clearest case because the cancellation is drawn and the failure of it is dramatic, but the pattern is not rare.

A pair of prestressing tendons on either side of a web balance each other’s transverse effects, until one is not stressed — during construction, or after a duct is found to be blocked.

A symmetric portal frame carries a symmetric load with no sway, until one of its two identical column bases turns out to be less fixed than the other.

Two shear walls either side of a core share a storey force in proportion to their stiffnesses, and one of them cracking is a change of stiffness that moves the resultant off the centre of rigidity — which is the corner that moves most arriving as a consequence of damage rather than of geometry.

And a floor slab’s two spanning directions share load in a ratio fixed by their stiffnesses, until cracking in one direction changes the ratio.

In every case the structure was designed on a balance, the balance held while everything was elastic and identical, and the question worth asking at design time is what the forces become when it does not. That question is answerable in an afternoon and is almost never asked, because the analysis that would have to be run is not the analysis the design is checked against.

Where the model stops #

The residual fraction is assumed. Thirty per cent is a convention drawn from tests and it varies with slenderness, with how many cycles the brace has been through, and with whether the brace has already been straightened once. Early cycles retain more; later ones much less, until the brace fractures at the hinge.

The tension brace was taken as yielding, not fracturing. A brace that fractures at a connection instead of yielding along its length delivers nothing at all, and the unbalanced force disappears — replaced by a complete loss of the bay, which is worse.

Nothing here is about the columns. The same overstrength argument applies to them: a column at the end of a braced bay carries the vertical components of every brace above it at their expected capacities, which is a much larger axial force than any analysis at design load produces.

The beam’s own axial force was ignored. It is not small: the beam in a braced bay is a collector, delivering the storey’s diaphragm force into the braced point, so it is carrying a large axial compression at the same time as the moment from the unbalanced force. Two ways to fail applies, and the interaction bites here rather than being negligible.

And the argument assumes the brace buckles out of plane, symmetrically. A brace that buckles in plane hits its neighbour, and a chevron whose two braces are on the same gusset can buckle in a coupled mode neither analysis contains.

The generalisation #

The habit worth taking away is to ask, of any member whose forces are said to cancel, what happens to the cancellation when one side of it changes state.

Cancellation is a statement about two quantities being equal, and equality is a property of a state rather than a property of a structure. Two braces cancel because they are the same size and both elastic. Two halves of a symmetric frame cancel because both are intact. A pair of prestress forces cancel because both tendons are stressed. In every case the cancellation is real, is worth relying on, and ends the moment the two sides stop matching — and a design that relies on it has an unstated dependency on a member it does not appear to involve.

The second habit is the one this collection keeps returning to under which failure arrives first. A structure has a sequence of failures, not a single capacity, and the sequence is a design output as much as any number is. A chevron’s sequence, chosen deliberately, is: compression brace buckles, tension brace yields, beam holds, frame drifts, and everything is repairable. Chosen by default, it is: compression brace buckles, beam hinges, floor drops, and the braces are still intact in the wreckage.