The brace that holds a mode

The arch that turns aside before it snaps found two ways for a shallow arch to fail under a load spread across its span. It can go down symmetrically until the load it carries begins to fall, a limit point, and snap through. Or, if it rises more than $\sqrt{22} = 4.69$ times its radius of gyration, it can take a second way out first: sideways, one half dropping while the other lifts. An arch rising eight radii of gyration turns sideways at a load parameter of 461. Its symmetric limit point, which it never reaches, is at 844.

The sideways mode needs the two quarter points to move in opposite senses, one down and one up. That essay ended on what happens if something stops them: a hanger to a deck, a diagonal across the arch, anything tying the quarter points together against that one motion. The arch would then have only its symmetric way out, at nearly twice the load. Whether a single member can do that, how stiff it must be, and what it carries, is the question here.

A brace on a shape rather than a point #

A brace is usually a member at a point, and it holds that point still. The brace wanted here is different: it should leave the symmetric motion alone, since that is the motion the arch fails in anyway, and resist only the antisymmetric one. The quantity it resists is the difference between the two quarter points’ deflections. A deck hung from the quarter points on two hangers and stiff in bending does this, and so, to first order, does a member that ties the two quarter points’ vertical movements together through any linkage that does not care if both go down.

The arch is modelled as before, by the first two terms of a sine series — a symmetric half-sine, which carries the crown down, and an antisymmetric full sine, which carries one quarter point down and the other up — with the supports unable to move apart, so that any shortening of the rib along its length is resisted by its axial stiffness. The brace adds a spring to the second term alone. Its stiffness is measured against the rib’s own bending stiffness in that mode, as a ratio κ: at κ = 1 the brace resists the sideways shape exactly as hard as the rib does.

Where the sideways path now leaves #

In the unbraced arch the sideways path leaves the symmetric one where the rib’s compression has grown enough to buckle it in its antisymmetric mode. A brace on that mode adds its stiffness to the rib’s, so the rib needs more compression before it will go. The branch now leaves where

$$(\hat e - u_1)^2 = \hat e^2 - 16(1 + \kappa),$$

$\hat e$ being the rise and $u_1$ the crown’s deflection, both in radii of gyration. The symmetric limit point does not move, because the brace does nothing to the symmetric mode. So the sideways branch comes first only while $\hat e^2 > 22 + 24\kappa$, and a brace of

$$\kappa^* = \frac{\hat e^2 - 22}{24}$$

is enough to return the arch to its limit point.

An arch built a tenth of a radius of gyration out of shape in its sideways mode shows the effect of the brace whole. Unbraced it peaks at 410 and turns aside. With half the brace the perfect arch needs, it peaks at 658. With the brace the perfect arch needs, 794. With twice that, 840, the limit point within half a per cent. The symmetric path beyond the peak — the snap — comes back, and the arch fails the way the earlier essay drew for a lower rise. The brace does not make the arch stronger than its own limit point. It removes the earlier way out.

Stiffness up to a point, and nothing after it #

For the perfect arch the effect is a kink.

As the brace stiffens, the critical load climbs, steeply at first, until it meets the limit point; from there it is flat. This is the shape a column’s brace makes: a critical load that climbs with the brace’s stiffness until the column buckles between braces instead, and then plateaus. The arch’s plateau is its symmetric snap, the column’s its next mode, and in each case the brace has a stiffness beyond which it is doing nothing but carrying its own weight. For a rise of 6 that point is at 0.58 times the rib’s antisymmetric stiffness and the brace buys 21% of load; for a rise of 8 it is at 1.75 and buys 83%.

The brace grows as the square of the rise #

The stiffness needed grows faster than the gain.

Below a rise of $\sqrt{22}$ the arch snaps symmetrically on its own and no brace helps. Above it, the brace needed grows as the square of the rise, and so does the reward: an arch rising twelve radii of gyration needs a brace five times as stiff as its own rib in that mode, and gains nearly four times the load. The limit point grows roughly as the cube of the rise, as the roof that jumps found for a two-bar frame, while the sideways buckling grows only linearly. So the higher the arch, the more a sideways brace is worth and the stiffer it has to be.

The brace’s stiffness here is against the rib’s own resistance in the sideways mode. For a sine-shaped rib of span $L$ and bending stiffness $EI$, a spring of stiffness $K$ on the difference between the quarter points’ deflections has $\kappa = K L^3/(2\pi^4 EI)$, so the brace for a rise of 8 is a spring of $2\pi^4 \times 1.75\,EI/L^3 = 341\,EI/L^3$. On a 60 m rib with $EI = 4.2\ \text{GN}\cdot\text{m}^2$, that is 6.6 MN/m: a hanger and a deck, or a pair of diagonals, have to hold the quarter points’ difference to a millimetre under 6.6 kN.

A deck as the brace #

The commonest brace on an arch’s sideways mode is a deck hung from it. Suppose the deck is a girder spanning the same span $L$ on its own supports, with a hanger from each quarter point of the arch down to it. When the arch tries to turn aside, the hangers pull one quarter point of the deck up and push the other down, and the deck resists that equal and opposite pair by bending in an S. Each half of the deck then behaves as a simply supported beam of span $L/2$ loaded at its middle, which deflects by $P(L/2)^3/48EI_d$, and the stiffness against the difference between the two quarter points is $192\,EI_d/L^3$. Divided by the arch’s own $2\pi^4 EI/L^3$, that is

$$\kappa \approx 0.99\,\frac{EI_d}{EI},$$

almost exactly the deck’s bending stiffness over the rib’s. A deck as stiff as the rib is a brace of $\kappa = 1$: enough for an arch rising up to $\sqrt{46} = 6.8$ radii of gyration, not enough for one rising eight, which needs a deck 1.78 times as stiff as its rib. The stiffer the deck is relative to the rib, the more of the arch’s sideways resistance it supplies, and a bridge with a deep deck and a slender rib is braced against the sideways mode by the thing it carries.

The hangers have to carry the force the brace carries, and the deck has to carry it in bending as well as its own traffic. The 12% of the arch’s load that a crooked arch puts through its brace at the peak goes through two hangers and into the deck’s S-bending, at the moment the deck is also carrying the load that brought the arch to its peak.

Built out of shape, the brace must be stiffer #

Every real arch is built slightly crooked, and the sideways mode is the one a crooked arch is already partly in.

The curves round off the perfect arch’s kink. With the brace the perfect arch needs, an arch built a hundredth of a radius of gyration out reaches 99% of its limit point, a tenth out 94%, three-tenths out 84%. With twice that brace the first two reach the limit point and the third 96%. The pattern is Winter’s: a brace sized at exactly the ideal stiffness delivers the ideal load only to a perfect member, and a real one wants about twice it.

The loss unbraced is the one the earlier essay found: the sideways buckling is unstable and symmetric, and an imperfection costs it as the two-thirds power of its size, a third of what the theory promised for a tenth of a radius of gyration. A brace stiff enough to suppress the bifurcation converts it back into a limit point, whose sensitivity to a sideways imperfection is far milder — which is why the curves for the larger braces converge on the limit point even for a crooked arch.

What the brace carries #

The argument for a sideways brace usually ends with the stiffness, on the reasoning that a perfect arch does not move sideways until it buckles, so the brace carries nothing until then and can be light. That is true of the perfect arch.

A crooked arch moves sideways from the first load, by an amount the growing compression amplifies, and the brace holds it with a force in proportion. At the peak, with the brace the perfect arch needs, a brace on an arch built a tenth of a radius of gyration out of shape carries 12% of the arch’s whole load. Twice as stiff, it carries 6%: a stiffer brace lets the arch move less and so carries less, the second half of Winter’s argument. But even at twice the stiffness, an arch built three-tenths out puts 17% of its load through the brace.

A brace on a mode is a structural member, not a tie. The force in it is the force needed to hold the arch’s own sideways imperfection, amplified by the rib’s compression. It is largest exactly when the arch is at its peak, and it must be designed for that peak, with connections to match. A brace sized for nothing but stiffness, and connected for a nominal force, fails at the moment it is needed, and the arch then turns aside at its unbraced load from a state already well along its peak.

How the braced arch fails #

The unbraced arch at its peak has barely moved down. Its crown has dropped 1.35 radii of gyration while its quarter points are already more than a radius of gyration out of level in opposite senses: it has failed by turning aside, with most of its rise intact. The braced arch has gone down more than twice as far, its crown 3 radii of gyration below where it started, with its quarter points only 0.65 out. It fails by flattening, and when it snaps it goes through as a shallow arch does, symmetrically.

That difference matters for what happens after. A sideways failure leaves the arch hanging asymmetrically, one half up, and it can sometimes be caught by the structure around it. A snap-through flattens the arch past straight, and in a roof it ends with the rib hanging in tension the wrong way up. The brace buys load and changes the failure into the more violent of the two.

Other modes, other braces #

The same reasoning runs through bracing elsewhere in this collection, and the arch only makes it plainest. A beam’s lateral–torsional buckling is a mode of twist and sideways bending together, and braces that stay where the framing puts them work by holding that combined shape at a few points; a brace on the wrong part of the section holds the wrong motion and does little. A network arch’s crossed hangers turn the rib and deck into a truss, which carries the shear twice and so stiffens precisely the sideways mode a half-span load would excite. A portal frame’s bracing holds its sway mode and leaves its columns free to bend.

In each case the useful question is not where the brace goes but which shape it holds, and how stiff the structure already is in that shape. A brace on a mode the structure was never going to fail in costs its weight and buys nothing. A brace on the mode it was going to fail in buys load until the next mode comes first, and carries a force set by how far out of shape the structure was built.

The arch in numbers #

For a rib rising 8 radii of gyration, the perfect arch turns sideways where $(\hat e - u_1)^2 = 64 - 16 = 48$, at $u_1 = 8 - 6.93 = 1.07$, and the load there is $16u_1 + 64(\hat e - u_1) = 17 + 443 = 461$. Its limit point is at $(\hat e - u_1)^2 = (64 - 4)/3 = 20$, $u_1 = 3.53$, and $p = 844$. A brace of $\kappa$ moves the sideways point to $(\hat e - u_1)^2 = 48 - 16\kappa$, which reaches the limit point’s 20 at $\kappa = 28/16 = 1.75$. With half that, $(\hat e - u_1)^2 = 34$, $u_1 = 2.17$ and $p = 16 \times 2.17 + 64 \times 1.875 \times 5.83 = 735$, against the crooked arch’s 658: the tenth of a radius of gyration out of shape costs that arch 10%.

Two modes and a spring #

Two modes. The sine series stops at its second term. A third, symmetric, term would let the symmetric path stiffen and lower the limit point a little for the higher rises; the sideways mode, which the brace is aimed at, is the second term exactly.

A spring on the mode, not a member. Real braces act at points, and a hanger at each quarter point to a deck resists the antisymmetric shape and, through the deck’s own bending, the symmetric one a little too. The spring here is the part that matters for the sideways mode alone; anything it does to the symmetric mode raises the limit point as well.

Elastic throughout. The rib does not yield, so these are elastic buckling loads. A steel rib rising eight radii of gyration under a load of 844 is likely to yield before it snaps, and its plastic capacity then caps what the brace can buy.

A load that does not move. The load is spread over the span and stays where it was put. A live load on half the span is already a sideways load, which a tied rib hung from vertical hangers counters with its deck; a brace on the sideways mode is the same idea applied to an arch with no deck.

What bracing a mode means #

Bracing as usually taught is about points: a lateral restraint at mid-height of a column, a fly brace at the compression flange of a beam. Each stops one point moving and turns one mode into a higher one. The brace here stops no point. It stops a shape, and leaves the arch free to deflect in every other way. Its stiffness is measured against the rib’s own stiffness in that shape, and its force against the load the shape would otherwise carry off. Both are the same quantities as for a point brace, read in the mode’s own coordinates.

The arch makes it visible because its two failures are so different: a sideways turn that leaves most of its rise and fails early, and a symmetric snap that uses all of it. Holding the first makes the second the failure, and the quantity a designer controls is the stiffness of whatever ties the two quarter points together.

Still open: the arch braced at its crown #

The brace here holds the quarter points against each other. A cheaper one holds the crown: a single prop or hanger at mid-span stops the symmetric mode instead, and leaves the sideways one free. On an arch that would have snapped symmetrically, a crown brace raises the limit point and may expose a sideways buckling the unbraced arch never reached. On one that already turned aside, a crown brace does nothing at first. Whether a single member at the crown and one at the quarter points, each sized against its own mode, can be traded against each other — and whether the arch then fails in a third mode neither holds, with a full wave between the supports — is the question of which points of an arch are worth tying to anything, and it belongs to the arch that is braced as cheaply as it can be.