The prop at the crown of the arch

The brace that holds a mode took a shallow arch that turns sideways before it snaps — one rising more than $\sqrt{22} = 4.69$ times its radius of gyration — and braced its quarter points against moving in opposite senses. That brace leaves the symmetric motion alone and resists only the sideways one, and a brace of $\kappa^* = (\hat e^2 - 22)/24$ times the rib’s own sideways stiffness returns the arch to its symmetric limit point, at nearly twice the load. On a perfect arch the brace carries nothing at all; on one built slightly out of shape, a few per cent of the load.

That essay ended on a cheaper brace. A single prop or hanger at mid-span — a column from a floor below, a hanger from a cable above, a strut to a deck — is one member at one point, where the quarter brace was a linkage between two. On an arch that would have snapped symmetrically, a crown prop stiffens the very mode the arch fails in. On one that turns aside, the essay suspected, a crown prop does nothing at first. This essay asks what it does after first, and whether the cheap brace can be traded against the clever one.

Three terms, because the crown is a point #

The arch is the earlier essays’ — a shallow pinned arch whose shape is a half sine wave, under a sinusoidal load, with supports that cannot move apart, so that any shortening of the rib along its length is resisted by its axial stiffness. The arch that turns aside before it snaps carried the arch’s deflection in two terms of a sine series: a symmetric half-wave that carries the crown down, and an antisymmetric full wave that carries one quarter point down and the other up.

A crown prop needs a third. The prop holds the crown’s deflection, and the crown deflects in both the first term and the third — a symmetric one-and-a-half waves whose middle crest is at the crown, with the opposite sign at the haunches. A prop that stops the crown going down lets the arch put the deflection into its haunches instead, the crown held and the two flanks dropping, and only a third term can draw that. So the deflection is $a_1 \sin \pi x/L + a_2 \sin 2\pi x/L + a_3 \sin 3\pi x/L$, the crown’s deflection is $a_1 - a_3$, and the prop is a spring on it.

The prop’s stiffness is measured as before, against the rib’s own bending stiffness in its first mode, as a ratio κc; physically $K_c = \kappa_c \pi^4 EI/2L^3$. The load is the load parameter $p$ that the earlier essays used, and everything else is in radii of gyration. With no prop the third term stays at zero and the arch is exactly the two-term arch: an arch rising six radii of gyration turns sideways at a load parameter of 311, and its symmetric limit point, which it does not reach, is at 375.

The crown does not move in the sideways mode #

The sideways mode is a full sine wave. It goes down over one half-span and up over the other, and at mid-span it passes through zero. The crown is its node. Whatever holds the crown — a prop of any stiffness, or a rigid one — does no work against the sideways mode and cannot resist it.

The figure shows what that means for the critical load. The arch rising 6 turns sideways with no prop, and turns sideways behind every prop: 335 at κc = 1, 357 at 2, 401 at 4. The sideways branch leaves at the same point every time, measured as the crown’s first-mode deflection, 1.53 radii of gyration. What moves is the load at which the arch reaches that point, and it moves only because the prop is taking part of the load. A crown prop does not brace the sideways mode. It is a third support, and the arch reaches its sideways buckling at a higher load only because it is carrying less of it.

The arch rising 4.5 shows the other half of the essay’s suspicion. Unbraced it snaps symmetrically, at 173, because its sideways branch would leave after its limit point. A crown prop stiffens exactly that symmetric mode, so the limit point rises — 182 at κc = 0.25, 191 at 0.5 — but the sideways branch, which the prop cannot touch, rises only by the prop’s share of the load, and somewhere near κc = 0.9 it overtakes the limit point. From there the arch rising 4.5 turns sideways too, at 209 for κc = 1. The two-term arch puts that switch at $\kappa_c = (22 - \hat e^2)/2 = 0.875$, and the third term moves it by a few per cent. A prop that stiffens the arch’s symmetric mode exposes a sideways mode the unbraced arch never reached, exactly as the previous essay suspected.

The branch that stops falling #

What the crown prop does to the sideways mode, it does after the arch has turned aside. The next figure follows the perfect arch along its sideways branch.

Unbraced, the sideways branch falls steeply: as the arch sways, the load it can carry drops by 48 load units for every radius of gyration the crown’s first mode goes down. That is the unstable bifurcation the first essay found, the reason the arch snaps the moment it turns aside and the reason an imperfection costs it so much. Behind a crown prop the branch falls more gently: −17.4 at κc = 2, −3.0 at κc = 3. Behind a prop of κc = 4 it rises, by 10.8 per unit, and the arch that turns sideways carries more load as it sways, not less.

The reason is in what the sway does to the crown. On the sideways branch the rib’s compression is held fixed — that is what being on the branch means — while the arch’s first-mode deflection keeps growing, so the crown keeps going down. A prop under the crown resists that, and its resistance grows as the crown descends. Unbraced, the load falls along the branch at a rate set by the rib’s own sideways stiffness, 64 units in the two-term arch; the prop adds back 16κc. The two-term arch’s slope is $16(1 + \kappa_c) - 64$, which is level at κc = 3; the third term’s slopes in the figure, −3.0 at κc = 3, cross zero at about 3.2.

The snap that goes, and its imperfection with it #

A level or rising sideways branch is a stable bifurcation, and an arch with a stable bifurcation does not snap. The next figure shows what that does to an arch built slightly out of shape.

Unbraced, the arch built a tenth of a radius of gyration out of shape carries 88 per cent of the perfect arch’s critical load, and one built three tenths out of shape 75 per cent — the steep loss of an unstable bifurcation, which grows as the imperfection’s two-thirds power. Behind a prop the loss shrinks: 92 and 84 per cent at κc = 2, 97 per cent at 3 for the smaller imperfection. Behind a prop of about κc = 3.1 the imperfect arch has no peak at all. It sways as it is loaded, with the load still rising, and the path in the hero figure for κc = 4 is the result: a smooth climb with no point at which the arch jumps.

The slope is the whole story, as it was for the column and the cylinder in a third of what the theory promised: a structure whose load falls after it buckles loses a large share of its strength to a small imperfection, and one whose load rises loses almost none. The crown prop moves the arch from the cylinder’s side of that line to the column’s.

That changes the kind of structure the arch is. An arch that snaps fails suddenly, at a load set by an imperfection nobody can measure, and its design has to carry a knock-down factor for the imperfection it might have. An arch that sways stably fails gradually, by yielding somewhere along a deflection that grows with the load, and its design is a strength check on a known deformation. The roof that jumps found the first kind in a shallow dome; the crown prop turns an arch from the first kind into the second.

The same prop at every rise #

The prop that stops the snap does not depend on the arch.

From an arch rising five radii of gyration to one rising twelve, the prop that stops the snap stays between κc = 3.05 and 3.17, a spread of four per cent while the rise more than doubles. The two-term arch says why: the sideways branch’s slope is $16(1 + \kappa_c) - 64$, which contains no rise at all. The prop competes with the rib’s sideways bending stiffness, 64 units, and both are properties of the rib’s bending; the rise sets where the branch leaves and how much load is on the arch when it does, but not whether the branch falls. The third term adds the small rise dependence the figure shows.

A designer can therefore state the stopping prop once, as a stiffness: about $3.1 \times \pi^4 EI/2L^3 = 150\,EI/L^3$ at the crown, for any shallow pinned arch of this shape under a load of this distribution. A real prop has a length and an area, $K_c = EA/\ell$, and the condition becomes one on its axial stiffness against the rib’s bending, which for a column from a slab below a few metres away is usually easy to meet and for a slender hanger from a cable often is not.

The prop carries what the brace did not #

The quarter brace and the crown prop can now be compared directly, at equal effect.

On the arch rising 6, the quarter brace of κ* = 0.58 returns the perfect arch to its limit point, 375. A crown prop reaches the same critical load at κc = 2.78. In physical stiffness the two are similar: the quarter brace is $K_a = 2\kappa^*\pi^4 EI/L^3 = 114\,EI/L^3$ on the difference of the quarter points’ deflections, the crown prop $K_c = 136\,EI/L^3$ on the crown’s. What they carry is not similar. The crown prop at its critical point carries 13.5 per cent of the arch’s load, on any arch, perfect or not, because it is a support and its force is the load it takes. The quarter brace carries nothing on a perfect arch and 8 per cent on one built a tenth of a radius of gyration out of shape, because its force is set by how far the arch has turned aside, which on a perfect arch is not at all before it buckles.

The quarter brace is sized for stiffness and the crown prop for strength. The brace that need not be strong is the general version of the first: a brace holding a mode carries a force set by the imperfection, a few per cent of the load it stabilises. A crown prop is not that kind of brace. It is a support, carrying a share of the load in proportion to its stiffness against the arch’s, and it gets its stability benefit — the level branch, the snap gone — as a consequence of carrying load at the one point the sideways mode does not move.

What the crown prop gives that the quarter brace does not is the end of the snap. A quarter brace of κ* returns the arch to its symmetric limit point, which is itself a snap — a limit point is a peak of the load — and an imperfection costs it less, but it still jumps. A crown prop past κc ≈ 3.1 leaves no peak to jump from. At equal stiffness, the quarter brace buys a higher snap and the crown prop buys no snap at all, at the price of a member that carries a seventh of the arch’s load.

Both at once, and the mode neither holds #

With a crown prop and a quarter brace together, the arch rising 6 with κc = 4 and κ = 1 has no sideways branch before its symmetric limit point, and that limit point, stiffened by the prop, is at 517 — the two braces each holding their own mode, and the arch failing in the stiffened symmetric one. With both stiffer still, nothing in three terms fails within six rises of deflection.

The next mode neither touches is the fourth term, two full waves, which has nodes at the crown and at both quarter points: a prop at one and a linkage between the others cannot see it. But in a sine series a mode of $n$ half-waves buckles where the rib’s compression reaches $S = -4n^2$ in these units — $-16$ for the sideways full wave, $-64$ for two full waves. The two-full-wave mode needs four times the compression that the sideways one did, and a shallow arch rising six radii of gyration with its crown and quarter points held cannot develop it before the rib yields. The third mode the essay feared is real, and it is far away.

Where arches already have a crown prop #

Many arches are propped at the crown without anyone calling it that. A deck arch carries its deck on spandrel columns, and over the middle of the span the deck often runs into the rib itself, joined to it over a short length so that the crown cannot move without the deck: a stiff prop, by any measure, if the deck is continuous and its supports are elsewhere. A tied arch carries its deck on hangers, and the hanger at mid-span ties the crown to the deck; whether that is a prop depends on how stiff the deck is in bending between its other hangers, which the rib that cannot lean on its own tie found matters a great deal for half-span loading. A network arch’s crossed hangers, which the hangers that carry the shear twice followed as a live load ran on, are quarter braces and crown props at once, which is part of why network arches are so stiff.

The calculation here says which of those propping arrangements matter for snap-through. A deck joined to the crown and stiff in bending is usually far past κc = 3, so on this calculation a deck arch with a crown connection does not snap sideways under a uniform load, whatever its rise. A single crown hanger to a flexible deck may be well short of it. And on a lightweight roof arch — a glulam or steel rib over a sports hall, where snap-through is a real failure mode because the rib is slender and the load is snow — a prop at the crown from a mezzanine or a gantry is a design choice with a calculable effect, as the braces that stay where the framing puts them found for cross-frames that cannot be moved: what matters is the stiffness at the point the framing provides, and here the point is the one place the sideways mode leaves alone.

By hand, for κc = 3 #

The two-term arch gives the stopping prop in one line. Along the sideways branch the rib’s compression is held at $S = -16$, so the load parameter is $p = 16(1 + \kappa_c)u_1 - 4S(\hat e - u_1) = 16(1 + \kappa_c)u_1 + 64(\hat e - u_1)$. Its slope with $u_1$ is $16(1 + \kappa_c) - 64$, zero at $\kappa_c = 3$. The branch leaves where $(\hat e - u_1)^2 = \hat e^2 - 16$, which for $\hat e = 6$ is $u_1 = 6 - \sqrt{20} = 1.53$, the same with any prop. There the load is $16 \times 4 \times 1.53 + 64 \times 4.47 = 98 + 286 = 384$ for κc = 3; the three-term arch, which lets the prop’s push go partly into the haunches, gives 379.

The prop’s share at that point is its force, $K_c$ times the crown’s deflection, over the arch’s whole load. In these units it is $4\pi\kappa_c u_1/p = 4\pi \times 3 \times 1.53/384 = 0.150$, against the three-term arch’s 0.14.

What the series leaves out #

Three terms of a sine series. A real prop applies a point force, which needs many terms to represent; three catch its effect on the first and third modes and leave out its local bending of the rib at the crown, which matters for the rib’s strength under the prop but not for which way the arch goes.

A linear prop. A column from a slab below can push but not pull, and a hanger from a cable can pull but not push. A pushing-only prop holds the crown against going down, which is the direction the load drives it, so for a downward load it acts throughout; a hanger under the same arch would go slack. And a prop’s own buckling, as a strut under 14 per cent of the arch’s load, is a check of its own.

One load distribution. The load here is sinusoidal, so it does no work on the sideways mode. A load over half the span drives the sideways mode directly, and then the crown prop, which cannot resist it, is no help at all; the quarter brace is the one that resists half-span loading, which is why decks hung from arches by many hangers behave as they do.

Still open: the arch on a column that shortens #

Every prop here is a linear spring. A crown prop that is a column from a floor below is a member that creeps, shrinks and carries its own load, and its stiffness against the arch changes over years as its concrete creeps. A prop of κc = 3.5 on the day it is cast may be one of κc = 1.5 after a decade of creep, back on the side where the arch snaps. Whether an arch propped at its crown by a concrete column keeps its stable sideways branch through the column’s life, or loses it as the column creeps, and whether the arch’s own creep, which relaxes the rib’s compression, offsets the column’s, is the question that the prop’s material puts to the stiffness this essay found it needs.