The column that creeps under the arch

The prop at the crown of the arch braced a shallow pinned arch at mid-span and asked what the prop could do. The answer was two things. It could not hold the arch’s sideways mode, whose node is the crown, and it raised the critical load only by carrying a share of the load itself — from 311 to 390 in that essay’s load parameter, at a rise of 6 radii of gyration, for a prop of κc = 3.5. But it changed the character of the failure. Unpropped, the sideways branch the arch takes when it bifurcates falls steeply, and the arch snaps; behind a prop of κc ≈ 3.2 the branch turns upward, and the arch, having turned aside, goes on carrying more.

It was the third essay to find that a shallow arch’s failure is a question about a path rather than about a load: the roof that jumps had found the limit point, and the sideways branch came after. That essay ended on the prop’s material. A crown prop that is a column from a floor below is concrete, and it creeps under the force it holds. A column of κc = 3.5 on the day it is cast may hold its force, ten years later, as if it were a spring of 1.5 or less — back on the side where the arch snaps. Whether an arch propped by a creeping column keeps its stable sideways branch through the column’s life, and whether the rib’s own creep offsets the column’s, was the question.

Two stiffnesses in one column #

Creep makes a concrete member’s stiffness depend on how long the force has been there. A force applied in a few seconds strains the column by $F/K$, its instantaneous stiffness. A force held for years strains it by $F(1 + \varphi)/K$, the creep coefficient φ adding to the elastic strain in proportion — the effective modulus, the simplest way of writing what the deflection that arrives three years late found in a slab. So a creeping column is not one spring but two: $K$ for a load that comes and goes, and $K/(1+\varphi)$ for a load that stays.

An arch carries both kinds. Its dead load — the deck, the finishes, its own weight — is on it for its whole life, and the column holds its share of that as a spring of $\kappa_c/(1+\varphi)$: the crown settles a little as the column creeps, the column’s force falls, and the rib carries the difference. A gust, a crowd, a snow load or a lorry arrives in seconds, and the column meets it with its full instantaneous stiffness — but from a rest length that creep has moved down, by $s = c\,\varphi/(1+\varphi)$ when the crown has deflected $c$ under the lasting load. In the energy of the arch the prop’s term becomes $8\kappa_c(u_1 - u_3 - s)^2$: the same stiffness, pushing only on what the crown moves beyond its crept position.

A quick load sees the column’s stiffness #

The three paths separate the two stiffnesses. The dashed one is the earlier essay’s: the arch on its column on the first day, rising to its critical load of 390. The light one is the arch holding a lasting load on a column crept to φ = 2 — a softer path, because the column now gives way in proportion, reaching its critical point at 338. The solid one starts where the lasting load of 273 has brought the crept arch, the dot, and follows a quick increment from there with the column at its instantaneous stiffness. It climbs at the day’s slope, not the crept one, and reaches 353.

The important property is not the height of that path but its shape where it ends.

The branch slope along the sideways path is $16 + 16\kappa_c(1 - \mathrm{d}u_3/\mathrm{d}u_1) - 64$ at the point where it leaves, and the rest-length offset s does not appear in it: moving the spring’s rest position moves where the path starts, not how steeply the branch falls. So for any quick load the column’s creep is invisible to the question the earlier essay answered. At κc = 3.5 the branch rises at +4.0 on the first day and on the hundredth year, and an arch that does not snap under a gust on the day it is propped does not snap under the same gust when its column is old. An old column is, if anything, stiffer: concrete’s instantaneous modulus goes on rising slowly for years.

The two pink dots are the stiffness the column holds the lasting load with: 1.75 after creep to 1, slope −21; 1.17 after creep to 2, slope −30. Both are deep on the snapping side. If the lasting load ever reaches the bifurcation, the arch will snap — but the lasting load reaching the bifurcation is a separate question, and it has a separate answer.

The lasting load loses only the column’s share #

For the load that stays on, the critical value falls as the column creeps: 352 at a creep coefficient of 1, 338 at 2, 332 at 3. It can never fall below the unpropped arch’s 311, because a column that has crept to nothing is no column, and the arch was always capable of 311 on its own. The column’s creep can take back only what the column was contributing, and the earlier essay found that contribution to be modest: the prop raised the critical load by carrying 6 to 18 per cent of the load, not by holding a mode.

For the arch to snap under its lasting load alone, that load would have to sit above 332 — 85 per cent of the day’s critical load — and no arch is designed to carry a permanent load within a sixth of its buckling load. At any ordinary ratio of dead load to critical load the lasting load stays well below the crept critical value, and the slope of the branch at that critical value is never tested.

What the column lets go of #

The share the column carries falls with its creep, from about 13 per cent of the arch’s load on the first day to 8 per cent at a creep coefficient of 1 and 5 per cent at 2. What it lets go of, the rib takes back in compression and bending. That is the mechanism of the lower critical load, and it is also a load path a designer should see: a column that was sized for its first-day force will be lighter-loaded for most of its life, and the rib will carry more than an elastic analysis at first loading says.

The transfer is not free for the rib’s own checks. The rib’s bending at its quarter points, which the crown prop was relieving, returns in proportion to the column’s lost share; a rib sized on the propped analysis will be stressed beyond it by the time the column has crept. It is the same direction the columns are shorter than the core found in a tall building, where the member that shortens hands its load to the one that does not.

The reserve the lasting load leaves #

The column’s creep moves the arch’s starting point for every later load, and that is where it costs something real.

Under a lasting load of half the critical load, the reserve for a quick increment falls from 0.50 of the critical load to 0.44 as the column creeps to 2 — about a ninth of it. Under 70 per cent, from 0.30 to 0.20, a third. Under 85 per cent, from 0.15 to 0.02. The heavier the lasting load, the larger the fraction of the reserve the column’s creep takes, because the lasting load does more of the creeping and the crept crown starts closer to the critical point. But the reserve is lost by the start moving, not by the snap coming back: what is left of it is still held behind a branch that rises.

A concrete rib creeps too #

The earlier essay’s question had a second half: whether the rib’s own creep, if it is concrete, offsets the column’s. It does keep the ratio. A concrete rib and a concrete column creeping at the same rate both hold their lasting forces as stiffnesses divided by $1 + \varphi$, so the column’s stiffness relative to the rib’s is unchanged, and the arch does not move along the κc axis at all. What changes is everything else.

With every stiffness in the arch divided by $1 + \varphi$, the arch’s energy under a lasting load q is exactly the elastic arch’s energy under $q(1+\varphi)$: the load parameter, which is $16qL^4/\pi^4 EI r$, scales with $1/EI$. So a concrete arch snaps under its lasting load when that load times $1 + \varphi$ reaches the critical load — at a creep coefficient of 1.0 if the lasting load is half the critical load, 0.43 if it is 70 per cent, 2 if it is a third. A creep coefficient of 2 is ordinary for concrete loaded young, so a concrete arch whose dead load is more than a third of its elastic critical load will, by this measure, snap under that dead load during its life, with no live load at all. That is the creep buckling the column that fails years later found in a strut, and a shallow arch’s limit point and bifurcation are more sensitive to it than a column’s, because they are already reached by deflection rather than by a load alone.

The curve is also steep where design lives. A lasting load of a third of the critical load snaps at φ = 2 and one of a quarter at φ = 3, so an arch whose creep coefficient is uncertain by a third — as every predicted coefficient is — has its life’s margin uncertain by the same proportion, and the only robust defence is to keep the lasting load well below the critical load that its creep will reduce it to. The concrete arch is designed against its crept self, not its new one.

Set beside that curve, the steel rib on its concrete column barely registers. The column’s creep matters to the snap only for lasting loads within a sixth of the critical load; the rib’s own creep matters for lasting loads above a third. Of the two creeps the lead asked about, it is the rib’s that governs, and it does not offset the column’s — it is a different and much larger effect that the column’s creep merely sits on top of.

Three checks, and the stiffness each one takes #

Put together, the arch on its creeping column asks three questions, and each is answered with a different stiffness for the column.

The first is whether the lasting load can be carried at all, for the whole life. That uses the crept stiffness, $\kappa_c/(1+\varphi)$, with the creep coefficient at the end of the life, and it is a check on the critical load: the lasting load against the 338 of a column crept to 2. It is the check that creep makes worse, and for a steel rib it makes it worse only by the column’s share.

The second is whether a quick load on top can be carried, and it uses the column’s instantaneous stiffness from the crept starting point. The critical total falls — 390 on the first day, 353 after creep under a 70 per cent lasting load — because the starting point has moved, and the reserve with it. That is the check that sets how much live load the arch may take late in its life, and it is the one the first-day analysis overstates.

The third is whether a quick load that reaches the critical point is the end of the arch or the start of a turn aside, and it is decided by the column’s instantaneous stiffness alone. Creep cannot touch it. The brace that holds a mode found that a brace’s value in a post-buckling question lies in its stiffness at the moment of buckling, and the moment of buckling under a quick load is seconds long — far too short for a column to creep.

Two of those three are unchanged in kind by creep and the third loses only a little, which is why the lead’s worry turns out to be smaller than it looked. A κc of 1.5 after a decade is a true statement about the column under its lasting force, and an irrelevant one about the column under a gust. The arch amplifies whichever load is applied to it, as the load that makes itself worse found every compressed structure does, but it amplifies a quick load through a stiffness that has not aged.

By hand, at a creep coefficient of 2 #

The column’s lasting stiffness is $3.5/(1+2) = 1.17$. Under a lasting load of $0.7 \times 390 = 273$, the arch braced by 1.17 sits where its crown’s first-mode deflection is 1.10 and its haunch term 0.02, so the crown has deflected $c = 1.08$ radii of gyration against 0.88 on the first day. The column’s rest length has crept down by $s = 1.08 \times 2/3 = 0.72$; its force is $\kappa_c(c - s) = 3.5 \times 0.36$ in the same units, and its share of the load is $4\pi \times 3.5 \times 0.36/273 = 6$ per cent, against 14 per cent on the first day.

For the concrete rib, the lasting load parameter at half the critical load is 195, and with the rib and the column both creeping, the arch behaves as an elastic one under $195(1 + \varphi)$. That reaches 390 at φ = 1. A designer using EN 1992-1-1’s effective creep ratio for slender members does exactly this, and would find the arch’s dead load limited to well under half its elastic critical load for that reason.

What the effective modulus and three sines leave out #

The creep is the effective modulus, applied at once. A lasting load whose creep develops over decades, on a structure whose geometry changes as it creeps, is a history; the effective modulus treats it as an instant at the end. For the column, whose share is small, the difference is small. For the concrete rib it matters more, and refined analyses of creep buckling generally find a little more time than the effective modulus gives — but they find the same snap.

The arch is the shallow sinusoidal arch of the earlier essays, three terms in a sine series, pinned at its springings, under a sinusoidal load. Its numbers are in its own load parameter; a real arch with a real load distribution moves them, as the arch that turns aside before it snaps showed for the mode that governs.

The rib is assumed not to shorten on its own. A concrete rib also shrinks, and shrinkage of a shallow arch’s rib lowers its crown by the same geometry the arch that gets shorter found — a fraction of a millimetre of shortening becoming millimetres of crown drop — which moves the arch towards its limit point without any load at all. For a shallow concrete arch that is a third creep-like effect, and it adds to the rib’s own.

The column has no shrinkage and no load of its own. A concrete column shrinks as well as creeping, which moves its rest length down whatever force it holds; it carries the floor it stands on; and a column standing on a floor that itself creeps puts that floor’s deflection under the arch. All three add to the crown’s offset s, and so to the starting point of every later load, without changing the instantaneous stiffness that holds the branch up.

Still open: the hanger that creeps the other way #

Every crown support here pushes up from below. A shallow arch can equally be braced at its crown by a hanger from something above — a cable to a mast, a tie to a stiffer arch — and a hanger that is a steel strand does not creep, but relaxes: its force falls at constant length, by a few per cent, and its anchorage slips a little in its first months. A strand hanger loses force without losing stiffness, which is the opposite of a creeping column, whose stiffness for a lasting load falls while its instantaneous stiffness holds. Whether an arch held up by a relaxing hanger has its lasting critical load untouched and its starting point lowered — the same reserve lost, by a different mechanism — or whether a slipped anchorage, which is a gap rather than a spring, reopens the snap that a stiff hanger closed, is the question the strand puts to the column.