The roof that makes the wind worse every storm

A frame that carries a load safely once may not carry it safely ten thousand times, and the map with three regions is the reason: below the shakedown load the frame yields a little in its first cycles and then stays elastic; above it, it either yields both ways at some section every cycle or adds a little rotation every cycle in a mechanism. The beam the wind walks without pushing found the second on a fixed-base portal under a wind that reverses, and the base that sends the wind to the knees pinned the bases and found that the pinned frame shakes down at its collapse load wherever gravity leads, and that its knees take over the alternating yield once the wind is large.

Every one of those frames was analysed as though its sway did not change its moments. A slender portal under a heavy roof is exactly the frame where it does. The roof’s weight, carried down the columns, acts through the sway: when the frame leans by Δ, the weight $N$ adds a moment $N\Delta$ to the columns, the extra moment leans the frame further, and the wind’s share of every moment grows. The load that makes itself worse is the general case, and its answer for a frame is the amplification $1/(1 - N/N_{cr})$ on the sway’s effects. A shakedown check is a check on elastic moments over a whole domain of loads, and a repeated wind brings the amplification on both halves of every storm. The question is what that does to the shakedown load, where it puts the knees’ alternation, and whether a check without it is still a check.

The roof and the critical load #

Take the portal of the earlier essays — 10 m wide, 5 m high, columns and beam of the same section with a plastic moment of 200 kN·m and a bending stiffness of $42{,}000\ \mathrm{kN\,m^2}$ — carrying a gravity load of 160 kN at midspan that comes and goes and a wind of 60 kN at the beam’s level that reverses. Add a roof whose weight is carried down the columns from the knees. The roof adds no first-order moment at all — its weight goes straight down the columns — but it is all there to act through the sway.

The frame’s elastic critical load for sway follows from its stiffness by Horne’s method, as EN 1993-1-1 gives it: a unit horizontal load at the beam’s level sways the frame by $\delta_1$, and the total vertical load at which the sway would grow without limit is $N_{cr} = h/\delta_1$. On pinned bases it is 5,040 kN; on bases of 5,000 kN·m per radian, 7,703; on fixed bases, 23,040.

The amplification belongs to each moment of the storm, not to the frame. A shakedown check reads the elastic moments at the four corners of the load domain: gravity off and on, wind each way. With the gravity off, only the roof acts through the sway, and the wind is amplified by $1/(1 - P/N_{cr})$; with the gravity on, both act, and the wind is amplified by $1/(1 - (P + V)/N_{cr})$. On the pinned frame under a 1,000 kN roof those are 1.25 and 1.30. The gravity’s own moments are not amplified, because gravity at midspan does not sway a symmetric frame.

The three bases differ in the factor by a lot. A pinned frame is four or five times softer in sway than a fixed one, its critical load four or five times lower, and its amplification correspondingly larger: at a 1,000 kN roof its $\alpha_{cr}$ is 4.3, where the fixed frame’s is 20. That is the frame the question was about.

The shakedown load, with the roof #

Run the shakedown check — alternation at each section, and Koiter’s incremental bound over every mechanism of the frame — on the amplified envelope, as the roof grows.

The first-order check does not see the roof at all, because the roof adds no first-order moment: the dashed lines are flat. The amplified ones fall, and they fall in proportion to how soft the frame is in sway. Fixed bases lose 0.6 per cent of their shakedown load to a 1,000 kN roof and 1.1 per cent to 2,000. Pinned bases lose 4.8 per cent and 9.6.

The fall is smaller than the amplification. At a 2,000 kN roof the pinned frame’s wind is amplified by 1.75 and its shakedown load falls by under a tenth, because on this frame gravity leads: the governing mechanism is incremental, and most of what drives it is the gravity’s moments, which are not amplified. The wind’s share is the part that grows, and it is a share.

What is unchanged is the pinned frame’s peculiarity. On pinned bases where gravity leads, the earlier essay found the shakedown load equal to the collapse load: the beam mechanism credited from both winds and the combined mechanism are the same number. With the wind amplified the identity survives — the amplified collapse load is 0.692 at a 1,000 kN roof, exactly the amplified shakedown load. Repetition still costs the pinned frame nothing. The heavy roof costs it the same whether the wind comes once or ten thousand times.

Where the knees begin to alternate #

The roof matters more where the wind does. Hold the roof at 1,000 kN and the gravity at 160, and raise the wind.

At small winds the two curves are one line, as they were on the pinned frame of the earlier essay: shakedown at collapse. Then the knees begin to alternate. A pinned base has no moment, so the wind’s whole moment goes to the knees, and once the wind is large enough a knee’s elastic moment swings by more than twice its plastic moment over a storm — no residual can fix a range — and the frame shakes down below its collapse load for the first time. First-order, that happens from a wind of 110 kN. With the wind amplified for a 1,000 kN roof, from 90 kN.

That is the first part of the question answered. Second-order sway moves the pinned frame’s knee alternation down to a wind about a fifth smaller, because alternation is decided by the range of the wind’s moment alone, and the wind’s moment is exactly what the amplification multiplies. Above the onset, the amplified shakedown curve runs below the first-order one by 8 to 9 per cent at 120 kN, and by about 6 at 200 kN, where the knees’ alternation governs and collapse is well above it.

The gap between shakedown and collapse is where a frame designed for strength alone is short. First-order, the knees’ alternation opens a gap of 7 per cent at 120 kN of wind — 0.533 against 0.571. Amplified, it is 0.488 against 0.539, a gap of nearly 10 per cent. The heavy roof makes the frame weaker under one storm and weaker still under many.

What fixity is worth #

The earlier essay found that base stiffness moves the shakedown load from the pinned frame’s toward the fixed one’s while leaving collapse almost alone. A stiffer base also raises the critical load, so under a heavy roof it should be worth more. How much more is the second part of the question.

First-order, a fixed base raises the shakedown load by 21 per cent over a pinned one, and the roof makes no difference to that, since the first-order check does not see the roof. Amplified, the gain grows with the roof: 26 per cent at 1,000 kN, 32 at 2,000. A partly fixed base of 5,000 kN·m per radian, the stiffness of a bolted plate on a pad, goes from 8 per cent to 13.

So a stiffer base is worth more under a heavy roof, by half as much again as a first-order check credits it with at the heaviest roof drawn. It is not worth twice as much, and the reason is the same as the reason the shakedown load fell less than the amplification: the base’s second benefit, a higher critical load, acts only on the wind’s share of the envelope, and on this frame gravity carries most of it. A frame in which the wind leads would gain more; a frame in which gravity leads, as this one does at 60 kN of wind, gains the first-order benefit and a modest bonus.

A roof is not a gravity load #

It is worth being exact about why the roof is the load that matters and the gravity at midspan is not. Both are weight; both act through the sway. But the gravity at midspan comes and goes, and when it goes it takes its share of the amplification with it, so the half of the storm with the gravity off is amplified only by the roof. The roof never goes. It is on the frame in every state of every storm, so its amplification is on both halves of the wind’s range, and a range — the difference between the moment one way and the moment the other — is what decides alternation.

That is why the alternation onset moves so much more than the incremental bound. Yielding one way and then the other is a failure of range: the section has been asked to carry more than twice its plastic moment between the two extremes of a cycle, and no residual stress can help it. A permanent weight that multiplies the wind’s moment in both directions multiplies the range directly. The incremental bound, by contrast, credits each hinge with its worst moment in one direction only, and there the gravity’s unamplified moment is most of the credit.

It also explains why Eurocode 3’s habit of checking a frame’s $\alpha_{cr}$ under its full design gravity is the right one for strength and slightly the wrong one here. Strength is checked at the instant of the worst combination, with all the gravity on. A shakedown check needs the amplification at every corner of the domain, and the corners without the live gravity have a higher $\alpha_{cr}$ — on this frame 5.0 at a 1,000 kN roof alone, against 4.3 with the gravity added. Using the lower one everywhere errs on the safe side, by a little.

How far a first-order check overstates #

The third part of the question is whether a shakedown check made without the amplification is still on the safe side. It is not, and the error can be read directly against the frame’s $\alpha_{cr}$.

At an $\alpha_{cr}$ of 3, the lowest at which EN 1993-1-1’s amplification is allowed, the first-order check overstates the pinned frame’s shakedown load by 7.8 per cent at 60 kN of wind and 13.8 per cent at 120. At an $\alpha_{cr}$ of 10, at which Eurocode 3 lets first-order elastic analysis stand, it overstates by about 2 and 3 per cent. The error is always less than the amplification itself, which is 50 per cent at $\alpha_{cr}$ of 3 and 11 at 10, because only the wind’s part of the envelope is amplified.

Two readings follow. A frame with $\alpha_{cr}$ above 10 can be checked for shakedown first-order with an error of a few per cent, which is Eurocode 3’s tolerance for strength and is reasonable here too. A frame with $\alpha_{cr}$ below 10 cannot, and the error is largest exactly where the wind leads — where the shakedown limit is alternation at the knees and the shakedown load is already furthest below collapse. The check that misses the amplification is wrong by the most where the frame is most exposed to repetition.

Three collapse loads and a shakedown load #

There is a natural objection: a designer of a slender portal does not use the first-order collapse load either. EN 1993-1-1 reduces a plastic collapse load for second-order effects by Merchant–Rankine’s rule, $1/\lambda_f = 1/\lambda_p + 1/\lambda_{cr}$, and a frame designed to that is designed to a lower load than either check here.

The rigid–plastic collapse load ignores the roof. The collapse load with the wind amplified as the shakedown check amplifies it falls with the roof, and the shakedown load lies on it, as it should where gravity leads on pinned bases. Merchant–Rankine lies below both, at 0.623 against 0.692 for a 1,000 kN roof, because it reduces every mechanism by the frame’s $\alpha_{cr}$, the gravity-only ones that sway does not touch included.

So a frame designed to Merchant–Rankine’s collapse load is, on this frame and where gravity leads, also safe for shakedown with margin to spare — not because the rule knows about repetition, but because it is conservative for gravity-dominated mechanisms by more than repetition costs. Where the wind leads, the knees’ alternation is below every collapse load: at 120 kN of wind the amplified shakedown factor is 0.488, Merchant–Rankine’s 0.505. The margin is gone, and only an amplified shakedown check finds it.

The knees’ alternation by hand #

The pinned frame’s alternation can be checked in two lines. On pinned bases the wind’s moment at each knee is $Hh/2$, from statics alone, so over a storm that reverses it swings through $Hh$; the gravity at midspan, coming and going, adds its own knee moment, $V m_g$ — 150 kN·m for this frame’s 160 kN — to one end of the swing. A knee alternates when that range passes $2M_p$, at a load factor of

$$\lambda_{alt} = \frac{2M_p}{V m_g + Hh} = \frac{400}{150 + 300} = 0.889.$$

Amplified, the half of the storm with the gravity off carries the wind’s moment times $A_0$, the half with it on times $A_1$, so the wind’s term becomes $(A_0 + A_1)Hh/2$; and since both factors depend on the load factor itself, one or two rounds of substitution find it. With a 1,000 kN roof, $\lambda = 0.783$: $A_0 = 1.18$, $A_1 = 1.22$, and $400/(150 + 1.20 \times 300) = 0.783$.

The critical load is one more line: $N_{cr} = h/\delta_1$, with $\delta_1$ the frame’s sway under a unit horizontal load by the unit-load theorem, which one column is enough to find on a symmetric portal.

What the frame leaves out #

The amplification is applied to the elastic moments, which is Eurocode 3’s simplified second-order method and is what a shakedown check reads. The residual moments the frame builds up as it yields in its first cycles are not amplified, and they should be: a frame that has ratcheted some way towards a mechanism has swayed, and the roof’s weight acts through that sway too. That is a second-order effect on the plastic response, and it makes the true shakedown load lower than the amplified check says, by an amount that depends on how far the frame ratchets before it shakes down. A frame whose incremental mechanism includes sway — the combined mechanism, on pinned bases — is the one most affected.

The roof’s weight is carried at the knees, so it adds no first-order moment; a roof whose weight is spread along the beam adds gravity moments of its own, which come and go with nothing — they are always there — and move the whole domain. The frame is a single bay with no imperfection: an out-of-plumb is really a lean, a horizontal force in proportion to the roof’s weight, which acts in every storm in the same direction and turns a reversing wind into one that pushes harder one way. And the frame leans in its own plane only: a portal braced out of its plane but slender in it is the frame drawn.

A check on the moments the frame will have #

A shakedown check is only as good as the elastic moments it is given, and a slender frame under a heavy roof has larger elastic moments than first-order analysis reports — in every storm, both ways. The amplification falls entirely on the wind’s moments, so it moves most what the wind decides: the pinned frame’s knee alternation, which begins at a fifth less wind, and the frames in which the wind leads, whose shakedown loads are overstated by up to 14 per cent at the lowest $\alpha_{cr}$ Eurocode 3 allows. Where gravity leads, the cost is smaller, the pinned frame still shakes down at its collapse load, and Merchant–Rankine’s conservatism covers it.

Base stiffness is worth more under a heavy roof than first-order analysis credits, because it raises the critical load as well as taking moment off the knees — a 21 per cent gain becomes 32 at the heaviest roof drawn. It is the cheaper remedy for the frame whose wind leads, since it attacks the alternation at the knees directly and the amplification with it.

Still open: the frame that ratchets as it sways #

The amplification here falls on the elastic moments alone. A frame above its shakedown load accumulates rotation in a mechanism, and when the mechanism includes sway the frame leans a little further every cycle — and the roof’s weight, acting through that lean, adds a moment that pushes the mechanism on. Whether a frame ratcheting in its combined mechanism accelerates rather than accumulating at a steady rate per cycle; whether the shakedown load with the plastic sway amplified is a limit at all, or whether a frame below it can still be walked over by the roof’s weight once an imperfection has started the lean; and how much lower that leaves the load a slender portal can carry for ever, is what the second order asks of the residual field rather than the elastic one.