The base that sends the wind to the knees

The beam the wind walks without pushing took a portal frame 10 m wide and 5 m high, fixed at both bases, with a plastic moment of 200 kNm everywhere, and loaded it with 160 kN at midspan that comes and goes and 60 kN of wind that blows from either side. Applied once, those loads are exactly the beam’s collapse load. Applied repeatedly, they are 12 per cent too much: the frame shakes down at a factor of 0.88, and above it the midspan gains plastic rotation every cycle without the frame ever swaying. The beam ratchets because each knee takes its worst hogging moment from the wind in a different direction, and the beam’s mechanism collects both.

That frame had fixed bases, and most portals do not. A great many stand on bases detailed as pins — a plate with two bolts near the column’s centreline, intended to carry shear and axial load and as little moment as possible. Many more stand on bases that are meant to be fixed but are a plate and four bolts on a concrete pad, with a stiffness that the bolts decide and a strength that may be less than the column’s. The question is what the bases do to the shakedown load: whether a pinned base, which cannot alternate because it carries no moment, makes the frame better or worse under repetition; whether a base of real stiffness sits between the two; and whether a base weaker than its column is a weak link or something more useful.

Where the wind’s moment goes #

On fixed bases the wind’s overturning moment is shared between four places. The columns bend in double curvature, with a point of contraflexure part-way up, and both the bases and the knees carry wind moment. On pinned bases the bases can carry none, so the columns bend in single curvature from a zero at the foot to a maximum at the knee, and the whole of the wind’s moment arrives at the two knees.

The difference is drawn section by section. Under the four extreme states — gravity with wind from the left, gravity with wind from the right, and wind from either side alone — each of the five critical sections sweeps a range of elastic moment. On fixed bases each base sweeps from −94 to +174 kNm and each knee from −216 to +56; the midspan reaches 240. On pinned bases the bases carry nothing, the midspan reaches 250, and each knee sweeps from −300 to +150: a range of 450 kNm at a section whose plastic moment is 200.

That number matters for a reason the map with three regions established for a different structure. A section whose elastic moment range exceeds twice its plastic moment cannot be made to stay elastic by any residual moment, because a residual shifts the range up or down without narrowing it. Such a section yields in one direction on one half of each cycle and in the other on the other half, for ever. Fixing the bases kept every range well inside 400 kNm. Pinning them pushes the knees’ range past it, at the full loads. So the alternation the fixed frame had at its bases has not been abolished by pinning them; it has been moved to the knees, and made larger on the way.

One line where there were two #

The shakedown domain is the set of load pairs the frame can carry for ever, drawn in the plane of the gravity load’s peak and the wind’s, each made dimensionless by the plastic moment: $v = VL/M_p$ and $\eta = Hh/M_p$. On fixed bases, the bound theorems give the familiar collapse boundary — beam mechanism at $v = 8$, sway at $\eta = 4$, combined at $v/2 + \eta = 6$ — and shakedown sits inside it, bounded by the beam’s ratchet when gravity leads and by alternation at a base when wind does.

On pinned bases the collapse boundary is smaller everywhere: the beam mechanism is unchanged at $v = 8$, but the sway mechanism needs only the two knees, $\eta = 2$, and the combined mechanism only the midspan and the leeward knee, $v/2 + \eta = 4$. The drawn loads, at $v = 8$ and $\eta = 1.5$, collapse at a factor of $4/5.5 = 0.73$ through the combined mechanism. And the shakedown boundary is not inside the collapse boundary at all. Wherever gravity leads, the two are one line. The pinned frame shakes down at 0.73, exactly its collapse factor.

The reason is a small identity, and it is worth seeing because it explains the fixed frame as well. On pinned bases the combined mechanism hinges at the midspan, rotating $2\theta$, and at the leeward knee, also rotating $2\theta$. The beam mechanism hinges at the midspan through $2\theta$ and at each knee through $\theta$. Under a wind that reverses, Koiter’s theorem credits each hinge of the beam mechanism with the worst moment it ever sees: the midspan its sagging peak under full gravity, and each knee its hogging peak under the wind blowing towards it. By symmetry, each knee’s worst hogging is exactly the leeward knee’s hogging under one storm. So the beam mechanism credited from both winds does the same work as the combined mechanism under one storm — the two leeward knees of two storms are, to the arithmetic, one knee rotating twice as far. The repetition penalty that took 12 per cent off the fixed frame is already inside the pinned frame’s collapse load.

On fixed bases that identity fails, because the fixed frame’s combined mechanism also hinges at both bases, and the bases dissipate energy the beam mechanism does not need. The combined mechanism costs $6M_p$ and the beam mechanism $4M_p$, so collapse is set by the beam alone and the beam’s credited moments — one knee from each wind — are new information that collapse never asked for. The penalty is the gap between those two numbers, and pinning the bases closes it by removing the bases from the combined mechanism, not by improving anything.

Better under repetition, worse in every other way #

It is tempting to read the coincident line as an advantage of the pinned frame, and it is one in a narrow sense: a designer of a pinned portal who checks its collapse load has, for gravity-led loading, checked its shakedown load too. But the pinned frame’s shakedown load is 0.73 against the fixed frame’s 0.88. The fixed frame, even after paying its penalty, is 20 per cent stronger under repeated loading than the pinned frame is under loading applied once.

Drawn against the wind, with the gravity load held at 160 kN, the two frames tell the whole story. With no wind both collapse and shake down at 1.00 — the beam mechanism, identical on both. As the wind grows, the fixed frame’s collapse stays at 1.00 until about 80 kN, because the beam mechanism does no work against wind, while its shakedown falls from the first newton of wind onwards. The pinned frame’s collapse and shakedown fall together, faster than either fixed line, and stay together up to about 102 kN of wind. There the knees’ elastic range reaches twice their plastic moment at the shakedown factor, a knee begins to alternate, and the pinned frame’s shakedown separates from its collapse and falls below it.

So the alternation the pinned frame was supposed to be free of arrives, and it arrives at the knee. On a fixed-base frame alternating plasticity at a base is a column bending back and forth just above a plate on a pad: visible, reachable, in a section designed for nothing but the moment it carries. At a knee it happens in a haunched, bolted or welded connection that carries the beam’s shear and the column’s axial force at the same time, at the top of the column, under the roof. A pinned base does not remove the place where the frame dissipates a reversing wind. It chooses a worse one.

A base of real stiffness #

No base is truly pinned and few are truly fixed. Every base is a rotational spring — neither pinned nor rigid, as every real connection is — and the frame’s elastic moments depend on how stiff that spring is compared with the column above it.

The shakedown factor rises smoothly with the base’s rotational stiffness, from the pinned 0.73 to the fixed 0.88. A base credited with a tenth of the column’s own stiffness $4EI/h$ — the allowance British practice has long made for a nominally pinned base — is 3,360 kNm per radian and shakes down at 0.77, a third of the way from pinned to fixed. A base at $30EI/h$, the stiffness EN 1993-1-8 requires before a base in a sway frame may be called rigid, is 252,000 kNm per radian and shakes down at 0.87. The base worked through in the essay on what the bolts decide, a 20 mm plate on two rows of anchor bolts at 9.1 times the column’s $EI/L$, would put this frame at 0.86: well short of the classification’s rigid threshold, and within 2 per cent of a rigid base’s shakedown load.

The collapse factor does not move at all. It is 1.00 for every base that has any stiffness and the column’s full plastic moment, and drops to 0.73 only at a true pin. That is the rigid-plastic theorems working as intended: collapse depends on the strengths of the hinges and the geometry of the mechanism, and a base that is soft but strong will eventually develop its plastic moment, given enough rotation. The joint that has to keep turning made the same point about beam-to-column joints and named the price — the rotation a soft joint must supply before it reaches its strength. Shakedown is less forgiving, because shakedown is decided by the elastic moments under the extreme states, and those depend on the stiffness at every section. A soft base sends wind moment to the knees whatever its strength.

So the base’s stiffness, which collapse never sees and which serviceability sees only as drift, is what decides how much of a frame’s collapse load survives repetition. That is a reason to care about a base’s stiffness beyond the deflection check, and a reason not to call a base fixed for strength and pinned for stiffness in the same analysis.

A base weaker than its column #

A base plate on four bolts is not only less stiff than a fixed base. It is often less strong than the column it carries: the bolts yield in tension, or the plate bends between them, at a moment below the column’s plastic moment. The usual reading is that such a base is a weak link. For shakedown that reading is wrong over most of the range that matters.

Take the fixed-base frame and lower the moment its bases can carry, keeping their stiffness. Down to 0.59 of the column’s plastic moment — about 117 kNm against 200 — the shakedown factor does not move from 0.88. The limit belongs to the beam’s ratchet, which hinges at the knees and the midspan and needs nothing from the bases, and the bases’ own elastic range under the extreme states stays inside twice their capacity. Below 0.59 the bases’ range catches up, a base alternates before the beam ratchets, and the shakedown factor falls in proportion to the base’s strength: 0.75 at a half.

The collapse factor tells a different story, because the fixed frame’s combined mechanism uses the bases. Below about 0.76 of the column’s moment, collapse starts to fall, and between 0.59 and 0.76 the frame’s collapse load is lower than a full-strength base would give while its shakedown load is untouched. That is the interval in which a designer checking only collapse would condemn a base plate that, under the loading it will actually see, costs the frame nothing.

The base as a fuse #

Below the threshold the base governs, and the question becomes what the frame does when it is loaded beyond its new limit. The fixed-base frame loaded 5 per cent above its shakedown load ratchets: its midspan gains about 25 milliradians of plastic rotation every cycle, the beam sags further with every storm, and nothing stops it. A frame whose bases are the limit fails differently.

With bases that yield at 0.40 of the column’s plastic moment, the shakedown factor is 0.60. Loaded to 1.05 times that, the bases yield, one way under wind from the left and the other under wind from the right. In the first two or three cycles each base finds its place; after that each swings through about half a milliradian every cycle, the permanent sway settles at three quarters of a millimetre and stops, and the beam and the knees never yield at all. Nothing walks. The damage each storm does is concentrated in one detail at the foot of each column, and it is the damage of low-cycle fatigue — the same small loop of yield traced over and over — rather than the accumulating deformation of a ratchet.

That is the trade the range of base strengths offers. A base at full strength leaves the frame’s repeated-load limit with the beam, at the highest factor, with ratcheting beyond it: a failure that grows a deflection everyone can see and no repair can undo short of replacing the beam. A base below 0.59 lowers the limit and moves it to the base, with alternation beyond it: a failure that grows nothing but fatigue damage, in a part that can be inspected from the floor and replaced. Seismic design has long chosen its fuses this way — a part chosen to yield so that the rest of the structure does not — and a portal’s base plate under wind can be chosen in the same spirit.

It has a cost the cycle simulation does not show. A base whose bolts yield in tension loses preload with every excursion, and the base that lets go and closes again found how quickly: a preloaded base that recovered its whole preload after every reversal up to its bolts’ yield kept only 83 per cent of it after the first cycle a little beyond, 62 per cent a little further. A fuse in the bolts is a fuse that loosens, and its stiffness falls as it does, which moves the knees’ share of the wind moment up. A plate that yields in bending, with the bolts kept elastic, is the better fuse.

What the bases decide #

Put together, the base decides three separate things, and only one of them is visible in a collapse check.

It decides the collapse load, through its strength alone: a pin removes two hinges from the sway and combined mechanisms, and a weak base reduces their dissipation, while stiffness does not enter.

It decides the shakedown load, through its stiffness and its strength together. Stiffness sets the elastic moments that each section sees under the extreme states, so it sets how much moment the reversing wind delivers to the knees that the beam’s ratchet credits. Strength sets whether the base’s own range can be absorbed by a residual moment. A base can be stiff and weak, or soft and strong, and the two have different shakedown loads for different reasons.

And it decides where the frame’s repeated yielding happens once the limit is passed: in the beam, accumulating; at the knees, alternating at the worst place in the frame; or at the bases, alternating at the most accessible. That third choice is the one a designer can actually make, by choosing the base’s strength, and it is the one a check of collapse alone cannot see at all.

What the frame here leaves out #

The frame is drawn to first order: the columns’ axial loads do not amplify the sway. A pinned-base portal is much more flexible in sway than a fixed one — here its knees take all the wind moment, and its lateral stiffness is about a fifth of the fixed frame’s for these sections, 1,010 kN per metre against 4,610 — so its sway moments are amplified by the gravity load more. That amplification acts on the knees, which on a pinned frame are already the alternating sections, and so it would bring the pinned frame’s knee alternation forward, at lower wind than the 102 kN found here. Nothing in the comparison of collapse and shakedown would reverse; the margin would shrink.

The loads are a point at midspan and a point at the knee. A distributed roof load, and wind on the walls as well as the roof, shift the elastic moments without changing which mechanisms exist. The plastic moment is the same throughout, so the knee hinges equally in the beam or the column; a haunched knee, much stronger than either member, moves the knee hinge into the rafter beyond the haunch, and makes the pinned frame’s knee alternation a rafter’s problem instead of a connection’s.

And every section is elastic–perfectly plastic, with an unlimited capacity to rotate. A base that alternates for a thousand storms is assumed to keep yielding at the same moment for the thousandth as for the first, which a bolt losing its preload does not.

Still open: the wind that brings its own amplification #

Every frame here is analysed as if its sway did not change its moments. A slender portal on pinned bases under a heavy roof is exactly the frame where it does: the gravity load acting through the sway adds a moment at each knee proportional to the sway, so the wind’s share of the knee moment grows by $1/(1 - 1/\alpha_{cr})$, and on a repeated wind that amplification acts on both halves of the cycle. Whether second-order sway moves the pinned frame’s knee alternation down to wind loads it would otherwise never reach, whether the base’s stiffness — which raises the elastic critical load as well as lowering the knee moments — is then worth twice what this frame credits it with, and whether a shakedown check without the amplification is still a check on the safe side, is what a frame with a heavy roof asks of this one.