The pipe held at every floor

The spectrum a floor hands on found what a building does to the motion it passes on to whatever is bolted to its floors: on the roof of an eight-storey frame, a rigid item feels 1.79 times the ground spectrum and one tuned to the building feels six times. Every component in that essay stood on one floor. It named the harder case among the ones left over: distributed systems — pipes, ducts, cable trays — that are held at several floors, each of which is moving differently.

A pipe held at two floors is loaded twice. Its own mass, and the water in it, is shaken by the floors’ accelerations; that is the load the floor spectrum was built to price. And the two floors are not in the same place, because the storey between them has drifted, so the pipe is made to follow a relative displacement it has no choice about. The first is a force. The second is an imposed deformation — the same kind of load a settling support puts into a continuous beam — and for a pipe the second is nearly always the one that matters. Which one matters, and by how much, is decided less by the earthquake than by how the pipe is held at each floor.

The building and the pipe #

The building is the one the floor spectrum was drawn for: eight storeys, each floor 300 tonnes on a storey stiffness of 400 MN/m, a first period of 0.93 s, driven by the same seeded ground record with a peak acceleration of 3.5 m/s². Its storeys are 3.5 m high, and under that record the ground storey drifts 23 mm — 0.67 per cent of its height — with the drift falling to 0.16 per cent in the top storey. The floor accelerations rise from 3.5 m/s² at the ground to 9.1 at the roof.

The pipe is a water-filled schedule 40 steel riser running from the ground to the roof. At every step of the record its displacement at each floor is the floor’s; between floors it bends as a beam, loaded by its own mass times the floor accelerations. At each floor it is held one of two ways: anchored, clamped against rotation as well as displacement, which is what a riser welded to a floor sleeve or clamped rigidly in a short collar is; or guided, held in line but free to turn, which is what a riser in a loose sleeve or a guide that bears on it from the sides is.

Anchored at every floor #

Anchor the pipe at every floor and each storey’s length of pipe is a beam fixed at both ends, one end moved sideways by the storey’s drift $\Delta$. The moment at each end is $6EI\Delta/h^2$, and the bending stress at the outer fibre, a distance $D/2$ from the axis, is

$$\sigma = \frac{6EI\Delta}{h^2}\cdot\frac{D}{2I} = \frac{3E\Delta D}{h^2}.$$

The second moment of area has cancelled. The stress a drift puts into an anchored pipe depends on its diameter and not on its wall: a thicker pipe of the same diameter takes a larger moment and has exactly the section to carry it, so it is stressed identically. What changes the stress is the outside diameter, in proportion.

For the ground storey’s 23 mm the formula gives $3 \times 200{,}000 \times 23.5 \times 114.3 / 3500^2 = 131$ N/mm² in a DN100 riser, which is what the whole time history gives, and it passes the yield stress of ordinary pipe steel at DN200. A 600 mm main anchored at every floor of this building would be asked for 697 N/mm² — nearly three times its yield — by a drift of two thirds of a per cent, about a third of the storey drift a code typically allows.

The inertia stress runs the other way. The moment a uniform load puts into a fixed-ended span is $wh^2/12$, and the load is the pipe’s mass per metre times the floor’s acceleration. The steel’s share of that mass grows with $Dt$ and its section modulus with $D^2t$, so the steel’s inertia stress falls as $1/D$; the water’s mass grows with $D^2$ and the section modulus with $D^2 t$, so the water’s falls as $1/t$. Both fall with size. At DN100 the inertia stress is 3 or 4 N/mm²; at DN600, 1.

So the floor spectrum, applied to an anchored pipe, prices the smaller of its two loads at every standard size, by a factor of three at the smallest and seven hundred at the largest. The spectrum is right about the acceleration. The pipe’s problem is somewhere else.

When the two parts add #

A pipe that is small enough to make the inertia matter raises a second question: whether the drift and the inertia arrive together. The drift in a storey and the acceleration of the floors either side of it are both products of the same modal responses of the building, so they are not independent — but neither are they the same signal, because each mode contributes to them in different proportions.

Near the top of the building, in the DN25 riser, the inertia is the larger part — the top storey barely drifts and the roof accelerates hardest — and the two parts are nearly in step. Under the first mode, a floor’s acceleration points back toward the building’s rest position while its displacement points away, so the pipe’s inertia force, which is its mass times the acceleration with its sign reversed, points the same way the storey is being racked. At the anchorage at the foot of the storey the two moments have the same sign. They add, and the combined peak is the sum of the two separate peaks to within a few per cent, where the square root of the sum of squares — the rule that treats them as unrelated — is more than a quarter short.

Lower down, the drift and the inertia are less alike: the ground storey’s drift is almost entirely first mode, while the lower floors’ accelerations carry a large share of the higher modes, and at the ground anchorage the two are correlated at only 0.31. There the drift is 38 N/mm² and the inertia 3, and it hardly matters how they are combined.

Guided instead of anchored #

Now keep the pipe in line at every floor but let it turn. The storey’s drift no longer has to be bent into the pipe, because the pipe can simply lean: a straight line from one floor’s position to the next is a perfectly good shape for a pipe held only in line. What it cannot do without bending is change its lean from one storey to the next. A continuous pipe leaning at 0.67 per cent through the ground storey and 0.64 per cent through the next has to kink by the difference at the floor between them — a hundredth of what it had to bend through when it was clamped.

The same pipe, the same building, the same record: anchored, 131 N/mm²; guided, 3. Nothing about the earthquake or the building changed; only the question the supports ask of the pipe did. An anchorage asks the pipe to be vertical at every floor, so every storey’s drift must be taken out of it in bending. A guide asks it only to be in the right place, and a continuous line through a set of floor positions can be almost straight.

This is the correction the statement in the essay on drift needs — that everything spanning between two floors “is racked through the drift of the storey it sits in”. It is true of a cladding panel fixed at its four corners and of a partition built tight to the slab above. It is not true of a pipe that is free to turn at its supports, which is racked through almost nothing.

A riser that is guided everywhere still has to be held somewhere. It needs a support for its weight and, in most systems, a point where its thermal expansion is anchored. If that point clamps the pipe against rotation, it brings back the anchored behaviour locally: the base anchorage in the figure takes 77 N/mm², more than half of what anchoring every floor put into the worst floor, and the effect has almost gone by the second floor.

What a guided pipe feels #

The figure separates the two quantities. The drift is large near the base and small near the roof, which is the shape of a shear building’s first mode. The change in drift from one storey to the next is small everywhere, because the first mode’s drift changes slowly with height — and most of what change there is comes from the higher modes, whose drifts reverse from storey to storey. A guided riser is bent by the building’s curvature, the second difference of its floor displacements, rather than by its drift, the first difference.

That makes the guided riser’s bending a property of the mode shapes in a way the anchored one’s is not. In a building whose drift is uniform with height, a guided riser would be bent by nothing at all except its own inertia. In a building whose drift changes abruptly, it is bent where the change is.

A soft storey puts the kink into the riser #

Weaken the ground storey and the building concentrates its drift there — 1.22 per cent against 0.42 in the storey above. A guided riser does not notice the larger drift itself; it notices the abrupt change at floor 1, where its lean has to go from 1.22 per cent to 0.42 in the space of one support. Its stress there goes from 2 N/mm² to 40, twenty times as much, while every other floor is barely touched. A soft storey is a kink in the building’s deflected shape, and a guided riser is a gauge of kinks.

The same thing happens wherever a building’s stiffness changes abruptly with height: at a transfer level, at the top of a podium, where a stiff core stops. Those are the floors where a guided pipe’s supports need the most attention, and the places a uniform-building calculation says least about. A tall building adds a slow version of the same kink that no earthquake is needed for: a riser fixed to the core and branched into floors carried by columns crosses the difference between the core’s shortening and the columns’ at every level, and it grows for years.

Where one anchorage costs least #

A riser needs at least one anchorage, and the choice of floor is free. Move a single rotation-clamping anchorage up the building, guided everywhere else:

The stress at the anchorage follows the drift of the storeys either side of it, because the anchorage is the one point where the pipe is forced back to vertical against the lean of the pipe above and below. It is 77 N/mm² at the ground and 11 at the roof. And it is a little over half of what the worst floor took when every floor was anchored: a single anchorage forces the pipe back to vertical against a lean it can partly shed by turning at the next guide, where an anchored storey had to take its whole drift between two clamps.

The rule that falls out is the one pipe designers use for thermal expansion, arrived at from the other end. Put the anchorage where the movement is least, and let the pipe take the rest of the movement by turning and sliding at guides. For a building’s drift the least movement is at the top, where the drift is smallest — which is not where a riser’s weight would most naturally be carried, and which is why a riser’s weight support and its lateral anchorage are often better as two different fittings at two different floors.

The same calculation as a thermal expansion loop #

The formula $\sigma = 3E\Delta D/h^2$ is older than seismic design of pipes. It is the guided cantilever of piping flexibility analysis: a straight leg of length $L$ between two fittings that hold it in line and against rotation, one end pushed sideways by a movement $\Delta$ that the pipe cannot resist, carries exactly that stress. Piping engineers use it to size the offset legs and expansion loops that let a hot line grow — inverting it for the leg length $L = \sqrt{3E\Delta D/\sigma_{allow}}$ that brings a thermal movement down to an allowable stress — and the diameter-not-wall result has been a rule of thumb in that trade for decades.

An earthquake’s storey drift is, to the pipe, a thermal expansion that arrives in a second and reverses several times. The leg needed to absorb 23 mm at 100 N/mm² in a DN100 pipe is $\sqrt{3 \times 200{,}000 \times 23.5 \times 114.3/100} = 4.0$ m, longer than the storey, which is why a riser anchored at every floor cannot be made safe by flexibility within its own storey and is made safe instead by not anchoring it. The movement nobody applied, the thermal one, and the movement the ground applied are one problem for a pipe.

The pipe’s own period #

Everything above treats the pipe as following its floors quasi-statically, with its inertia simply its mass times the floor’s acceleration. That is right only if the pipe’s own span between supports is much stiffer than the building.

For a riser at a storey’s spacing it is, except for the smallest sizes. A DN100 span between guides 3.5 m apart is at 20 Hz and a DN300 at 52, above every mode an eight-storey building has, so the floor spectrum hands them essentially the floor’s peak acceleration and no resonance — the rigid end of a spectrum that is a property of the structure feeling it. A DN25 span at 6.3 Hz sits among the building’s higher modes, and on a horizontal run with supports 10 m apart it comes down past the first mode — which is where the tuned component, six times the ground spectrum, reappears. The floor spectrum is the right tool for a small pipe on long supports and the wrong one for a large pipe held at every floor, and the two are distinguished by the pipe’s own span frequency against the building’s.

The numbers at the ground storey, by hand #

The ground storey drifts 23.5 mm over 3.5 m. A DN100 riser has an outside diameter of 114.3 mm and a wall of 6.02 mm, so its second moment of area is $\tfrac{\pi}{64}(114.3^4 - 102.3^4) = 3.01 \times 10^6$ mm⁴.

Anchored at both floors, the end moment is $6 \times 200{,}000 \times 3.01\times10^6 \times 23.5/3500^2 = 6.9$ kN·m, and the stress $6.9 \times 10^6 \times 57.15 / 3.01\times10^6 = 131$ N/mm². The water-filled pipe weighs 24.3 kg/m; at floor 4’s peak acceleration of 7.1 m/s² it carries 172 N/m, a fixed-end moment of $172 \times 3.5^2/12 = 176$ N·m, and a stress of $0.176 \times 10^6 \times 57.15/3.01 \times 10^6 = 3.3$ N/mm² — which is what the anchored riser carries from its inertia at mid-height. Set beside the drift’s 131, it is forty times smaller, and that ratio is the whole argument in two lines.

A shear building, a straight pipe and a quasi-static span #

The building is a shear building — floors that translate without rotating — so its drift profile is a first-mode shape with higher modes added. A frame or a wall rotates its floors as well, and a riser anchored against rotation then has to follow the floor’s rotation too; a guided one does not care.

The pipe is straight, elastic and continuous from ground to roof. Real risers have joints, branches at every floor and valves whose mass is concentrated; a branch taken off at a floor ties the riser to the floor’s own movement in a second direction, and a grooved or flanged joint is a hinge that changes where the pipe can turn. The pipe is taken to stay elastic, so the 697 N/mm² at DN600 is a statement that the anchored arrangement is impossible, not a prediction of what the pipe would do.

The pipe follows its floors quasi-statically, which the span frequencies justify for everything but the smallest sizes on long spans. The supports themselves are rigid — a guide with clearance lets the pipe lag its floor, and a clamp that slips is an anchorage only until it slips.

What the pictures cannot show #

That a guide is not a clean mathematical pin. A pipe in a sleeve with a few millimetres of clearance moves freely until it touches one side, then is struck by it, and the figures here, which treat every guide as perfect, cannot show the impact. They also cannot show the pipe’s other directions: the record here shakes the building one way, and a real riser at a corner of a real building is racked in two directions at once and twisted by the building’s torsion.

Nor can they show where a pipe actually fails. The peak stresses here are in a plain pipe; the failures that are seen after earthquakes are at threaded joints, at the branch connections where a floor’s pipework joins the riser, and at couplings, each of which is weaker than the pipe and each of which is where the pipe’s forced shape concentrates.

Still open: the branch that ties the riser to a floor #

A riser rarely runs uninterrupted. At every floor a branch leaves it horizontally, runs across the floor and is held there — and the branch is a second pipe whose far end moves with its own floor, a few metres from where it joins a riser that is moving with the drift of the storeys above and below. Whether the branch connection, which is where pipework is seen to fail, is loaded by the riser’s kink or by the difference between the riser’s movement and the floor’s, and how long the branch has to be before it is flexible enough to be left alone, is the question the guided riser leaves at every floor it passes.