The branch that holds the riser to a floor

The pipe held at every floor found that how a riser is held decides what an earthquake does to it far more than the earthquake does. Anchored at every floor, a DN100 riser in an eight-storey building is bent by each storey’s drift and carries 131 N/mm². Guided — held in line, free to turn — it simply leans with each storey and carries 3, bent only where the drift changes from one storey to the next. That essay stopped at the place where risers are actually seen to fail. A riser rarely runs uninterrupted: at every floor a branch leaves it horizontally, runs across the ceiling or the floor and is held there, and the branch’s far end moves with whatever it is held to.

So a branch is a second pipe tied to the riser at one end and to a floor at the other, and the two ends are not moving together. The riser at the tee is wherever its lean puts it — partway between the floors that guide it. The branch’s brace is wherever its slab is. Between them the branch is asked to take up a difference in position and a difference in angle, and it is not at all obvious which of the two pipes that difference ends up in.

A riser that leans, and a branch that does not #

The building is the one the floor spectrum and the riser were drawn for: eight storeys of 3.5 m, each floor 300 tonnes, a first period of 0.93 s. This time it is shaken in two directions at once, by two independent components of the same seeded ground record with the same peak, 3.5 m/s². Its ground storey drifts 23 mm in one direction and 22 in the other.

The riser is a water-filled DN100 pipe guided at every floor. The branch is a DN50 pipe taken off it at a tee 3.0 m above the ground floor, half a metre below the slab of floor 1 — a branch at ceiling level, which is where most branches run — and taken 2 m horizontally to a seismic brace on that slab. The brace holds the branch’s end in line with the slab in every direction and lets it turn. Both pipes are modelled as three-dimensional beams, continuous over the storeys either side, and the branch’s and the riser’s moments at the tee are traced through the whole record.

The difference the branch has to take up is easiest to see in the storey’s elevation. Between the ground and floor 1 the guided riser is close to a straight line from one floor’s position to the next: it leans by the storey’s drift ratio, about two thirds of a per cent at the peak. At a point half a metre below floor 1 it is therefore a seventh of the storey’s drift behind floor 1 — about 3 mm — and it is turned by the full lean, about 0.0064 radians. The brace is on floor 1. So the branch, if it simply followed its two ends, would have to accommodate 3 mm of movement between them and a turn of 0.0064 radians at one end.

Two directions, two different loads #

Which of those the branch accommodates depends on the direction of the drift, because a pipe is very differently stiff along its axis and across it.

Drift across the branch moves the tee sideways relative to the brace, which the branch takes as bending in plan, like the guided cantilever of piping flexibility analysis. And it turns the tee about the branch’s own axis, which the branch takes as a twist. Both are soft actions for a 2 m length of 60 mm pipe, and the branch absorbs them with modest stress. Drift along the branch is different. It turns the tee about a horizontal axis across the branch, which bends the branch in its vertical plane. And it tries to move the tee along the branch’s length relative to the brace — 3 mm of shortening or stretching in a pipe whose axial stiffness is enormous.

A DN50 pipe 2 m long has an axial stiffness of about 69 kN/mm. Three millimetres of it would take about 210 kN, far more than the riser it is attached to can resist. So the branch does not stretch; it holds. The 3 mm is taken out of the riser instead.

The strut #

Braced so that it cannot slide along its length, the branch is a strut. It ties the riser at the tee to the slab of floor 1, half a metre below that floor’s guide, and in the branch’s direction the riser now has a support it did not have: a point that moves with floor 1 at a height where the riser wanted to be a seventh of a storey behind it. The riser is a continuous beam whose supports used to be the floors; now it has one support too many, and it is half a metre from the next, and that short span is forced to carry the whole difference between the lean of the storey below and the near-zero movement between the tee and the floor above. The riser, which carried 2.8 N/mm² at that height when it was alone, carries 71 at the tee. Almost all of that comes from the shaking along the branch.

The branch itself is comfortable. Its far end and its near end both move with floor 1 in its own direction, so it is not bent by the tee’s movement along it; it carries the riser’s turn at the tee, partly, and the sideways movement across it. At the tee it carries 9.9 N/mm², and the force it passes along its length as a strut peaks at 15 kN, which the brace must also carry.

This is the case the earlier essay’s rule of thumb — guide a riser and it is barely bent — does not cover, and the reason is the one the support that moved gave for a continuous beam: a support added between two others attracts the curvature that the supports either side impose. A branch braced to a slab is a support, added at whatever height the branch happens to leave the riser.

Which slab to brace it to #

The 3 mm a branch at ceiling height forces into the riser is the storey’s drift times the fraction of the storey between the tee and the brace’s slab. A branch at ceiling level braced to the ceiling slab has half a metre of that; the same branch braced to the floor below — by a stand off the floor slab, say — has three metres of it. The difference is not a matter of degree. Braced to the slab below, the tee at ceiling height leaves the riser at 480 N/mm² and the branch at 340, and the strut force rises to 98 kN: both pipes are past yield, by a drift that is a third of what a code allows.

The figure’s two families of curves are mirror images. A branch near the floor wants to be braced to the floor; a branch near the ceiling, to the ceiling; a branch at mid-height is badly placed either way, with the riser at 140 N/mm² whichever slab it is braced to. The rule is the one the riser’s own anchorage obeyed: brace where the movement relative to the pipe is least, and for a branch leaving a guided riser that is the slab nearer the tee.

Longer helps the branch, not the riser #

The question the riser essay left — how long a branch has to be before it is flexible enough to be left alone — turns out to have two answers, one for each pipe. For the branch, length is flexibility, and it works quickly: the sideways movement across a branch is taken in bending that falls as the square of the branch’s length, and the turn at the tee in bending that falls as its length. A held branch carries 70 N/mm² at 0.5 m, 28 at 1 m and 10 at 2 m.

For the riser, length does nothing. What loads the riser is the branch’s stiffness along its own axis, and a pipe’s axial stiffness falls only as one over its length, from a value so far above the riser’s bending stiffness that halving it changes nothing. A 6 m branch is still a strut. Every length of branch, held along its length, kinks the riser by about the same amount.

So the branch connection, where pipework is seen to fail, fails on the riser’s side of the tee when the branch is held. A rule that looks after the branch by making it long enough to be flexible, as the movement nobody applied would size a thermal expansion leg, protects the branch and leaves the riser exactly where it was.

Letting it slide #

The remedy for the strut is to stop the branch being one. A brace that holds the branch’s end sideways and vertically but lets it slide along its own length — a guide, in piping terms, rather than a fixed point — leaves the riser’s lean untouched in the branch’s direction. The riser goes back to 4.3 N/mm² at the tee, close to its 2.8 alone.

But the riser’s lean does not vanish; it moves. With the branch free to slide, the tee turns with the riser by the full 0.0064 radians, and the branch, held at its far end against turning only by its own bending, is bent in its vertical plane by that turn. The moment at the tee is three times the branch’s bending stiffness times the turn over its length, and its stress is

$$\sigma = \frac{3EI\theta}{L}\cdot\frac{D}{2I} = \frac{3E\theta D}{2L},$$

which for θ = 0.0064, D = 60 mm and L = 2 m is 58 N/mm²; the whole time history gives 54. The diameter-not-wall result returns, now for an imposed rotation instead of an imposed displacement, and now falling as one over the branch’s length rather than one over its square. A branch that slides needs to be longer than a branch that is held, by a good deal: to bring this one below 30 N/mm² it needs about 4 m.

That is the trade, and it does not have a free side. The storey’s lean has to be taken out of one pipe or the other. Held, the branch protects itself and kinks the riser through a span half a metre long. Sliding, the riser is spared and the branch is bent through its whole length. Of the two, the sliding branch is the one a designer can fix with length and the held branch the one no length fixes, which is an argument for the sliding brace and a long, flexible branch — the same arrangement thermal design reaches by a different route.

Where the building drifts most #

Moved up the building, the arrangement follows the drift, because the drift is the riser’s lean and the lean is what the brace takes out of it. The riser’s stress at the tee is near 70 N/mm² in each of the lower three storeys, worst in the second, and falls to 14 in the top storey, which drifts least — mostly in the first mode, with the higher modes adding little to a drift near the base. The branch follows at an eighth of the riser’s values.

This is the reverse of what the floor spectrum would say, read as a load rather than a property of what feels it. The floor accelerations rise up the building, from 3.5 m/s² at the ground to 9.1 at the roof, and a branch designed for its own inertia would be designed hardest at the top. Its connection to the riser is loaded hardest at the bottom, by a displacement the floor spectrum does not describe. Two motions with one name separated a storey’s drift from its floors’ accelerations; a branch connection is a place where the first decides and the second barely registers.

Two directions of shaking #

The two directions of shaking load the branch through different moments: the shaking across it through bending in plan and a twist, the shaking along it through bending in its vertical plane. They add as vectors, and their peaks seldom coincide, because the two ground components are independent and the two directions of the building respond to them separately. So the combined peak, 9.9 N/mm², is close to the larger of the two alone. The rules of thumb that combine two directions without a time history — the square root of the sum of squares, which combines modes for the same reason, or the larger plus 30 per cent of the other — say 13.4 and 12.4, a third too high. They are made to be safe for a quantity both directions load in the same way, and here they are not.

For the riser the question hardly arises. Its stress at the tee comes almost entirely from the shaking along the branch, through the strut, and the shaking across the branch contributes a few N/mm². That is the most useful thing the two directions show. The floor spectrum in the branch’s own direction decides the riser; the one across it decides the branch. In a building that is stiffer in one direction than the other — shear walls one way and frames the other, the common case — the two pipes at one tee are governed by two different spectra. Stiffen the building in the branch’s direction by two and a half times, so that its period in that direction falls to 0.59 s and the ground storey drifts 10 mm instead of 22, and the riser’s stress at the tee falls from 71 N/mm² to 32 while the branch’s hardly moves.

The sliding branch and the strut, by hand #

The sliding branch’s moment is a beam turned at one end and pinned at the other: 3EIθ/L, with θ = 22.4/3,500 = 0.0064, giving 58 N/mm² for a DN50 pipe 2 m long against the history’s 54. The difference is the riser: the tee is not quite rigid, and the riser turns a little less at the tee than the storey’s average lean.

The held branch’s sideways bending can be bracketed the same way. Across the branch, the tee is a seventh of the storey’s drift behind the brace, 3.3 mm; a branch fixed against turning at the tee and pinned at the brace takes that at a stress of 1.5EδD/L², about 15 N/mm²; the history gives 9.6, because the riser gives a little at the tee and the peak of the drift across and the peak of the turn do not coincide.

And the strut’s stiffness: a DN50 pipe has a steel area of about 690 mm², so over 2 m its axial stiffness is 200,000 × 690 / 2,000, about 69 kN/mm per pipe — enough that the 3 mm difference would need over 200 kN to stretch it, against a riser that the 15 kN the branch actually carries is already bending to 71 N/mm².

A junction, a rigid brace and an elastic pipe #

The tee is a beam junction. A real tee is a fitting with its own flexibility and a stress intensification factor that piping codes apply to it — a little under two for a butt-welded tee of these proportions by ASME B31.3’s formula, and more for a fabricated or threaded one — and the stresses here are nominal stresses in the pipes either side of it. At the fitting itself the figures should be read as multiplied by that factor, which makes the strut’s 71 N/mm² in the riser something like 130 or more.

The brace is rigid. A real seismic brace has a stiffness, often not much more than the pipe’s, and some free play in its fittings; both soften the strut, and the play lets the branch take the first few millimetres before it bears. A brace with 3 mm of play would make this branch behave as though it could slide, until it could not.

The pipes are elastic and quasi-static. The building’s drift is taken as imposed on the pipes, which follow it without dynamics of their own; the riser essay showed that DN50 and DN100 spans of this length are far above the building’s frequencies. Nothing slower is imposed: the shortening of columns against a core, which a branch running between the two would meet as a strut in exactly the same way, is left out. And the stresses past yield mean that arrangement is impossible, not that the pipes would carry those numbers.

The fitting that actually breaks #

They cannot show the fitting that actually breaks: the threaded or grooved joint at the tee, whose strength and flexibility are not those of the pipe. They cannot show a branch with an elbow, which is the usual way a designer gives a branch flexibility along its own axis without a sliding brace — a branch that turns a corner is a strut no longer, and its two legs bend instead. And they cannot show what the water inside does when a branch connection does break, which is the reason the connection is worth this much attention.

Still open: the branch with an elbow #

A branch that leaves a riser, runs a short way and turns through a right angle before it reaches its brace carries the riser’s movement along its first leg into bending of its second. Whether a short second leg is enough to stop the first being a strut — and whether the elbow, which piping codes treat as several times as flexible as straight pipe and several times as highly stressed, then becomes the connection that fails instead of the tee — is the question an elbow puts to the trade between kinking the riser and bending the branch.