The elbow that takes the strut away

The branch that holds the riser to a floor followed a DN50 branch taken off a water-filled DN100 riser half a metre below the first floor of an eight-storey building, run 2 m horizontally to a seismic brace on that floor’s slab. The riser is guided at every floor and leans with each storey’s drift. Between the ground and the first floor it is a seventh of the storey’s drift behind the slab at the tee’s height, about 3 mm at the peak of a two-directional earthquake. The branch, braced so that it cannot slide along its length, does not stretch 3 mm. It holds, and holds the riser to the slab with it: the riser, carrying 2.8 N/mm² at that height alone, carries 71 at the tee.

There were two remedies. A brace that lets the branch slide along its length spares the riser, at the price of bending the branch by the riser’s lean. Or the branch can be given flexibility along its own axis the way pipework usually is — by turning a corner. That essay ended on the corner: a branch that leaves the riser, runs a short way and turns through a right angle before its brace carries the riser’s movement along its first leg into bending of its second. The questions were whether a short second leg is enough to stop the first being a strut, and whether the elbow, which piping rules treat as several times as flexible as straight pipe and several times as highly stressed, becomes the connection that fails instead of the tee.

A branch with a corner in it #

The branch is the earlier one with a corner put in. It leaves the riser along the building’s y axis, runs 2 m, turns through a 90° long-radius elbow — a bend of 1.5 times its nominal size, 76 mm — and runs along x to a brace on the same slab, which holds its end in every direction and lets it turn. The second leg’s length is the variable: from 100 mm, barely more than the elbow’s own size, to 3 m.

An elbow is not a piece of straight pipe bent round. Under bending, its circular cross-section flattens into an oval — the Kármán effect — so it is more flexible than a straight pipe of the same section, and the ovalling concentrates stress at its flanks. Piping practice carries both effects through two numbers computed from the elbow’s flexibility characteristic $h = tR/r^2$, its wall thickness times its bend radius over its mean radius squared: a flexibility factor $k = 1.65/h$, by which the elbow’s bending stiffness is divided, and a stress intensification factor $i = 0.9/h^{2/3}$, by which the nominal bending stress there is multiplied for a fatigue or displacement check. For this DN50 long-radius elbow $h = 0.37$, so $k = 4.4$ and $i = 1.73$. The welding tee has its own intensification by the same rules, 1.58.

The riser, branch and elbow are three-dimensional beams — the elbow as six short pieces with a bending stiffness divided by $k$ — and the two directions of the earthquake are applied together through the building’s response, as before.

The strut goes, and the stress moves #

The riser is spared almost at once, which is the support that moved undone: the tee stops being a support the riser did not have. With 100 mm of second leg it carries 19 N/mm² at the tee, a quarter of the straight branch’s 71; with 300 mm, 4; with any longer leg, 3, which is what it carries with no branch at all. A second leg of a few diameters is enough to stop the first being a strut.

The force the first leg carries along its length shows why. Straight, the branch’s axial stiffness was so large that it could not take up the 3 mm, and it pushed 15 kN into the riser. With an elbow, the 3 mm along the first leg becomes 3 mm across the second leg’s end, and a leg in bending is soft where a leg in tension is stiff: its stiffness falls as the cube of its length. With 300 mm of second leg the first leg carries 1.2 kN; with 1 m, 0.15. The riser no longer has a support half a metre below its guide.

But the storey’s lean is still the storey’s lean. With nothing holding the tee to the slab, the tee turns with the riser by the storey’s full lean, about 0.0064 radians, and the branch’s first leg is bent by that turn exactly as the sliding branch was. The tee, which carried 10 N/mm² nominal on the straight held branch, carries 52 nominal with the elbow — 82 intensified. And the elbow, bending in the plane of its own turn as the second leg takes the first leg’s movement, carries more than the tee while its leg is short: 146 N/mm² intensified with 100 mm of leg, 84 with 300 mm.

The elbow moves the stress from the riser to the branch’s fittings. It does what the sliding brace did, with no sliding brace, and for a short second leg it adds a new peak at the elbow on top of what the sliding brace already put into the tee.

Which fitting governs #

The two fittings cross. With a short second leg the elbow is bent hard, because the second leg is stiff in bending and the elbow is where its moment and the first leg’s meet; with a long one the second leg is soft and the elbow is lightly loaded, while the tee still carries the lean’s bending of the first leg. With this long-radius elbow they meet near 300 mm, both at about 82 N/mm², and from about 400 mm the tee carries more. At 1 m the tee is at 53 and the elbow at 39.

So the answer to the question the elbow was asked is that it depends on the leg past it. An elbow with a short leg after it is the fitting that fails; one with a long leg is not, and the tee is again. The crossover is where the second leg’s bending flexibility matches what the first leg’s turn at the tee asks of it.

The bend’s radius #

The elbow’s flexibility and its intensification both come from how tightly it is bent, and in opposite senses for the two fittings’ contest.

A short-radius elbow is more flexible, with $k = 6.7$, and more intensified, with $i = 2.3$: it carries 147 N/mm² with a 200 mm leg, above the tee’s 84, and stays above the tee until its leg is nearly a metre long. A bend of three diameters — a pipe bent rather than an elbow welded in — has $k = 2.2$ and $i = 1.09$, and carries 64 with a 200 mm leg and 25 with 1 m, below the tee at every length drawn. The tee hardly notices which corner is downstream of it.

The intensification matters more than the flexibility. A more flexible elbow lets the second leg take up the movement with less moment, which lowers its nominal stress; a more intensified one multiplies whatever nominal stress is left. For these proportions the multiplication wins, and the gentler the bend, the less the elbow carries. A piping layout that turns its branches with pulled bends rather than welded elbows is choosing the fitting that does not become the weak point.

The first leg’s length #

Once the elbow has taken away the strut, what loads both fittings is the riser’s lean at the tee, and the leg that takes it is the first one. Its stress falls as one over its length, as the sliding branch’s did: with 600 mm from tee to elbow the tee carries 115 N/mm² and the elbow 113; with 4 m, 45 and 37. So the length a designer gives the first leg relieves both fittings, and the length of the second leg decides only which of them carries more.

That is the movement nobody applied in a new place. A pipe between two points that move apart is bent by their relative movement, and flexibility is length; an expansion leg is sized to take a thermal movement in bending at a stress the pipe can live with. The riser’s lean at the tee is the same kind of imposed movement, a rotation rather than a translation, and the first leg is the expansion leg for it.

Which slab, again #

The earlier essay’s worst case was a branch at ceiling height braced to the floor below, where the 3 mm became 22 mm and both pipes went far past yield.

An elbow rescues the riser even there: braced to the farther slab, the riser falls from 484 N/mm² to 21. It does not rescue the branch. The 22 mm the branch must take up between the tee and the brace now goes into the second leg’s bending, and 500 mm of leg cannot take 22 mm: the elbow carries 347 N/mm² intensified, well past the pipe’s 241 N/mm² yield. An elbow takes away the strut; it does not take away the movement, and the brace’s slab still decides how much movement there is. The rule from the straight branch stands: brace to the slab nearer the tee.

Up the building #

Moved up the building, every stress follows the storey’s drift, as the straight branch’s did: the elbow is worst in the ground storey, 59 N/mm², and falls to 10 in the top storey. The riser is between 1 and 7 at every storey, where the straight held branch put up to 78 into it. As before, the floor accelerations, which rise up the building, say nothing about any of this. Two motions with one name separated a storey’s drift from its floors’ accelerations, and a branch connection is governed entirely by the first.

Two directions, now both soft #

The straight branch was loaded by the two directions of shaking through two different mechanisms: drift across it bent it in plan and twisted it, which it took easily, and drift along it tried to stretch it, which it refused, making it a strut. The elbow changes the second and leaves the first. Shaken only along the first leg, the riser carries 2.7 N/mm² at the tee instead of 71, and the branch’s fittings carry the lean: the tee 66 intensified and the elbow 52. Shaken only across the first leg, nothing much changes from the straight branch — the riser carries 2.6, the tee 28 and the elbow 20 — because the first leg already took that direction in bending, and the second leg, now lying along the shaking, adds only its small axial give in series with the first leg’s bending.

Together they give the tee 74 and the elbow 59, close to the larger direction alone, for the same reason as before: the two ground components are independent, the building responds to them separately, and their peaks seldom coincide. The square-root rule would say 72 for the tee, near enough here because one direction dominates. Where the two directions load one fitting through the same moment — they do not here — the rule that combines modes by the square root of the sum of squares overstates the combination in exactly the way the straight branch’s peaks showed.

The elbow therefore does more than end the strut. It makes the branch soft in both horizontal directions, which is what a seismic branch connection needs: a building’s drift, unlike a spectrum read as a load, arrives in whichever direction the ground happens to send it, and a connection stiff in one direction will find that direction.

The elbow, by hand #

The elbow’s characteristic: a DN50 schedule-40 pipe has an outside diameter of 60.3 mm and a wall of 3.91 mm, so its mean radius is 28.2 mm. Bent to 76 mm, $h = 3.91 \times 76/28.2^2 = 0.374$. Then $k = 1.65/0.374 = 4.4$ and $i = 0.9/0.374^{2/3} = 0.9/0.519 = 1.73$.

The first leg’s bending by the riser’s lean: the tee turns by $\theta = 0.0064$ and the leg, 2 m to the elbow, is held there against turning by the second leg’s stiffness, roughly a beam turned at one end and pinned at the other. Its stress is $3E\theta D/2L = 3 \times 200\,000 \times 0.0064 \times 60/(2 \times 2\,000) = 58$ N/mm² nominal, against 47 from the history with 500 mm of second leg; intensified by the tee’s 1.58, 74.

The second leg’s stiffness against a movement across its end: a cantilever of length $\ell$ fixed at the elbow’s end, $3EI/\ell^3$. For 300 mm of DN50 that is about 6 kN/mm; for 1 m, 0.17 kN/mm. The straight branch’s axial stiffness was 69 kN/mm. The elbow replaces the larger number with the smaller, which is the whole of what it does for the riser.

Beam pipes and code factors #

The elbow’s factors are the piping rules’. The flexibility factor and the stress intensification factor are empirical and conservative, calibrated on fatigue tests of fittings under reversed bending. They are applied here as the rules apply them, to bending in and out of the elbow’s plane alike, which overstates the out-of-plane effect a little.

The tee is a welding tee. A threaded or grooved fitting has a much larger intensification and its own flexibility, and its joint, not its body, is usually what fails. A threaded tee would govern at every second-leg length drawn.

The brace is rigid and has no play, as before. A real brace’s play, a few millimetres, is a sliding brace for the first few millimetres of movement.

Only drift is imposed. The building drawn is the one the floor spectrum was computed for, and its drift is taken as imposed on the pipes. A branch between a riser and a column-mounted brace in a tall building would also see the columns shortening against the core, a slow movement an elbow takes up in exactly the same way.

The stresses are elastic and nominal. Past yield — the far-slab elbow’s 347 — the numbers say the arrangement is impossible, not what it would carry, and an elbow that yields first concentrates its rotation at the flanks the intensification factor is pointing at.

What the corner buys #

The straight branch traded one pipe for the other: held, it kinked the riser; sliding, it bent itself. The elbow is the sliding brace built into the pipe. It has the sliding brace’s virtue — the riser is spared — and the sliding brace’s cost, the first leg bent by the lean. It adds a fitting with its own stress intensification. That fitting governs for a short second leg and disappears from the reckoning for a long one or a gentle bend.

The useful rule is a layout rule. Give the branch an elbow, give the leg past it at least half a metre, prefer a bend to an elbow, and make the first leg as long as the room allows, because that leg is the one that takes the storey’s lean. One support too many was the straight branch’s fault; the elbow removes the support, and length removes the stress.

Still open: the branch that runs to another riser #

Every branch here ends at a brace on a slab. Many branches end at another pipe — a cross-connection between two risers, a loop between a riser and a header — and the two ends of such a branch are moved by two risers that lean together in the storey’s drift but sit at different points of the plan. In a building that twists as well as sways, two risers at opposite corners lean by different amounts, and the branch between them is stretched, compressed and bent by the difference. Whether an elbow in a cross-connection does for a twisting building what it does for a swaying one — or whether the twist, which moves the two ends along the branch’s length in opposite senses, needs an elbow in each direction — is the question a branch between two moving things puts to the corner that freed it from one.