Settlement: Uniform vs Differential

The tank that settled 180 mm and was fine — and the pump that settled 12 mm and was not

Two settlement readings from the same tank farm, taken the same week.

A 40 m storage tank has come down 180 mm during hydrotest, evenly, all round the shell. The tank is accepted, the piping is connected afterwards, and nobody loses a night's sleep.

Three hundred metres away a transfer pump on a small pad has settled 12 mm while the rack it is piped to, founded on piles, has not moved at all. The pump is now out of alignment, its suction nozzle is complaining, and the seal is on its second replacement.

Guess first: which one is the settlement problem?

The 12 mm one. Fifteen times less movement, and it is the one that costs money.

The misconception, stated plainly: "Settlement is bad. Less settlement is better. Get the total settlement down and you are safe."

Half of that sentence is true and the half that is false is the expensive half. What damages structures, cracks foundations and destroys nozzles is not settlement — it is differential settlement: the difference between two points that are connected to each other. A building, a rack or a pipe run that moves down uniformly has had no strain imposed on it at all. It is simply lower than it used to be.

Why uniform settlement is (mostly) harmless

Take a pipe run on two supports. Drop both supports by the same 40 mm. What has happened to the pipe?

Nothing. Every point moved 40 mm down, so no point moved relative to any other point, so no curvature was imposed, so there is no bending moment, no stress, and no nozzle load. The run is just 40 mm lower.

That is the whole mechanism, and it holds for foundations, structures, tanks and piping alike. Strain comes from relative displacement. Rigid-body motion is free.

See it directly: ▶ open the interactive: civil settlement 3d — a vessel and a pipe support on their own foundations. Drag the slider with Both settle equally on and the assembly rides down like a lift, pipe dead straight, nozzle moment zero. Turn it off and drop the same amount into one foundation only: the run bows, the nozzle moment arrow spins up, and the pipe goes red.

Uniform settlement is not completely free, and the exceptions are worth naming because they are where real uniform-settlement failures happen:

• Anything you connect to that did not move — buried lines entering the paving, a tie-in to an older unit, a flare header on piles, a ladder to an adjacent structure.
• Drainage and grade. A bund, a sloped floor or a gravity drain that settles uniformly still ends up at the wrong level.
• Freeboard and clearances — under-rack road clearance, tank nozzle elevations against a pump centreline, flood level.
• It rarely stays uniform. Soil is not uniform, so "uniform" settlement is really "the part of the settlement that happened to be the same everywhere". Bjerrum's rule of thumb for footings on sand is that the differential is roughly ¾ of the total — which is exactly why codes limit total settlement: it is a controllable proxy for the differential you cannot predict.

Symbol key — every symbol on this sheet

• δ (delta) — settlement: downward movement of a point · mm
• Δδ — differential settlement: |δ_A − δ_B| between two connected points · mm
• L — distance between the two points (span between supports, or footing spacing) · mm, m
• β (beta) — angular distortion = Δδ/L, quoted as 1/N · dimensionless
• ω (omega) — tilt/rotation of a whole rigid structure (planar, no distortion) · rad
• E — Young's modulus, 203 GPa for carbon steel at ambient · MPa
• I — second moment of area of the pipe section · mm⁴
• Z — section modulus = I/(D/2) · mm³
• c — end-restraint constant in M = c·EIΔδ/L²: 6 fixed/guided, 3 fixed–pinned, 0 pinned–pinned
• M — bending moment the settlement forces into the run · N·mm, kN·m
• V — end shear = the vertical force landing on the nozzle or support · N, kN
• σ (sigma) — bending stress from the imposed displacement · MPa
• S_A — B31.3 allowable displacement stress range = f(1.25 S_c + 0.25 S_h) · MPa
• D1 — the displacement load vector a pipe stress engineer applies at a node · mm

Angular distortion and the limits everyone quotes

Normalise the differential by the distance it happens over and you get the number the whole subject runs on:

β = Δδ / L                 (angular distortion, always quoted as 1/N)

The classic bands come from Skempton & MacDonald (1956) and Bjerrum (1963), and they have survived sixty years of use:

Typical total settlement allowances that sit behind those: 25 mm for isolated footings on sand, 40–65 mm for rafts on sand, 65 mm for footings on clay, 65–100 mm for rafts on clay. Note the pattern — rafts are allowed more total settlement because they enforce less differential.

What settlement does to pipe and nozzles

Pipe is a beam. Impose a relative support movement Δδ over a length L and, if the run is held against rotation at its ends, you get:

M = 6·E·I·Δδ / L²          (end moment, fixed or guided both ends)
V = 12·E·I·Δδ / L³         (end shear — the force that lands on the nozzle)
σ = M / Z = 3·E·D·Δδ / L²  (bending stress; I/Z = D/2 cancels beautifully)

Three things fall straight out of that last form:

1. σ does not depend on wall thickness. Only on E, the outside diameter, the settlement and the span. A heavier schedule does not help — it makes the force worse, not the stress better.
2. σ ∝ D. Big lines suffer. A 24″ run picks up nearly twice the stress of a 12″ run for the same settlement over the same span.
3. σ ∝ 1/L². This is the lever you actually have. Double the distance to the first support and the stress falls by four. It is why "move the first support further from the nozzle" is the standard fix, and why a short, stiff spool between a settling vessel and a piled rack is the worst geometry in the plant.

And the fourth thing, which is the one people get wrong: if the run is genuinely free to rotate at both ends, c = 0 and there is no stress at all. The pipe just tilts. Settlement only bites when the run is restrained — which, with a nozzle at one end and a guide at the other, it almost always is.

Run the numbers: ▶ open the interactive: civil settlement calc — set the span, the settlement at each support and the pipe size, and it gives you β against the Bjerrum bands, the bending stress against S_A, and the moment against the nozzle allowable. The σ-versus-span plot underneath is the one to stare at: the 1/L² collapse is far steeper than intuition expects.

Where the pipe stress engineer picks it up

Settlement arrives in the stress model as an imposed displacement, not a force: a D-vector (ΔX/ΔY/ΔZ) applied at the anchor, restraint or equipment node. ASME B31.3 §319.2.1 classes "movements of piping supports or terminals" as displacement strains, so the resulting stress is secondary, self-limiting and checked against S_A, not against S_h.

Three habits that separate a good settlement case from a bad one:

• Model the difference, not the absolute. If the whole unit settles 40 mm, apply zero. Apply only what one end does relative to the other.
• Run best-estimate and upper-bound. The geotechnical report gives a range; the code case is the range, so run a no-settlement case too — the un-settled condition often governs something else (support lift-off, for example).
• Shakedown protects the pipe, not the nozzle. The secondary/self-limiting argument lets the pipe stress be compared with S_A. It gives you nothing at the equipment: the vendor's nozzle allowable (API 610 for pumps, WRC/vendor data for vessels and exchangers) is an absolute force-and-moment limit and the settlement moment counts against it at full value. In practice the nozzle check fails long before the pipe check does.

Worked example

A 12″ line (D = 323.9 mm, STD wall, I = 1.164 × 10⁸ mm⁴, Z = 7.19 × 10⁵ mm³) runs 6 m from a vessel nozzle to its first rack support. The vessel foundation settles 25 mm; the rack is on piles and does not move. The run is guided at the support and anchored by the nozzle, so c = 6.

β  = 25 / 6000                                    = 1/240   → past first cracking (1/300)
M  = 6(203 000)(1.164e8)(25)/6000²  = 9.84e7 N·mm = 98.4 kN·m
V  = 12(203 000)(1.164e8)(25)/6000³               = 32.8 kN
σ  = 3(203 000)(323.9)(25)/6000²                  = 137 MPa

Against the allowables:

σ  = 137 MPa   vs  S_A = 207 MPa (A106 B, f = 1)        →  66 %   PASS
M  = 98.4 kN·m vs  vendor nozzle allowable ~25 kN·m     → 394 %   FAIL

The pipe is comfortable. The nozzle is nearly four times over. That is the normal result — and it is why "the stress analysis passed" is not an answer to a settlement question.

Now the fix, and note that only one of these is cheap:

• Move the first support from 6 m to 12 m → σ and M fall by 4× → M = 24.6 kN·m. Done, for the price of a drawing revision.
• Put the vessel on piles → Δδ = 0. Done, for the price of piles.
• Add a loop or an expansion joint → reduces c toward 0, but adds cost, leak paths and its own analysis.
• Heavier wall → makes it worse: σ is unchanged, V and M go up with I.

Where differential settlement comes from, and how to kill it

Sources, roughly in order of how often they bite:

• Two different foundation systems joined by pipe — piled rack, shallow-founded pump or vessel. This is the single most common one, and it produces the full settlement of the shallow side as differential.
• Different bearing pressures. Two footings on the same soil settle by different amounts if they are loaded differently. The trick is not equal size — it is equal contact pressure.
• Different loaded widths. A tank at 200 kPa settles far more than a footing at 200 kPa, because settlement depends on how deep the stressed zone reaches.
• Variable ground — fill against natural, an old channel, a lens of soft clay, a buried foundation.
• Time. Sand settles immediately; clay consolidates for years. Two structures built the same week on different strata diverge slowly, long after handover.
• New loads nearby. A new tank surcharges the soil under the existing rack next to it.
• Hydrotest. A large tank fills, settles 100–300 mm, and empties — a big one-off movement that the connected piping must either avoid or absorb.

Mitigations, from most to least permanent:

• Piles to a competent stratum — eliminates most of it, and is the honest answer when the differential is between a heavy item and a light one.
• Raft / mat — converts differential into tilt by tying everything to one stiff plate.
• Match bearing pressures across adjacent footings at design stage. Free, and almost never done.
• Preload / surcharge / staged loading — settle the ground before you connect anything. A tank hydrotest is exactly this: API 650 lets the hydrostatic test act as the preload, which is why the rule is hydrotest the tank, let it settle, then make the piping tie-ins.
• Flexibility by layout — more span, a loop, a dog-leg. The 1/L² curve is doing the work.
• Flexible connections — expansion joints, hoses. They work, and they are also a leak path and a maintenance item; use them when geometry genuinely cannot give you the length.
• Design the re-levelling in — shim packs at tank-farm supports, jacking points, grout pads. Cheap on the drawing, expensive to retrofit.
• Monitor. Settlement markers and a reading schedule turn a surprise into a trend.

Common pitfalls

• Specifying a total-settlement limit and calling it done. The structure feels β, not δ.
• Applying the absolute settlement as the displacement load case when the whole unit moved — a large fictitious load on a piece of pipe that never strained.
• Passing the B31.3 displacement check and stopping. Shakedown does nothing for the nozzle.
• Stiffening the pipe to "resist" settlement. Imposed-displacement stress is independent of wall thickness, and the nozzle force increases with I. Add flexibility, never stiffness.
• Connecting large-tank piping before hydrotest — and then being surprised by 150 mm.
• Forgetting consolidation time. A clay site keeps moving for years; the ten-year differential, not the day-one differential, is the design case.
• Ignoring uplift. A support that settles less than its neighbours can unload completely and lift off, which changes the whole restraint scheme in the stress model.

Outcome

• Uniform settlement imposes no strain — it is rigid-body motion. It matters only at interfaces with things that did not move, and at grade/clearance.
• Differential settlement, normalised as angular distortion β = Δδ/L, is the damaging quantity. Target 1/500; 1/300 is first cracking; 1/150 is structural damage; machinery wants 1/750.
• For pipe: M = 6EIΔδ/L², σ = 3EDΔδ/L². Stress is independent of wall thickness, proportional to diameter, and falls as 1/L² — so span is the lever, and a short stiff spool is the enemy.
• The nozzle check fails before the pipe check. B31.3 lets the pipe stress be secondary; the equipment allowable is absolute.
• Settlement reaches the stress model as an imposed displacement (D-vector) of the difference between connected points, run as best-estimate and upper-bound.
• Interactive: ▶ open the interactive: civil settlement calc — β bands, σ vs S_A, moment vs nozzle allowable, and the 1/L² span curve.
• 3D: ▶ open the interactive: civil settlement 3d — drop one foundation and watch the nozzle; toggle "both settle equally" to see the stress vanish while the movement stays.

Open items

• A tank-specific sheet: API 653 Annex B settlement categories, the cosine-curve fit, edge settlement and the shell-distortion criteria.
• Support-stiffness effects — the calculator assumes rigid ends; a real support spring sheds a useful fraction of the moment.
• A worked CAESAR II load-case set for a settlement study (which combinations, and why a no-settlement case still has to be run).

Educational teaching material — simplified and illustrative. Not for engineering design use.
© EPiC Circle · provided by PipeAxis