Structural Engineering & Statics
where the load goes
Concept

Core — where it appears

The stack of walls around a building's lifts and stairs, which is usually the whole of its resistance to wind. It behaves as a vertical cantilever, so its drift profile is largest at the top while the frame around it is largest at the bottom.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

Two differences up the same building, peaking in different places. Differential shortening between a perimeter column and the core of a 40-storey building, plotted up the height. The part driven by load peaks at level 20 — exactly half way up, because a floor near the top has almost nothing built above it to shorten what is beneath, and a floor near the bottom has almost nothing beneath it to shorten. The part driven by shrinkage does not care what is above it at all and accumulates all the way to the roof. Their sum is worst at level 40, at 43 mm, which across a 9 m bay is a floor out of level by one in 208.

The columns are shorter than the core

Every column in a tall building gets shorter as the building is built on top of it, and the core beside it gets shorter by a different amount. The floors between them tilt by the difference — and the difference is largest exactly half way up, because a floor near the top has almost nothing built above it and a floor near the bottom has almost nothing beneath it.

deflection · Differential shortening A couple applied to the core, and two columns to make it. A 20-storey core with one outrigger at 59% of its height. The arm is stiff in bending and the perimeter columns are stiff in tension and compression, so between them they resist the core's rotation at that level — a couple of 35283 kNm here, carried as a 294 kN pair in the columns at 120 m centres. The compatibility is one equation: the core's rotation at that level, less what the couple takes back out of it, equals the rotation the arm and its columns allow. The top drift falls from 360 mm to 74, which is 80% of it, and the base moment from 73500 to 38217 kNm. The deflected shape is drawn hugely exaggerated: the real top drift is about one five-hundredth of the height.

The arm that makes the columns work

The perimeter columns of a tall building are already there, already carrying gravity, and already the furthest thing from the centre. They take almost none of the overturning, because a floor slab transmits shear and not moment — and one storey-deep arm at the right height changes that by nearly a half.

structures · Outrigger Two faces, a couple, and a core that does none of it. A sandwich section 61.4 mm deep: two 0.7 mm faces separated by 60 mm of core. The bending is carried as a couple between the faces — 41 N/mm² of tension in one and compression in the other, over a lever arm of 60.7 mm — and the core carries a shear stress of 0.047 N/mm² and nothing else. The parallel-axis term is 98.8% of the section's second moment; the faces' own bending about their own centroids is 0.004% of it, and the core's is 1.2%. Separated by nothing at all the same two faces would be 2.3e+4 times less stiff.

Two skins and the space between them

A sandwich panel is a section made of a material that carries the bending and a material that carries none of it. The parallel-axis term is not a correction here — it is 98.8 per cent of the second moment — and the shear deflection is not a correction either.

sections · Sandwich section The shear deflection is not a correction. The share of a sandwich panel's deflection that is shear rather than bending, against how slender the panel is. A solid beam at a span-to-depth ratio of 20 spends about a per cent of its deflection on shear; this panel spends 25% at the same ratio, because its core is 2800 times softer in shear than its faces are in tension. At the 37 of the panel drawn it is 9%. The curve falls as the square of the span because bending grows as the fourth power and shear as the second, so the term that is negligible for a long panel is the whole answer for a short one.

The stiffness that belongs to the span

A section's flexural rigidity is a property of the section, and a member's is not. Once shear deformation is counted the effective stiffness contains the span, so the same panel is a different member at three metres and at six — and for a sandwich the correction is not a correction.

deflection · Shear deflection What the second, third and fourth arms are worth. Top drift removed against the number of outriggers, each arrangement at its own optimum levels, on a 40-storey core 200 m tall. One arm at 59 per cent of the height removes 82.3 per cent of the drift. A second, with both moved to 35 and 71, takes it to 91.6 — a gain of 9.3 points, which is half of what was left. The third is worth 3.0 and the fourth 1.5, and each one costs a storey of the building's most valuable height.

What the second arm is worth

One outrigger at its best height removes five sixths of a tall core's drift, which sounds like the end of the argument. A second removes half of what is left, a third half of that, and each of them costs a storey of the most valuable floor area in the building — so the question is not where to put an outrigger but how many the arithmetic still justifies.

structures · Outrigger How far a local load spreads, and what is nearest its limit. Along a steel-faced polyurethane cladding panel, 0.5 mm faces of 320 N/mm² steel on an 80 mm core crushing at 0.12 N/mm², from the middle of a strip load of 3.0 N per mm of width spread over 10 mm: the face's deflection over its value under the load, the core's compressive stress over its crushing strength, and the face's bending stress over its yield stress. The load spreads over a length set by the fourth root of the face's bending stiffness over the core's, 1/β = 20.5 mm; the deflection is 1.44 mm under the load and changes sign beyond 48 mm. The core is at 0.60 of its crushing strength and the face at 0.89 of its yield stress: this panel's face yields at 3.36 N/mm and its core crushes at 5.00.

The face that dents and the core that crushes

A sandwich panel carries bending as a couple between its faces, and that is a calculation about the whole panel. Put a local load on one face — a foot, a fixing, a dropped tool — and the face becomes a thin beam on a soft bed, spreading the load over a length the fourth root of their stiffnesses sets. Then either the face yields and dents, or the core crushes beneath it, and which comes first is decided by one thickness: below it the face gives, above it the core does, and a steel-faced roof panel is on the wrong side of it for anyone who walks on it.

sections · Sandwich section

Named alongside it

The objects these essays reach for when they reach for this one.

ServiceabilityStiffnessTall buildingWrinklingCompatibilityDifferential shorteningDriftLateral systemOptimisationOutriggerShear deflectionShear modulus

All concepts