Structural Engineering & Statics
where the load goes
Generator

Two centres, and the distance between them is a torque

Rendered here at the parameters it defaults to, with every essay that calls it — which is the same list as the blast radius of changing it.
Two centres, and the distance between them is a torque. A storey 30 by 18 m with its walls drawn heavy, pushed in one direction by 1000 kN. The force acts through the centre of mass and the storey turns about the centre of rigidity — the stiffness-weighted centroid of the walls, at x = 15.0 m — and the distance between the two is an eccentricity of 0.00 m before the 5% that has to be assumed anyway. The table below the plan splits each wall's force into its direct share and its torsional one. Torsion relieves the walls near the centre of rigidity and loads the far ones, so the wall in trouble is not the wall carrying the most: west wall is asked for 8% more than its direct share, and the walls at right angles to the push carry 19 kN each with nothing applied along them at all.

Two centres, and the distance between them is a torque. A storey 30 by 18 m with its walls drawn heavy, pushed in one direction by 1000 kN. The force acts through the centre of mass and the storey turns about the centre of rigidity — the stiffness-weighted centroid of the walls, at x = 15.0 m — and the distance between the two is an eccentricity of 0.00 m before the 5% that has to be assumed anyway. The table below the plan splits each wall's force into its direct share and its torsional one. Torsion relieves the walls near the centre of rigidity and loads the far ones, so the wall in trouble is not the wall carrying the most: west wall is asked for 8% more than its direct share, and the walls at right angles to the push carry 19 kN each with nothing applied along them at all.

4 essays call plan-rigidity. The drawing above is what it returns with no arguments at all; every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this generator changes every one of these figures.

Two centres, and the distance between them is a torque. A storey 30 by 18 m with its walls drawn heavy, pushed in one direction by 1000 kN. The force acts through the centre of mass and the storey turns about the centre of rigidity — the stiffness-weighted centroid of the walls, at x = 15.0 m — and the distance between the two is an eccentricity of 0.00 m before the 5% that has to be assumed anyway. The table below the plan splits each wall's force into its direct share and its torsional one. Torsion relieves the walls near the centre of rigidity and loads the far ones, so the wall in trouble is not the wall carrying the most: west wall is asked for 8% more than its direct share, and the walls at right angles to the push carry 19 kN each with nothing applied along them at all. Structural form

The corner that moves most

A lateral force is shared out in proportion to stiffness only if it passes through the centre of rigidity, which is not the centre of the plan and not the centre of mass. The distance between the two is a torque, and the wall that pays for it is the one furthest away and carrying least.

 Whether the floor shares the load out by stiffness or by area. The share of a uniform storey force taken by each of three equally stiff walls, against the stiffness of the floor plate that spans between them. A plate far stiffer than the walls translates almost rigidly, every wall deflects the same and the share is the ratio of stiffnesses — 60% to the middle wall here, the same as everyone else. A plate far softer than the walls behaves as a continuous beam over them and the middle wall takes 33%. Neither end is the tributary-area answer of 50%, which assumes a plate that is both soft and discontinuous over the wall, and which nothing here ever reaches. Structural form

The floor is a beam lying down

A floor plate spans horizontally between the walls that resist a lateral load, carries a distributed inertia load, and has chords, a web and a span-to-depth ratio like any other beam. Its stiffness decides whether the walls share the load by their stiffness or by the area of floor nearest them — and the familiar tributary answer turns out to be neither limit.

 The ratio stops growing and the twist does not. For a 30 × 18 m floor with a torsional radius of 10.6 m (0.35 of its length), against the distance of the centre of rigidity from the plan's middle as a share of the length, with the accidental eccentricity of 5 per cent of the plan: solid, the codes' ratio of the worst edge's displacement to the average of the two edges; dashed, the worst edge's displacement over the translation of a plan with no eccentricity. Dotted, the thresholds 1.2 and 1.4. The ratio rises to 1.76 at an eccentricity of 0.30 of the length and then falls, to 1.72 at 0.45; the edge's movement rises the whole way, from 1.20 to 4.83. The core-and-façade plan, at 0.14, reads 1.63 and moves 1.98. Structural form

A measure of twist that divides by the twist

The seismic codes decide whether a building is torsionally irregular by one ratio: the worst edge's displacement over the average of the two edges'. For a plan with no natural eccentricity that ratio is exactly how much further the corner moves than a plan without torsion would. For an eccentric plan it is not, because the twist moves the average as well as the corner, and the ratio divides one by the other. It saturates: at a torsional radius a third of the plan's length it never passes 1.76, however far the stiffness is moved off-centre, while the corner's movement keeps growing past four times. Past a point a more eccentric plan reads more regular, and the amplifier the codes build from the ratio inherits its ceiling.

 One quantity, three names. The middle braced bay's share of the storey force (solid) and one end bay's (dashed), for a 48 by 12 m plate, 200 mm of concrete, spanning two 24 m bays between three braced bays of 100,000 kN/m each, under 40 kN/m, against a factor on the plate's stiffness as built, on a logarithmic scale. The share moves continuously from 61 per cent, a continuous plate on unyielding supports, to 33 per cent, the rigid plate. Left of 0.02 (shaded) ASCE 7 calls the plate flexible and assigns tributary shares, 50 per cent to the middle; the plate there actually gives it 59 per cent at the boundary. Right of 1.54 (shaded) EN 1998-1 calls it rigid and assigns stiffness shares, 33 per cent; at that boundary the plate gives 37. The plate as built, factor 1, gives 38 per cent to the middle and 31 to each end. Structural form

Rigid by one code and not by the other

A floor plate between shear walls shares its storey force out in proportions that move continuously with its stiffness, from a continuous beam's to a rigid body's. The codes put it in one of two boxes and use a different model in each. ASCE 7 and EN 1998-1 draw their boxes' edges a hundred times apart in plate stiffness, each at a point where its own licensed model already misstates the middle wall by about a tenth — and an ordinary 200 mm slab sits in the gap between them.

The library, page 4 of 7 — where plan-rigidity sits