Structural Engineering & Statics
where the load goes
Concept

Centre of mass — where it appears

The point where the whole of a floor's mass can be taken to act, and so where its inertia force passes in an earthquake. When it stands apart from the centre of rigidity the difference is a lever arm, and the floor must twist as well as sway.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The dynamic ratio peaks where the modes couple, and falls where the static one climbs. The standards' ratio, the larger edge's displacement over the average of the two edges', for a 30 × 18 m floor swaying with a period of 0.5 s, against Ω, the torsional radius over the floor's radius of gyration, for natural eccentricities of 2.1 and 4.2 m: from a static push at the centre of mass (dashed) and from a response spectrum combined by CQC (solid); the dotted lines are 1.2 and 1.4. At 2.1 m, static 2.05 at Ω = 0.5 falling to 1.08 at 2; dynamic 1.08 at 0.5, highest at 1.42 near Ω = 1.15, and 1.10 at 2. At 4.2 m, static 2.46 at Ω = 0.5 falling to 1.15 at 2; dynamic 1.17 at 0.5, highest at 1.55 near Ω = 1.20, and 1.19 at 2.

The floor that twists in its own time

Seismic design standards flag a twisting floor by the ratio of its worst edge's movement to the average of its two edges', and a static push saturates that ratio. Give the floor mass and shake it, and it sways and twists in two modes that share the motion. A response-spectrum analysis then reads the same ratio from combined peaks — and gives a different verdict on a third of the plans tried. Floors torsionally stiffer than they sway read worse than a push says, because the coupled modes put twist at the edges. Floors torsionally flexible read better, because combining the modes keeps how far each edge moves and forgets which way, so a twisting floor's two edges both look large.

structures · Plan torsion The wall that pays is the one the elastic push said to spare. The mean peak ductility demand on each of the four walls of a 30 × 18 m floor on four walls that yield, its centre of rigidity 4.2 m off the centre of mass and its torsional radius 10.6 m, across the plan, over twenty records at a strength reduction of 4.0. Strength follows elastic force: 9.6, 6.1, 4.8, 4.3; Strength follows stiffness: 4.2, 4.7, 6.1, 8.2; Four equal walls: 8.5, 6.7, 5.4, 3.7.

The wall the twist was said to spare

Give a floor's walls the strengths an elastic push says they need and the far wall, which the push loads most, gets the most. Shake it past yielding and the floor does stop twisting: the twist falls by a third, the standards' ratio drops below its limit, and the plan reads regular. The demand has not gone. It has crossed the floor to the stiff wall on the other side, whose strength the same push had reduced, and which is now asked for more than twice the ductility of the far wall it was thought to be sheltered by.

structures · Plan torsion

Named alongside it

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

Centre of rigidityPlan torsionTorsional irregularityDampingDuctilityModal analysisNatural frequencyPlastic hingeResponse spectrumShear wallStiffness distribution

All concepts