Structural Engineering & Statics
where the load goes
Concept

Soft storey — where it appears

A storey markedly less stiff or less strong than those above it, usually because its walls or infills were omitted for an entrance or parking. It concentrates the whole structure's inelastic drift into one level, and every global measure of the building improves as it does so.

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

Three modes of a five-storey frame. The first three mode shapes of a five-storey shear frame, from the eigenvalue problem rather than sketched. Mode 1 has a period of 0.6 s, no node and carries 88.0% of the mass; Mode 2 has a period of 0.21 s, one node and carries 8.7% of the mass; Mode 3 has a period of 0.13 s, two nodes and carries 2.4% of the mass. The nth mode crosses the axis n−1 times, which is a theorem rather than a drawing convention.

A structure has more than one period

One mass on one spring has one period. A building with eight floors has eight, each with a shape of its own, and the second one bends the building into a curve nobody drew. They are not harmonics, they do not interfere, and each behaves as though the others were not there.

dynamics · Mode shapes Where the drift went. Storey drift at the target displacement, for the same frame with and without a soft ground storey at 50 per cent of the others' stiffness and strength. The regular frame spreads 219 mm over every storey; the soft one reaches 266 mm and puts 5.58 per cent of it into the ground storey against 0.53 next to it — a concentration of 5.9 against 1.7. The roof goes 22 per cent further, and where that extra displacement lands is the whole of the difference between the two buildings.

Weaker in one place, and better on every average

Take an eight-storey frame and make its ground storey half as stiff and half as strong. Its ductility demand falls, its first mode carries more of the mass, and its period lengthens into a gentler part of the spectrum. Three global numbers all improve, and the building is the one that collapses.

dynamics · Pushover Three quantities converging at three different rates. How much of the exact answer a truncated modal analysis of a ten-storey building reaches, against how many modes it keeps. The mass count is the rule — ninety per cent, reached at two modes. The base shear is ahead of it: one mode carries 85 per cent of the mass and 98 per cent of the base shear. The force in the top storey is behind it, at 80 per cent on one mode and 94 on two. The rule is written in the quantity that converges fastest, and it is checked against a quantity nobody plots.

The modes that were left out

Nobody runs every mode a model has, and the rule for how many is a mass count — enough to account for ninety per cent of the structure. The rule is written in the one quantity that converges fastest. On a twenty-storey frame two modes give the base shear to within one per cent and the force in the top storey to within twenty.

dynamics · Mode shapes Two capacity curves that agree, from two answers that do not. Base shear against roof drift for the same eight-storey frame with a soft ground storey, pushed with a fixed triangular pattern and with one recomputed from the tangent stiffness at every step. At a roof displacement of 426 mm they differ by 8.7 per cent — which is the number a capacity-spectrum procedure reads, and it is the number that is nearly the same. The storey drifts underneath these two curves differ by a factor of 3.8 at the fourth floor, and the drift is what the analysis was run for.

The pattern that stopped describing the building

A pushover analysis pushes with a load pattern chosen to resemble the first mode, and by the time the structure has done anything worth analysing it no longer has that mode. Recomputing the pattern as the frame softens changes the base shear by nine per cent and the drift at the fourth floor by a factor of four.

dynamics · Pushover The same mode's curvature, which can. The first mode's drift in each storey — the difference between the floors above and below it, which for a shear building plays the part a beam's curvature plays — intact, solid, and with storey 3 10 per cent less stiff, dashed, each scaled to the roof. The damaged storey's drift rises by 9.6 per cent; the storeys above it change by at most 1.8 and below it by at most 1.5. The largest change is in storey 3: the drift finds the damage and says where it is, which the frequency's fall of 0.91 per cent cannot.

What a mode shape notices that a frequency does not

Take a tenth of the stiffness out of one storey of a ten-storey building and its first frequency falls by a per cent at most, and by almost nothing if the storey is near the top. The mode shape looks unchanged, its match to the old one 0.99997. But the mode's storey drift — its curvature — rises by about a tenth in the damaged storey and hardly anywhere else, wherever that storey is. The instrument that sees the damage is the one almost nobody installs.

dynamics · Mode shapes The same riser, held three ways. The peak bending stress at each floor of a water-filled DN100 riser running the full height of the eight-storey building, under the same record, held three ways. Anchored against rotation at every floor it reaches 131 N/mm², at the lower floors where the drift is largest. Guided at every floor — held in line and free to turn — the same pipe never exceeds 3 N/mm². Guided everywhere but anchored at its base, it carries 77 N/mm² at the base and 21 N/mm² one floor up, and the guided values from there on.

The pipe held at every floor

A riser runs the height of a building and is fixed at every floor, so each of its supports moves with a different floor. The floor spectrum prices the pipe's inertia, and for a pipe anchored at every floor the inertia is nearly irrelevant: the drift puts 131 N/mm² into a 100 mm riser at two thirds of a per cent, thirty times its inertia, and passes yield at 200 mm. Guide the same pipe instead of anchoring it and the drift almost vanishes — the pipe then feels only how much the drift changes from one storey to the next.

dynamics · Floor spectrum

Named alongside it

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

Mode shapeDegrees of freedomModal analysisEigenvalueNatural periodStiffness matrixCollapse mechanismConvergenceModal massOrthogonalityPushoverResponse spectrum

All concepts