Structural Engineering & Statics
where the load goes
Concept

Response spectrum — where it appears

The peak response of a family of single-mass oscillators to one ground motion, plotted against their periods. It is a summary of an earthquake rather than a load, and reading a force off it requires a structure's period and its mass.

Named by 14 essays across 2 fields — each of them below, with the objects they name alongside it.

How much of the mass each mode carries, over eight modes. The effective modal mass of each of the eight modes of a frame of eight storeys, as a percentage of the total. Mode 1 carries 85.6% and mode 2 9.1%; two modes are needed to reach 90% of the mass. The masses sum to exactly the total, which is a property of the eigenvectors rather than a normalisation applied afterwards.

Most of the mass moves together

A sixteen-storey frame has sixteen modes, and the first one carries eighty-four per cent of the mass. That is why a hand calculation on one mode gets the base shear right to within a tenth — and why the same calculation gets the acceleration at roof level wrong by a factor of two.

dynamics · Modal mass What a point on a response spectrum is: three structures, three integrations, three points. Three oscillators of periods 0.3, 0.8, 1.8 s, each integrated through the whole of the same ground motion, and the peak of each one plotted against its own period on the curve at the right. The peaks are 12.54, 46.2, 108.95 mm. The complete spectrum is that done 44 times. Nothing in the curve is a property of the earthquake alone: every point on it carries a period and a damping ratio that belong to a structure.

The spectrum is not a load

A response spectrum looks like a load curve and is not one. Every point on it is the peak of a complete time integration of one particular structure, and the curve is what you get by doing that again for every structure there could be.

dynamics · Response spectrum The demand falls and the movement rises, by the same factor. One elastic spectrum read twice: as an acceleration on the left and as the displacement that goes with it on the right. A fixed-base building at 0.5 s sits on the plateau and is asked for 1.05 g. Put it on bearings soft enough to make its period 2.54 s and the demand falls to 0.103 g — a base shear 10.2 times smaller, bought with no strength whatever. The same shift on the right-hand plot goes the other way: displacement is S_a T²/4π², so the demand rises from 65 mm to 165. That number is the design. It is a gap all the way round the building, a moat every service has to cross, and a detail that a later contractor will fill in unless somebody says what it is for.

Made weaker on purpose

Everything else in this collection resists a load by being stiff or strong enough for it. A base-isolated building resists an earthquake by refusing to hear it — a layer of bearings under the whole structure with a lateral stiffness a twentieth of the frame's, bought with almost no strength at all.

dynamics · Base isolation The force falls, the drift rises, and the damping goes the wrong way. What a compliant foundation does to a 0.6 s building on a 8 × 8 m footing, against the stiffness of the ground under it. Three curves, all normalised to the fixed-base answer. The period lengthens — 1.31 times at 200 m/s — because the swaying and rocking of the foundation are flexibilities in series with the structure's own, and the rocking term carries an h², so it is the tall building that feels it. The base shear falls with the period, which is why a fixed base is usually called conservative. The displacement rises, by 51% here, and that is what breaks the cladding, the services and the gap to the building next door. And the effective damping falls rather than rises: the structure's own is divided by the cube of the lengthening — 2.2% of an original 5% — while a slender building's foundation radiates only 0.28% back, because rocking radiates almost nothing at these frequencies.

The ground is a spring

Every dynamic result in this collection has assumed a structure rising from something that does not move. Nothing does. A foundation can slide and it can rock, both are flexibilities in series with the structure's own, and the rocking one carries a square of the height — so the period lengthens, the force falls, the drift rises, and the damping goes the wrong way.

dynamics · Soil-structure A broad tank sloshes and a tall one does not. The liquid's division into the part that moves with the wall and the part that sloshes, against the tank's proportion. The convective masses come from the potential-flow solution and the impulsive mass is whatever is left, so the two sum to the liquid's mass exactly at every proportion rather than approximately over part of the range. A tall tank at H/R = 3 is 84% impulsive and behaves almost like a solid; a shallow one at H/R = 0.5 is 72% convective and most of its contents never notice the earthquake. This tank sits at H/R = 1.33, which is 65% impulsive.

The liquid has a period of its own

Shake a tank and its contents do not all go with it. Part of the liquid moves as though it were bolted to the wall and part sloshes at a period fixed by gravity and the radius, which the tank's stiffness has no influence over whatever. The split is decided by one proportion, and the two parts then take entirely different amounts of the earthquake.

dynamics · Sloshing The ground is a structure, and it has a period. The transfer function of 38 m of soil at 75 m/s over bedrock at 800 m/s: how much the surface moves for a given motion in the rock, at every frequency. It is a column fixed at the bottom and free at the top, so its resonances are the odd harmonics — the peaks stand at 1, 3, 5, 7 times the first, which is a fixed-free column and nothing else. The fundamental is at 0.493 Hz, a period of 2.03 s, and it is 4H/v_s exactly. The peak amplification is 7.5 against the bound 1/(α + πξ/2) = 7.5, which agrees to 0.2% — and the α in it is the impedance ratio, 0.0554 here. That is the term that keeps the answer finite: assume rigid bedrock and α is zero, the bound becomes 13 and the model is predicting an amplification set by damping alone. What limits the surface motion is that the energy can leave downward.

The ground has a period of its own

An earthquake is measured on rock and felt on soil, and between the two is a layer that behaves exactly like a structure — a column fixed at bedrock, free at the surface, with a fundamental period of four times its depth over its shear wave velocity and a set of odd harmonics above it. The motion a building receives is the rock motion through that filter, and the filter is sharp.

dynamics · Site response Buildings that sway alike need almost no gap between them. The separation two adjacent buildings need, against the ratio of their periods. The obvious answer is the sum of what each can do — 340 mm — and it is wrong, because the two peaks do not occur at the same instant. The right combination is the one modal responses use, √(u₁² + u₂² − 2ρu₁u₂), with ρ the cross-correlation coefficient of the two responses. ρ depends on the period ratio and behaves the opposite way to intuition: at a ratio of one the two buildings sway together, ρ = 1, and the gap collapses to the difference of the two, 100 mm. At the 0.57 drawn ρ is 0.029 and the gap is 248 mm — 27 per cent less than the sum, and 99 per cent of the square root of the sum of squares.

The gap between two buildings

Two towers side by side in an earthquake need a gap. The obvious answer is the sum of what each can move, and it is wrong — because the two peaks do not happen at the same instant. What decides the answer is the ratio of the two periods, and buildings that sway alike need almost no gap at all.

dynamics · Pounding The building is a filter with one very sharp tooth. What a component mounted on the roof of a eight-storey building feels, divided by what the same component would have felt on the ground. A rigid item sees the floor's peak acceleration rather than the ground's, which is already 2.29 times as much. An item whose own period matches the building's sees 6.1 times as much, at 0.94 s — a resonance inside a resonance, and the two amplifications multiply rather than add. Far above the building's period the ratio falls back toward one, because a component much softer than its support simply follows the ground. The shape of this curve is a property of the building and not of the earthquake, which is why equipment is qualified against a floor spectrum rather than a ground one.

The spectrum a floor hands on

Most of what breaks in an earthquake is not the structure. It is a transformer, a chiller, a rack, a ceiling, a pipe — and every one of them is bolted to a floor rather than to the ground. The motion it feels has already been through a filter with one very sharp tooth in it.

dynamics · Floor spectrum 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 The cross term has the sign of the two contributions. Four responses of the floor, each combined three ways and divided by the complete quadratic combination at ρ = 0.50. Base shear: modal 927 and 839 kN, so the root-sum-square is 0.82 of it and the absolute sum 1.15. Base torque: modal -8640 and 8640 kNm, so the root-sum-square is 1.41 of it and the absolute sum 2.00. Flexible edge: modal 60 and -6 mm, so the root-sum-square is 1.05 of it and the absolute sum 1.15. Stiff edge: modal -5 and 47 mm, so the root-sum-square is 1.05 of it and the absolute sum 1.15. Where the two modes push the same way the root-sum-square is short; where they push opposite ways it is long.

The twist the combination rule invents

Two modes close together respond together, and the square root of the sum of squares assumes they do not. The error has the sign of the two modal contributions: where they agree, as they do in base shear, the rule comes up short, and where they oppose, as they always do in torque, it comes up long — by a factor of five for a floor whose stiffness sits twenty centimetres off its mass, a torque the building does not have.

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 On a flexible tower the full tank is not the worst one. The tower's base shear against how full the tank is, for a 12 m tank holding up to 8 m of water on a 25 m tower: a stiff tower (period 1.20 s with the tank full) and a flexible one (2.50 s), the liquid and tower coupled (solid) and with all the water bolted to the container (dashed). The stiff tower's shear rises all the way to full, 2,141 kN, against 2,728 kN rigid. The flexible tower's peaks at 84% full, 688 kN, and falls to 665 kN full; rigid water says 884 kN, flat from 37% full, because past the spectrum's last corner a heavier mass on the same spring draws the same force.

The tank that sloshes the tower

On the ground a tank's liquid splits into a part that moves with the wall and a part that sloshes at its own long period, and the two answers can be added afterwards because they are orders apart. Lift the tank onto a tower and the tower's period moves up towards the sloshing period, the two masses stop being separate, and the system has two modes that belong to neither. Adding the ground answers then overstates the base shear by a third, treating the water as solid by as much again, the wave grows by half, and a flexible tower is worst with the tank about five-sixths full.

dynamics · Sloshing A branch braced along its length holds the riser to a floor. Elevation along the branch: a water-filled DN100 riser guided at every floor of the eight-storey building, with a DN50 branch taken off 500 mm below the first floor's slab and run 2.0 m to a brace on the slab above. The riser leans with the storey's drift between its guides; the branch's brace holds the branch's far end to its slab. Under the building's two-way record the riser alone carries 2.8 N/mm² at the tee's height. With the branch braced so that it cannot slide along its length, it becomes a strut tying the riser to the slab: the riser carries 71 N/mm² at the tee and the branch 10. Let the branch slide at its brace and the riser is back to 4.3, while the branch, now turned by the riser's lean, carries 54.

The branch that holds the riser to a floor

A riser guided at every floor leans with each storey's drift and is barely bent by it. Take a branch off it and brace the branch to a slab, and the branch becomes a strut that holds the riser to that slab partway up the storey. The riser, which was carrying 3 N/mm², carries 71 at the tee. Let the branch slide at its brace and the riser is spared, but the branch is then bent by the riser's lean. The lean has to go somewhere, and the brace decides where.

dynamics · Floor spectrum 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

Named alongside it

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

Natural periodDampingMode shapeModal massBase shearModal analysisServiceabilityFloor accelerationResonanceSecondary systemAmplificationBase isolation

All concepts