Structural Engineering & Statics
where the load goes
Concept

Storey drift — where it appears

The horizontal movement of one floor relative to the one below it, expressed as a fraction of the storey height. It is the deformation imposed on everything spanning between two floors, and it is a poor relation of the roof drift rather than a fraction of it.

Named by 11 essays across 4 fields — each of them below, with the objects they name alongside it.

Two shapes that are the wrong way up for each other. Deflected shapes of a 20-storey building under a uniform wind, drawn to the same scale. The wall alone bends: its shape is flattest at the base and steepest at the top, reaching 146 mm. The frame alone shears: it is steepest at the base where the storey shear is largest, reaching 140 mm. Tied together at every floor they reach 58 mm — less than a quarter of either, and less than the 72 mm two springs in parallel would give, because each is stiff exactly where the other is not.

How a tall building stands still

A shear wall bends and a framed tube shears, and the two deflected shapes are the wrong way up for each other. Tie them together at every floor and the pair is stiffer than the sum of their stiffnesses — because near the base the wall holds the frame back and near the top the frame holds the wall.

structures · Lateral system 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.

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.

structures · Plan torsion Every path to the ground goes through the link. A braced bay 8 m by 4 m whose two diagonals stop 800 mm apart instead of meeting. The storey shear reaches the ground through the diagonals, and the vertical components they deliver to the beam have to pass through the segment between them: the link carries 47% of the applied shear as a shear force, at a lever arm short enough that its ends reach 0 kNm while the rest of the beam carries 0. The deflected shape drawn is the solved one, magnified — the real drift under this load is 0.008 mm. Everything outside the link is designed to stay elastic while the link is yielding, which is what makes the mechanism a choice rather than a hope.

The part that is meant to be weak

A braced frame is stiff and has nowhere to yield. A moment frame yields everywhere and is soft. Move the two diagonals a metre apart along the beam and the whole storey shear has to pass through the segment between them — which keeps most of the stiffness and puts every yielding in one member the designer chose.

structures · Eccentric brace One drift, two motions, opposite curvatures. The sideways movement of a 120 m building under a uniform wind, drawn as the sum of the two mechanisms that produce it. The bending curve is a cantilever's: flat at the base, steepening upward, concave one way. The racking curve is a stack of parallelograms: steepest at the base and flattening, concave the other. They add to 366 mm at the roof, of which 61% is bending. The one group that decides the split is αH = H√(GA/EI) = 2.48: below one the racking dominates and the building behaves as a frame, above about six the bending does and it behaves as a cantilever, and everything interesting is in between.

Two motions with one name

A tall building's sway is two movements added. A frame racks like a stack of parallelograms, worst at the bottom; a cantilever bends about its base, worst at the top. The total at roof level says nothing about which storey is worst, and on this building it is neither.

deflection · Drift components 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 Where two buildings touch, and what is there when they do. Closure between an 8-storey building 24 m tall and a 4-storey one 15 m tall, drawn against height, with the 50 mm gap between them. The two swaying out of phase close on each other more the higher up they are, so they first touch at 7.6 m and are in contact above it. The dots are the taller building's floors: its storeys are 3.00 m and the other's are 3.75, so two of them arrive part way up a column rather than at a slab. A blow at mid-height of a column asks it for a shear of half the impact and an end moment of Fh/8, neither of which is a demand any part of the design contains.

The floor that arrives at a column

The gap between two buildings is computed from their roof displacements, which is where each of them moves most. It is not where they touch, and it is not what is there when they do — a slab edge meeting a column part way up its height is a different event from two slabs meeting, and it is the one that appears in the photographs.

dynamics · Pounding A link that keeps two buildings apart has made them one. The largest closing movement between a 500 t building with a 0.8 s period and a 300 t building with a 1.2 s period, 50 mm apart, under one 1.0 s sine pulse of 0.50 g, and each building's largest displacement, against the size of a viscous damper joining them across the gap, from 0.01 MN·s/m to 541.3 MN·s/m. With no link they close by 515 mm, and the stiffer building moves 220 mm and the softer 419 mm. The link that takes the most energy out, 0.64 MN·s/m, still lets them close by 214 mm. The least that keeps the 50 mm gap is 4.8 MN·s/m, where the stiffer building moves 269 mm and the softer 280 mm. At the largest link the two move together, 281 mm and 281 mm.

The damper that ends up as a joint

A damper across the gap between two buildings acts on exactly the motion the gap is sized for, and it can be sized for two different things. The size that takes the most energy out of the pair still lets the buildings collide. The size that keeps them apart has nearly stopped moving: it has joined them into one building, and the stiffer of the two pays for it in drift.

dynamics · Pounding 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 Wrong in both directions until the building racks. Across buildings from a pure bending tower to a pure racking frame (αH on a logarithmic scale), each 20 storeys with gravity putting its own critical factor at 5.0: solid, the least storey sway factor over the building's own, which is what taking the least storey as the building's costs; dashed, the ground storey's second-order increase estimated from its own sway factor, over the true increase. At αH = 0.1 the least storey factor is 0.77 of the true one and the ground storey's estimate 0.09 of the true increase; at αH = 10, 0.86 and 0.37; at αH = 100, 0.94 and 0.98. The first error is on the safe side and the second is not, and both vanish only for a building that racks.

The storey that cannot see the building lean

A sway check made storey by storey asks each storey how much it drifts under a push and how much gravity sits on it, and reads a critical load factor for the storey from the two. For a frame whose storeys rack like a stack of shelves that is exact. For a building that bends — a braced core, a wall — it is wrong twice. Taking the least storey as the building's reads a critical factor of 3.9 for a building whose own is 5. And amplifying each storey by its own factor finds a 2 per cent second-order increase at the ground storey, where the truth is 20, because the ground storey hardly drifts and carries the lean of everything above it.

stability · Sway stability 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.

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.

structures · Plan torsion The buckling mode moves up to where the spine is not. The building's first buckling mode, each floor's sideways movement as a fraction of the roof's, up the height of a 20-storey building of a bending spine and a racking frame (αH 1.0), its gravity scaled to an elastic critical factor of 5.0, under an equal lateral load at every floor: the spine to the roof, and stopping at storeys 15 and 10. With the spine to the roof the mode is the bending tower's, growing all the way up. With the spine stopping, at storey 15 its top moves less than a thousandth as far as the roof, and at storey 10 its top moves less than a thousandth as far as the roof: the building buckles in the first storey of frame above the spine, every floor above that storey moving with it as one, and the spine takes no part.

The storey above the core decides

A building whose bending core runs to the roof shares one sway mode between its core and its frame, and the storey-by-storey stability check reads it wrongly twice. Stop the core five storeys short and the check becomes exact: the first storey of bare frame above the core reads the building's own critical factor to three figures, because the whole building buckles there and nowhere else. The check stays wrong below the core, and it goes wrong again when the frame above is more than about twice as stiff.

stability · Sway stability

Named alongside it

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

Lateral systemMode shapeShear wallStiffnessAccidental eccentricityAmplification factorBuckling modeCentre of rigidityCollapse mechanismCritical load factorDiaphragmEnergy dissipation

All concepts