Structural Engineering & Statics
where the load goes
Concept

Shear wall — where it appears

A wall used to carry horizontal load in its own plane, working as a deep vertical cantilever. Its deflected shape is a bending one, largest in slope at the top, which is the opposite of a frame's and is why the two together are stiffer than the sum of their stiffnesses.

Named by 6 essays across 3 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 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.

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.

structures · Diaphragm 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 Two cantilevers, or one wall, and the beams decide which. The deflected shape of a coupled pair of 6 m walls, drawn against the two limits it lies between. Release the coupling beams entirely and the pair is two independent cantilevers, deflecting 111 mm. Make them rigid and it is one composite wall of the full width, deflecting 16 mm — 6.8 times stiffer, because the lever arm between the wall centroids is 8.40 m and everything inside either wall is smaller than that. Real beams of 600 × 350 mm over a 2.4 m opening land at 23 mm and carry 63% of the base overturning as an axial couple rather than as wall bending. The degree of coupling never reaches one, because a beam of finite depth cannot suppress the walls' curvature entirely.

Two walls that agreed to be one

A pair of shear walls with a row of doors between them is the commonest lateral system there is, and it has two readings that differ by a factor of seven. What decides which one applies is a beam 600 mm deep over a 2.4 m opening — and most of the overturning ends up as an axial couple that no bending diagram contains.

internal-forces · Wall coupling 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.

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.

structures · Diaphragm

Named alongside it

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

Lateral systemDiaphragmLoad sharingStiffnessStorey driftAccidental eccentricityCentre of rigidityDriftIn-plane stiffnessPlan torsionShear deflectionShear stiffness

All concepts