Structural Engineering & Statics
where the load goes
Concept

Lateral system — where it appears

The parts of a building arranged to carry wind and earthquake to the ground, as distinct from the parts carrying gravity. Separating it from the gravity system is a design decision rather than a fact, and the members frequently belong to both.

Named by 17 essays across 5 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 A couple applied to the core, and two columns to make it. A 20-storey core with one outrigger at 59% of its height. The arm is stiff in bending and the perimeter columns are stiff in tension and compression, so between them they resist the core's rotation at that level — a couple of 35283 kNm here, carried as a 294 kN pair in the columns at 120 m centres. The compatibility is one equation: the core's rotation at that level, less what the couple takes back out of it, equals the rotation the arm and its columns allow. The top drift falls from 360 mm to 74, which is 80% of it, and the base moment from 73500 to 38217 kNm. The deflected shape is drawn hugely exaggerated: the real top drift is about one five-hundredth of the height.

The arm that makes the columns work

The perimeter columns of a tall building are already there, already carrying gravity, and already the furthest thing from the centre. They take almost none of the overturning, because a floor slab transmits shear and not moment — and one storey-deep arm at the right height changes that by nearly a half.

structures · Outrigger 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 The corner columns take what the middle ones did not. Axial stress in the columns across one flange face of a 30 by 40 m framed tube, at the base. Plane sections says the flat line: every column on the face at the same distance from the neutral axis, therefore at the same stress. The solved distribution is the curve — 69.4 N/mm² at the corner against 18.2 in the middle, a ratio of 3.81. The middle columns lag because the only route the axial force has into them is the in-plane shear of the spandrel frame, bay by bay from the corner. The face is carrying its resultant on an effective width of 51 per cent, and the tube deflects as though its second moment were 72 per cent of the gross.

The corner columns take more than their share

A framed tube is a hollow cantilever, and a hollow cantilever's flange ought to be uniformly stressed. It is not, and the reason is that the only route the axial force has into a column in the middle of a face is the in-plane shear of the frame — one bay at a time, from the corner inwards.

structures · Framed tube 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 The wind pushes on one face and pulls on three. A 30 × 20 m building in plan, with the measured pressure coefficient on each face and the arrows drawn in the direction the pressure acts. Only the windward face is pushed; the other three are sucked, and the side faces are sucked hardest of all at c_p = -0.7. The horizontal resultant is 842 kN at a velocity pressure of 0.9 kPa, and the arithmetic of it is the whole point: the leeward suction pulls the building downwind, so it ADDS, supplying 38% of the answer, while the two side faces cancel each other exactly and supply none of it. The coefficients are wind tunnel data; what is computed is the free body they are applied to.

Most of it is suction

A wind load is drawn as arrows pressing on the windward face, which is where about three fifths of it comes from. The rest is a pull on the back. The two side faces carry the largest suctions on the building and contribute nothing at all to the answer — and the inside of the building, which nobody draws, decides whether the roof stays on.

equilibrium · Wind pressure The load that makes itself worse. The amplification of a deflection against the ratio of applied load to buckling load. A structure at half its buckling load deflects twice as far as first-order analysis predicts, and the curve runs away well before the load is reached.

The load that is really a lean

No frame is ever plumb. The columns are out of upright by something like a three-hundredth, and every tonne of gravity load standing on that lean has a horizontal component. The force that represents it is not a safety allowance — it is an exact statics substitution for a geometry nobody drew.

equilibrium · Notional load The corner columns take what the middle ones did not. Axial stress in the columns across one flange face of a 36 by 36 m framed tube, at the base. Plane sections says the flat line: every column on the face at the same distance from the neutral axis, therefore at the same stress. The solved distribution is the curve — 78.5 N/mm² at the corner against 13.4 in the middle, a ratio of 5.88. The middle columns lag because the only route the axial force has into them is the in-plane shear of the spandrel frame, bay by bay from the corner. The face is carrying its resultant on an effective width of 45 per cent, and the tube deflects as though its second moment were 64 per cent of the gross.

The columns that lean

A framed tube carries its wind load by bending the spandrel beams between its columns, and it does it badly — the corner columns take nearly six times what the middle ones do. Tilt the columns instead, so the perimeter is triangulated, and the same shear is carried axially. The concentration falls to 1.21 and the tube recovers most of the stiffness the plan said it had.

structures · Diagrid The two braces balance until one of them buckles. An inverted-V brace after the compression member has gone. While both braces are elastic they carry equal and opposite forces and their vertical components cancel on the beam above, which is why the beam in a chevron bay is usually sized for gravity alone. The compression brace buckles at 445 kN and then sheds most of what it was carrying — 30% is left here — while the tension brace goes on to yield at 1065. The difference between the two vertical components is 659 kN, applied at the middle of the span with no help from either brace, and it asks the beam for 1317 kNm against the 200 kNm the gravity load asks for — 6.6 times as much. The beam drawn does not: 1517 kNm against a capacity of 731. The force is not a load case anybody applies; it is what the frame leaves behind on its way to the state it will actually be in.

The force the brace leaves behind

Two braces meeting under a beam carry the storey shear as a tension and a compression whose vertical components cancel, so the beam above sees nothing. They cancel only while both braces are elastic. Once the compression brace buckles it sheds most of its force, the tension brace goes on to yield, and the difference is a point load at midspan that nobody applied.

structures · Chevron brace Effective length is a property of the storey. The effective length factor of the one column that resists sway, against the total gravity load on the storey as a multiple of its own. At the left-hand end it carries the storey alone and its K is 1.99 — the 2.0 every chart gives a column fixed at the base and free to sway at the top, reproduced here by a route that never mentions a chart. Then columns are added that have pinned bases and therefore no lateral stiffness whatever. They contribute load and nothing else, so they cannot buckle on their own and they lower the load at which everything buckles together. K rises as the square root of the load ratio, exactly, and at the storey drawn — three leaning columns carrying 69% of the gravity load — it is 3.57. That is off the end of every published alignment chart, and the leaning columns themselves, which a designer would take at K = 1.0 for pinned ends, are at 2.54.

The column that leans on its neighbours

A column with a pinned base and a pinned top has no lateral stiffness at all and cannot stand up alone, and yet thousands of them do. What holds them is the rest of the storey, and what it costs is paid by whichever columns do have stiffness — whose effective length rises as the square root of the load being leaned on them, straight off the end of every chart.

stability · Storey buckling The hinge is at mid-height in exactly no storey. The height of the point of contraflexure in each column of a 4-storey, 3-bay frame under lateral load, as a fraction of the storey height, against the portal method's assumption that it is at the middle. The exact solution is a plane-frame stiffness analysis of the same frame. In the bottom storey the zero sits at 0.609 of the height, because a fixed base is stiffer than the joint above it and takes more of the column's moment; in the top storey it sits at 0.359, because there is no column above to share that joint. The average over the whole frame is 0.475, which is why the assumption survives — it is right on average and wrong everywhere. The worst error in the storey shear share is 25%, and the column shears still add to the storey shear to 0e+0 of it, because the method is exact statics applied to an assumed structure.

The analysis that assumes the answer

A rigid frame is indeterminate, so statics cannot finish it. The hand methods finish it anyway, by assuming where the bending moment is zero and treating those points as hinges. That is not a shortcut around the analysis — it is a different kind of answer, exact in equilibrium and wrong in compatibility, and knowing which half is which is what makes the error a bound rather than a mystery.

internal-forces · Portal method What the second, third and fourth arms are worth. Top drift removed against the number of outriggers, each arrangement at its own optimum levels, on a 40-storey core 200 m tall. One arm at 59 per cent of the height removes 82.3 per cent of the drift. A second, with both moved to 35 and 71, takes it to 91.6 — a gain of 9.3 points, which is half of what was left. The third is worth 3.0 and the fourth 1.5, and each one costs a storey of the building's most valuable height.

What the second arm is worth

One outrigger at its best height removes five sixths of a tall core's drift, which sounds like the end of the argument. A second removes half of what is left, a third half of that, and each of them costs a storey of the most valuable floor area in the building — so the question is not where to put an outrigger but how many the arithmetic still justifies.

structures · Outrigger 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 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.

StiffnessDriftStorey driftBracingFree bodyLoad pathShear wallP-deltaBucklingCompatibilityDuctilityLoad-sharing

All concepts