Structural Engineering & Statics
where the load goes
Concept

P-delta — where it appears

The second-order effect in which vertical load acting on a displaced structure adds to the moment that displaced it. It appears in a stiffness matrix as a geometric term with the opposite sign to the elastic one, so the critical load is where the total stiffness reaches zero.

Named by 11 essays across 2 fields — each of them below, with the objects they name alongside it.

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 makes itself worse

A structure that has leaned carries its weight off the axis, which makes it lean further. The amplification is one over one minus the load ratio, and it runs away long before the buckling load.

stability · Second-order 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 check that everything adds up, and the error it cannot see. Four versions of the same 2-bay, 3-storey frame, with the global equilibrium residual each one produces — the sum of the reactions against the sum of the applied loads, as a fraction of the applied total. It is the first thing every analysis prints and it is worth having: a lost restraint and a load entered in the wrong unit both show up immediately, at 6% and 24%, because both change what the structure is carrying. The fourth bar is the point. A member whose stiffness is wrong by a factor of ten redistributes the internal forces completely — the second bar shows the change in the member forces, 18% — and the global residual is exactly zero, because the wrong answer is still in equilibrium with the same loads. Equilibrium is one equation per degree of freedom of the whole body, and a stiffness error lives entirely in the many equations underneath it. A model can satisfy every equilibrium check ever devised and be a model of a different structure.

The stiffness the load takes away

Buckling is usually taught as an event — a critical load, a bifurcation, a mode. Written as a matrix it stops being an event at all. A compressive load subtracts a stiffness from the structure, the subtraction grows with the load, and the critical load is simply where what is left reaches zero.

stability · Geometric stiffness 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 load being amplified is not the load doing the amplifying. The sway amplifier 1/(1 − ΣP/P_cr) against the storey's total gravity load, as columns are added that carry load and provide no lateral stiffness. The critical load of the storey is fixed at 8203 kN by the bracing that exists, and every leaning column moves the structure along the axis without changing it. At the storey drawn the amplifier is 1.57, so the second-order sway moment is 57% on top of the first-order one — and none of that 70% of the load which is causing it appears in any stability calculation done column by column. The storey stays stable across the whole of this axis. That is the practical reason a gravity-only column is drawn on the frame model rather than designed on its own: it is not being checked, it is being counted.

Counted, not checked

A column with pinned ends and no bracing cannot buckle on its own, so nothing about it fails a stability check. It still carries load, and load with no stiffness attached lowers the buckling load of everything around it — which is why a gravity-only column is put on the frame model and never designed by itself.

stability · Second-order The push that tips it, in every direction. The tipping push in each plan direction, drawn as a distance from the centre, for a 6000 kN body with its weight 15 m up and the push at 18 m. On rigid ground the curve is the hull's: weakest along the axes at 1333 kN. On four equal pads it shrinks, and more toward the corners. With the pads as built — stiffnesses 20000, 20000, 8000, 20000 kN/m — it is no longer symmetric: the weakest direction is 15°, on the soft pad's side, at 1039 kN.

It tips inside its own hull

On rigid ground a body tips when its resultant reaches the edge of its base, and how stiff its supports are has nothing to do with it. On pads that settle, the body leans as it is pushed, the lean moves its weight, and the push that tips it falls by one number a site engineer already has: settlement times the height of the weight, over the square of the half-width. Toward a corner the loss doubles, and one soft pad makes the weakest direction one nobody checks.

equilibrium · Overturning Three grounds, three tipping pushes. The push against the lean it produces, in thousandths of a radian, for the 6 m square, 3,000 kN block on ground that bears 300 kPa at most, half of it mobilised at 7.5 mm of settlement, pushed 8 m up, its weight 5.0 m up, on the three grounds matched at their limit and at half of it; dashed, the rigid-plastic limit, 813 kN. Linear, capped: 776 kN at a lean of 13.2 thousandths; hyperbolic: 701 kN at a lean of 26.2 thousandths; S-shaped: 770 kN at a lean of 15.2 thousandths. The S-shaped ground behaves almost exactly like the capped one; the hyperbola, which never quite reaches its limit, tips 10 per cent sooner at twice the lean.

The ground that never quite gives way

On a bed of springs capped at the ground's bearing limit, a block pushed sideways tips at a definite push, below the limit the rigid-plastic parabola gives. Real ground is curved: it softens all the way to its limit and never quite reaches it. On such ground the block has no tipping push of its own. Its resistance creeps toward the limit, and the peak it does have is made by its own weight leaning with it — 14 per cent below the limit with the weight 5 m up, nearly at it with the weight at the base, where the lean simply runs away.

equilibrium · Overturning 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 A hat makes the top storeys look stiff. Each storey's own sway factor α up the height of a 20-storey building of a core to the roof 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: with no hat, with a hat truss at the roof holding the core's rotation with 10 times the core's EI/H, and with the same restraint at storey 10; the building's critical factor 5.0 as a vertical line. With no hat the top storeys read the least, 4.06 at the roof. With the hat the roof storey reads 15.47 and the least, 4.14, has moved down to storey 9. With the outrigger at mid-height the least is back at the top, 3.85.

The hat that hides the lean

A hat truss on a core-and-frame building holds the top of the core, nearly doubles the building's critical load, and makes its top storeys look three or four times as stiff as they did. The storey check believes them. Under the hat the roof storey reads a sway factor of 15 and estimates 7 per cent of second-order drift; the building still buckles leaning, and the roof storey's true increase is 24 per cent. The hat turns the storey check from pessimistic to unsafe at exactly the storeys it was built to stiffen.

stability · Sway stability Caught from both sides without a mode shape. The elastic critical load factor of a one-bay frame of six storeys, 3.5 m high and 6 m wide, columns of 40,000 kN·m² and beams of 60,000, 400 kN on every floor, 14.96 (level line), and the bounds the traces of its flexibility give with no assumed shape, against how many traces are taken: from below, 5.12, 10.82, 13.11, 14.07 for the first four; from above, 22.90, 19.23, 17.39, 16.39. Dotted, the storey estimate of EN 1993-1-1, 12.86, and Rayleigh's quotient on the first-order sway, 15.88.

The trace that adds up every storey

A column's buckling load can be bounded from below by the traces of its flexibility, with no shape guessed. A sway frame has a flexibility too — one number per pair of floors — and its traces bound the frame's critical load factor from below and from above, closing in as more are taken: on a six-storey frame whose factor is 14.96, the first four give 14.07 to 16.39. But the first trace is Dunkerley's sum over the storeys, counting each as though its sway mode were the frame's, and it bounds a well-balanced frame worst of all: one over the number of storeys when every storey is alike, which is exactly when Eurocode 3's storey-by-storey estimate is exact.

stability · Stability energy

Named alongside it

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

Critical loadSway stabilityAmplification factorEffective lengthLateral systemNotional loadSecond-orderBracingBuckling modeFree bodyGeometric stiffnessImperfection

All concepts