Structural Engineering & Statics
where the load goes
Concept

Amplification — where it appears

The factor by which a second-order analysis multiplies a first-order deflection, equal to one over one minus the load ratio. It is more usefully read as a softening than as a magnification, because a stiffness that has fallen changes how load divides between parallel systems and a magnified deflection does not.

Named by 14 essays across 4 fields — each of them below, with the objects they name alongside it.

The deflection goes on growing, and sometimes it does not stop. Second-order deflection of a sustained-loaded concrete column against age, on a log time axis. Creep takes the effective modulus down, which takes the buckling load down with it — from 11580 kN on the day to 3309 in the long term, 29 per cent of it — so the amplifier 1/(1 − N/N_cr) grows even though nothing was added to the load. At 2200 kN the column settles: 25 mm of eccentricity on the day and 60 mm at the end, a factor of 2.4 for a load that never changed. At 5294 kN — still only 46 per cent of the day-one critical load — it does not settle, and the divergence arrives at 55 days for no new reason at all.

The column that fails years later

A concrete column under sustained load goes on straining at constant stress, so its deflection grows — and because the second-order moment is the load times that deflection, the demand grows with it. There is a load below which the two settle and one above which they never do.

stability · Creep buckling 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 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 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 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 A built-up column has a second way to bend. A 24 m column of two chords 800 mm apart, joined by single lacing. On the left it buckles the way a solid column does, by bending; on the right the chords stay straight and the lattice racks, which a solid column cannot do at all. Neither happens alone, and the two flexibilities add rather than the two stiffnesses — so the critical load is 3832 kN against an Euler load of 4629 kN, which is 83% of it, and the column behaves as though its slenderness were 66 rather than 60.

The lacing decides the force it has to carry

A perfectly straight column carries no shear, so the diagonals of a built-up column are resisting a force that exists only because the column is crooked. Computing it turns out to be circular — the shear is amplified by the very flexibility the lacing supplies — and the rule of thumb that replaces the calculation is wrong by a factor that grows as the lacing gets worse.

stability · Built-up column The bow the frames have to hold. A chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames every 4 m, carrying 930 kN critical at 1548 kN, drawn with its sideways movement exaggerated; its critical load is 1.66 times the chord force. Dashed, the worst of the bows tried: four half-waves, each 12.0 mm high, a five-hundredth of its own length. Solid, the chord once the compression has grown the bow; each arrow is the force the frame there must supply, the largest 13.6 kN — 1.47 per cent of the chord force.

What the frames hold is the bow

A row of U-frames is sized for a force, and the rules give that force as a percentage of the compression chord's force that nobody derives. Derived, it is not a property of the frames at all. It is the chord's initial bow, grown by how near the chord works to buckling — and it is least, not greatest, near the stiffness at which the chord stops using the frames. Stiff frames tend to a limit that is π²/500 of the chord force: the familiar two per cent, arriving from a bow of one five-hundredth.

stability · Continuous restraint Two quotients from one guessed shape. The critical load of a pin-ended column whose outer quarters keep 10% of the middle's flexural rigidity from four guessed shapes, each worked two ways: Rayleigh's quotient, strain energy of the guess's own curvature over the work of the load, and Timoshenko's, which uses the curvature the guess's bending moment would cause instead. The exact load is 3.225 EI₀/L². a half sine: 8.256 by Rayleigh and 3.745 by Timoshenko; its own sag shape: 8.041 by Rayleigh and 3.712 by Timoshenko; a mid-span sag: 8.875 by Rayleigh and 3.847 by Timoshenko; a parabola: 6.600 by Rayleigh and 3.492 by Timoshenko. Both are upper bounds, and from the same shape the second is never the worse of the two.

The bound from underneath

Rayleigh's quotient turns a guessed shape into a buckling load that is always too high. Divide the same guess differently — use the curvature its bending moment would cause instead of its own — and a parabola that was 21.6 per cent high is 1.3 per cent high. Add one more number that needs no guess at all and the load is caught from below as well, which is the only side of it an amplifier can safely use.

stability · Stability energy The rib yields long before it would buckle. The largest combined stress, axial plus bending, anywhere in a 60 m two-pinned parabolic rib at a 12 m rise (EI 250,000 kN·m²) with a 12,000 mm² section 1,000 mm deep, against a factor on 15 kN/m of dead load over the span and 6 kN/m of live load over the left half together. Dashed: first order; solid: second order; dotted: the 355 N/mm² yield stress. First order, the rib reaches yield at a factor of 1.73; second order, at 1.18. The same load pattern, spread uniformly, would bifurcate the rib at a factor of 2.97, the right-hand edge — which is the number the eigenvalue reports and a number the rib never reaches.

The weight that bends a rib it cannot bend

A parabolic rib carries its own uniform dead load as pure thrust, with no bending in it at all. Put a live load on half the span and the rib bends in the shape of its own buckling mode from the first kilonewton, and the whole thrust, most of it from the dead load, multiplies that bending. The rib never reaches its buckling load. It yields well short of it, at a load its first-order check says it can carry.

stability · Arch buckling The deck moves the frames' feet, and the chord goes with them. A chord 24 m long, EI 2.52 × 10¹² N·mm², held by U-frames of 1,400 N/mm every 4 m, carrying 930 kN, in plan, sideways movements drawn 6 times the scale of the length: each frame's foot pushed 7.5 mm inward by its cross-girder's rotation under the deck (arrows), the chord's movement under that push alone (solid), and under the worst bow alone, four half-waves at 12.0 mm (dashed). The chord follows the push, and under its compression goes a little further — 9.1 mm at its middle frames against the 7.5 mm the feet moved — because a chord 24 m long is far softer sideways than the frames holding it; the frames that carry most are the end ones, beside the chord's fixed ends.

The frames the deck pushes first

A half-through girder's U-frames are the ends of the cross-girders that carry its deck. Load the deck and every cross-girder sags, its ends rotate, and every frame's foot swings inward by millimetres before the compression chord has asked the frames for anything. It looks like a second imperfection on top of the chord's bow. It is a much weaker one: the frames all move together and the chord simply goes with them, so a 7.5 mm push adds 1.4 kN to frames that the bow loads with 13.6, and stiffer frames make it smaller, not larger. What the push leaves is mostly in the frames' own members, not in what they ask of the chord.

stability · Continuous restraint Where the outlet sits decides whether the roof ponds at all. The load the water puts in the more heavily loaded member of a square bay, over what the same head of water would put in it on a rigid roof, against the bay's side, with girders and joists each sized to deflect L/240 under 1.5 kN/m². With the level fixed by a scupper it is 2.01 at 12 m and 3.92 at 18 m, and the bay ponds at 24.5 m. With a drain on the girder line it is 1.24 and 1.41; with a drain at the bay's middle, 0.82 and 0.75, falling as the bay grows, because the water's surface goes down with the roof.

The roof that drains towards its sag

The depth of water a ponding check takes as given is set by the drainage: an overflow's height above the roof, plus the head a storm needs to leave through it. Which head a roof can carry depends less on how stiff it is than on where the overflow sits. With the level fixed by a scupper in the parapet, a 12 by 9 m bay allows 85 mm of water and an 18 m square bay 39. With a drain at the middle of the bay, whose inlet goes down as the roof sags, the same bays allow 182 and 205 mm — more than a rigid roof would — because the water's surface follows the sag down instead of filling it.

deflection · Serviceability A milliradian of slack costs the brace more than a milliradian of crookedness. For the 24 m plate girder under a uniform load, braced at midspan and 6.0 m either side by braces of 5,691 kN·m/rad — 2.0 times the 2,846 that just reaches 5,627 kN·m — erected with 5.0 mrad of initial twist: the moment in the middle brace when the girder reaches 5,627 kN·m, against each brace's slack (solid), and against the same amount added instead to the girder's initial twist, the braces tight (dashed). With tight braces and no extra twist, 34.1 kN·m; each milliradian of slack adds 8.0, each milliradian of initial twist 6.8. At 10.0 mrad, 114.0 against 102.4.

The braces with slack in their bolts

A cross-frame bolted into clearance holes resists nothing until the girder it braces has twisted far enough to take up the clearance, and then resists fully. It is a two-way brace that starts late. The girder, erected a little crooked, twists freely into its braces while the load is still small, and from then on the braces hold a girder that is more crooked than the one they were fitted to. The slack acts as an imperfection, and a costly one: each milliradian of it adds more to the middle brace's force than a milliradian of extra crookedness would, and no stiffness takes it back out. On a 24 m plate girder braced at twice the stiffness it needs, three milliradians of slack — a pair of ordinary clearances over its depth — raise the middle brace's force by 70 per cent.

stability · Beam bracing How much of the drain has to work. The load in the more heavily loaded member, with the roof's own 0.5 kN/m², over its 2.0 kN/m² capacity, for a 12 by 9 m bay in a 150 mm/h storm, with a drain at its middle that passes the storm at 30 mm when clear and a 400 mm scupper at its edge whose sill is 50 mm above the supports, against the share of a clear drain's flow the drain still passes. Members at L/180, L/240, L/360 and rigid: with the drain fully blocked, 1.25, 0.99, 0.84, 0.66; a tenth open, 0.80, 0.77, 0.71, 0.62; half open, 0.44, 0.45, 0.46, 0.48; clear, 0.37, 0.37, 0.38, 0.40. A softer roof is the worse with the drain blocked and very slightly the better once about a fifth of it works.

The drain that is mostly blocked

A roof's ponding check is made with its primary drains blocked, and a drain is seldom blocked outright: leaves and gravel choke it to a fraction of its flow. A fraction turns out to be nearly enough. On a 12 by 9 m bay sized to L/240, a drain passing a seventh of its clear flow still takes the whole of a 150 mm/h storm, because the water standing over it at the bottom of the sag is the bay's own deflection plus the level. A softer roof's drain works harder still. Only near complete blockage does the scupper take over, and only there does stiffness help.

deflection · Serviceability Where the knees begin to alternate. The shakedown factor (solid) and collapse factor (thin) of a portal 10 m wide and 5 m high, plastic moment 200 kNm in the columns and 200 in the beam, on pinned bases, with 160 kN of gravity that comes and goes and a 1,000 kN roof carried at its knees, against the peak of a wind that reverses, first-order (dashed) and amplified. First-order, the shakedown and collapse factors are one line until the knees alternate, from a wind of 110 kN; amplified, from 90 kN. At 60 kN: shakedown 0.727 first-order and 0.692 amplified, collapse 0.727 and 0.692. At 120 kN: shakedown 0.533 first-order and 0.488 amplified, collapse 0.571 and 0.539. At 200 kN: shakedown 0.348 first-order and 0.326 amplified, collapse 0.400 and 0.366.

The roof that makes the wind worse every storm

A portal frame's shakedown check is made on its elastic moments, and the elastic moments a heavy roof sways are larger than first-order analysis says: each storm's wind is amplified by 1/(1 − N/Ncr), on its way out and on its way back. On pinned bases under a 1,000 kN roof that takes the wind at which the knees start to alternate from 110 kN down to 90, and the shakedown load down by 4.8 per cent where gravity leads and 8.4 where the wind does. Fixed bases are worth more under a heavy roof than without one, but by half again, not twice.

materials · Shakedown

Named alongside it

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

ImperfectionSecond-orderBucklingBracingCritical loadLoad pathMode shapeServiceabilityStabilityContinuous restraintCreepDamping

All concepts