Structural Engineering & Statics
where the load goes
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27 September 2026

16 essays on stability, dynamics, equilibrium, connections, materials, deflection, sections and stress, structural form and internal forces

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. Stability

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.

9 figures 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. Dynamics

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.

7 figures Weakest the day it is finished. The factor of safety against tipping of a 15,000 kN tower on a 10 m square base over soft clay (undrained strength 35 kPa, gaining 0.22 of the effective stress the tower puts into it, consolidating with a 90 per cent time of 7.1 years), through twelve years, for construction times of three months, one year and four years; its wind push grows as it rises. Built in three months, it bottoms out at 1.14 on the day it is finished; built in one year, it bottoms out at 1.33 on the day it is finished; built in four years, it bottoms out at 1.60 on the day it is finished. After that every curve climbs toward 1.93, the factor once the clay has consolidated. Had the finished weight arrived in an instant the factor would be 0.92; on ground that could not fail it would be 3.00. Equilibrium

The ground that arrives after the weight

A block's resistance to tipping on ground that can yield is a parabola in its weight, with its peak at half of what the ground can bear. On soft clay the ground's capacity is not fixed: it grows as the clay drains under the weight, and it grows years after the weight has arrived. A tower on such ground is at its weakest on the day it is finished — 1.33 against the 1.93 it will have once the clay has drained — and a silo filled in a week walks over the top of the parabola and down the far side, which is the other way a structure can be built.

6 figures A butt throat is strongest a little off square. The resultant a millimetre of throat can carry in S355 against the direction of the load, from straight across the weld (0°) to straight along it (90°). The fillet's falls throughout, from 385 to 314 N/mm². The butt's rises from 441 to 478 at 22.7°, because its cap limits only the normal stress and a little shear costs it nothing, and then falls to 314. So the butt's advantage over the fillet is 1.15 straight across, 1.29 at 22.7°, and none straight along. Connections

The shear that helps a butt weld

A partial-penetration butt throat pulled straight across its weld is capped at 0.9fu, which in S355 makes it 15 per cent stronger than a fillet throat. Add a little shear along the weld and the butt throat gets stronger, not weaker, because the cap limits only the normal stress — until the shear is 42 per cent of the pull, where it is 29 per cent stronger than the fillet and carries its largest resultant, 478 N/mm² against 441 straight across. Along a moment connection's web weld the mix of pull and shear sweeps through that peak, and in the direction that is all shear a single-sided throat is as good as two.

6 figures One steel, one crack, three plates. The strip-yield failure stress of plates with an edge crack, at −20 °C in a grade 355 steel whose master-curve reference temperature is −38 °C, against the crack length on a logarithmic scale, for plates 12, 25 and 60 mm thick, each with the toughness of its own crack front at the 5 per cent level: 88, 77 and 66 MPa√m. At a 20 mm crack the 12 mm plate fails at 267 N/mm², the 25 mm plate at 242 and the 60 mm plate at 214. Dashed, the simple check with the 25 mm toughness, the lower of fracture and yield: 274 N/mm² at the same crack, for every thickness. Materials

The toughness that belongs to the plate

A steel's cleavage toughness is measured on a specimen 25 mm thick, and a crack through a plate starts at the weakest spot its front passes through — so a crack through a 60 mm flange samples more weak spots than the test did, and is less tough, and one through a 12 mm web samples fewer. Put each plate's own toughness into the strip-yield assessment and the simple check, which overstates a cracked plate by 23 per cent at the specimen's thickness, overstates a 60 mm flange by 36 per cent and a 100 mm plate by 45, while a thin web pays much of the strip-yield debt back.

6 figures Fully stressed, with a choice of diagonals. Two trusses sized so that every member is at the allowable stress under the full load, each member drawn as wide as its area, tension and compression in two colours. Above, an eight-panel Pratt truss as deep as a panel is long, with a counter-diagonal in each of its six interior panels, loaded by 10 at every bottom joint, found by resizing and reanalysing until nothing changes; below, the same truss without its counters. The counters in the two middle panels have shrunk to nothing (dotted) and the four nearer the supports have stayed, working in compression beside the diagonals in tension. The truss that kept them needs 1,200 units of steel against the Pratt's 1,220, and deflects 24.0 at mid-span against 26.0. Deflection

The truss whose forces follow its sections

In a determinate truss each member's force is fixed before its section is chosen, so sizing for strength and sizing for stiffness can be done in either order. Put a counter-diagonal in every panel and they cannot. The fully stressed design becomes an iteration that starves some counters to nothing and keeps others, lands on a different truss from every start, and — whichever it lands on — weighs the same and deflects the same. Then enlarge one group of members to stiffen it, and a vertical nobody touched is overloaded by 58 per cent.

6 figures The film over a keyed shaft, and the corner it climbs. Contours of Prandtl's stress function — the height of a soap film blown up over a hole of the section's shape — for a 50 mm shaft with a keyway 14 mm wide and 5.5 mm deep, its bottom corners rounded to 0.40 mm; the shear stress is the slope of the film, so it is highest where the contours crowd. Left, the whole shaft: the contours follow the circle and bend round the keyway, and the keyway costs 11 per cent of the plain shaft's torsional stiffness. Right, a window 3.0 mm across around the keyway's corner, solved on its own fine grid with its edges taken from the whole film: the contours crowd into the fillet, where the stress is 2.81 times the plain shaft's surface stress under the same torque. Sections and stress

The corner a soap film cannot finish

Cut a keyway into a round shaft and its torsional stiffness falls by only 11 per cent. The stress does not fall; it moves. It leaves the surface beside the keyway's mouth almost unloaded and crowds into the keyway's bottom corners, where a soap film stretched over the section would have to stand vertical. Computed on finer and finer grids, a sharp corner's stress never settles — it grows by the cube root of two every time the grid is halved — and a fillet of a few tenths of a millimetre is what turns an infinite answer into three times the plain shaft's.

5 figures Three members after a diagonal goes. The forces in three members of the counter-braced truss when the diagonal 9–2, carrying 242 kN, is removed instantaneously, with 2 per cent damping, over one and a half of the damaged truss's first periods (0.57 s); dashed, the static force each settles to. The counter-diagonal 1–10 goes from −112 kN to −354 kN and peaks at −491 kN, 1.57 times its change. The bottom chord 3–4 ends exactly where it started, 750 kN, and on the way peaks at 1,115 kN. The bottom chord 2–3 settles lower, at 556 kN, after a first swing the other way, to 905 kN. Structural form

The factor of two belongs to one mode

A member that fails suddenly hands its force to the structure around it all at once, and the convention is to double the static answer: a load applied suddenly to a spring overshoots to twice its static deflection. A truss is not one spring. Take a diagonal out of a counter-braced truss in an instant and some members swing to three times their change of force, one swings the wrong way first, and a bottom chord whose force does not change at all passes through half as much again as it carries — because every mode overshoots by two, at its own time, and a member is a sum of modes.

5 figures A soft layer over stiff ground, and a moment neither has. The bending moment down a 20 m pile of bending stiffness 120 MN·m² with a free head, pushed sideways by 150 kN (its characteristic length in ground at 5.00 MN/m³ being 1.89 m), in three grounds: all of it at 5.00 MN/m³; all of it at 0.50 MN/m³; and soft ground 2.0 characteristic lengths (3.8 m) thick over the other (dotted: the interface). The largest moments are 219 kN·m, 346 kN·m and 427 kN·m; the head deflections 20.4 mm, 81.4 mm and 61.6 mm. The layered ground gives a larger moment than either ground on its own: the pile bends as a pile in the soft layer would, and then meets ground that holds it. Internal forces

The layer a pile feels

A pile pushed sideways in uniform ground has one characteristic length, the fifth root of its stiffness over the ground's, and every answer is a multiple of it. Put a soft layer over stiff ground and that length stops existing. The head feels the top of the ground in proportion to the square of its own deflection, so an average weighted that way gets the head deflection within eight per cent. It gets the moment a quarter too low: with soft ground two characteristic lengths thick over stiff, the pile bends harder than it would in either ground alone, and the peak sits where the stiff ground takes hold.

6 figures 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. Equilibrium

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.

5 figures What the actuator pushes against decides its limit. The largest feedback gain, as the damping it would add to the bare mode, at which the loop stays stable, against the delay as a fraction of the period, on a logarithmic scale, for a mode of 1.00 per cent damping carrying a tuned mass of 2.00 per cent of its modal mass at Den Hartog's tuning. Pushing against the ground, fed by the structure's velocity: 2.41 at a delay of 0.05, 0.25 at 0.2. Pushing against the tuned mass, fed by the same velocity: 0.0156 at almost no delay, rising to 0.091 at a quarter of a period. Pushing against the tuned mass, fed by the velocity across it: unlimited with no delay, 0.048 at 0.05, 0.0073 at 0.2. Dynamics

The actuator that pushes against the mass

An actuator that damps a structure by feeding back its velocity has to push against something. Against the ground, it is stable up to large gains until a delay of a quarter period removes its damping. Against a tuned mass — the arrangement meant to keep a passive damper underneath — the same feedback goes unstable at a gain of a per cent and a half with no delay at all, makes the structure worse than the passive mass at every gain below that, and drives the mass three or four times as far. What limits it is not when it acts but what it pushes against.

5 figures A column curve with no plateau. The tangent-modulus buckling stress of perfect pin-ended columns of an aluminium alloy of 0.2 per cent proof stress 250 N/mm² and modulus 70,000, as a share of the proof stress, against the slenderness λ̄ — the square root of the proof stress over the Euler stress — for Ramberg–Osgood exponents of 5, 10, 20 and 40; dashed, the sharp envelope a material with a plateau at the proof stress would give, the lesser of the proof stress and Euler's. n = 5: 1.54 at λ̄ = 0.2, 0.66 at 1, 0.41 at 1.5; n = 10: 1.16 at λ̄ = 0.2, 0.74 at 1, 0.45 at 1.5; n = 20: 1.04 at λ̄ = 0.2, 0.82 at 1, 0.45 at 1.5; n = 40: 1.00 at λ̄ = 0.2, 0.88 at 1, 0.45 at 1.5. Stocky columns buckle above the proof stress, because the curve goes on rising past it; slender ones follow Euler; in between every curve sags below the envelope. Stability

The column with no plateau

A steel column's curve is the lesser of two lines, its yield stress and Euler's, because mild steel keeps its whole stiffness up to yield and then has a plateau to stop on. Aluminium and stainless steel have no plateau: their stiffness starts falling well below the proof stress and never stops. Their column curves are not the steel curve moved; they are a different shape — four tenths below the corner for a stainless steel, above the proof stress for a stocky column — and the shape is set by two numbers, only one of which describes the knee.

6 figures How far a local load spreads, and what is nearest its limit. Along a steel-faced polyurethane cladding panel, 0.5 mm faces of 320 N/mm² steel on an 80 mm core crushing at 0.12 N/mm², from the middle of a strip load of 3.0 N per mm of width spread over 10 mm: the face's deflection over its value under the load, the core's compressive stress over its crushing strength, and the face's bending stress over its yield stress. The load spreads over a length set by the fourth root of the face's bending stiffness over the core's, 1/β = 20.5 mm; the deflection is 1.44 mm under the load and changes sign beyond 48 mm. The core is at 0.60 of its crushing strength and the face at 0.89 of its yield stress: this panel's face yields at 3.36 N/mm and its core crushes at 5.00. Sections and stress

The face that dents and the core that crushes

A sandwich panel carries bending as a couple between its faces, and that is a calculation about the whole panel. Put a local load on one face — a foot, a fixing, a dropped tool — and the face becomes a thin beam on a soft bed, spreading the load over a length the fourth root of their stiffnesses sets. Then either the face yields and dents, or the core crushes beneath it, and which comes first is decided by one thickness: below it the face gives, above it the core does, and a steel-faced roof panel is on the wrong side of it for anyone who walks on it.

5 figures A deck holds the middle of the span. The twist along an 8 m open section on forks, free to warp, carrying 12 kN/m at 75 mm from its shear centre — 900 Nmm of torque per mm of span. With nothing fastened to it the beam twists 6.25° at mid-span; a deck resisting the top flange's rotation at 5.0 kNm per metre per radian holds it to 4.20°, and one four times as stiff to 2.10°. The deck works where the beam is weakest, in the middle; near the supports the beam's own torsional stiffness does most of the work whatever is fastened to it. Deflection

A deck is a spring, not a wall

An open-section beam loaded off its shear centre twists, and the usual reassurance is that the deck fastened to its top flange will stop it. The deck resists the flange's rotation with a stiffness per metre of span, and that stiffness has to be compared with the beam's own. On an 8 m beam a screwed deck of ordinary stiffness removes a third of the twist; on a 16 m beam the same deck removes three quarters, because the beam's torsional stiffness falls with the square of its span and the deck's does not.

6 figures Ductile until it was welded. A 2.0 m 6082-T6 member pulled in tension, as supplied and with one weld across it whose heat-affected zone is 60 mm long: the load per unit of parent section against the member's overall strain, to the maximum load, where the weaker part begins to neck. The plain member reaches 310 N/mm² at 7.0% strain, 140 mm of stretch. The welded one reaches 185 N/mm² at 0.62%, 12.3 mm — 8.8% of the plain member's, and the parent never reaches its proof stress of 260 N/mm². (uniform elongations assumed: 7.0% for the parent, 12% for the zone) Materials

The soft zone that takes all the stretch

The metal beside a weld in heat-treated aluminium is weaker than the rest of the member and more ductile, and it is tempting to read the second fact as a consolation. It is not. A member welded across stretches only where it is weak, and the weak zone is sixty millimetres long; a two-metre 6082-T6 tie that would extend 140 mm before necking extends 12 mm once welded, and the parent metal never yields at all. What decides it is one ratio, the zone's ultimate strength over the parent's, and the ductility returns only when that ratio reaches one.

6 figures The modes that tilt, and the ones that do not. Johansen's mechanisms for a 12 mm bolt with washers in single shear through 40 and 40 mm of timber of density 350 kg/m³, each with the rope term the fastener's axial resistance of 6.64 kN adds to it. Modes a and b, where the fastener only translates, gain nothing; the modes in which it tilts gain a quarter of its axial resistance, capped at a quarter of their own Johansen value. The joint's capacity rises from 5.02 kN (mode c) to 6.28 kN (mode c), 25 per cent. Connections

The pull that Johansen left out

Johansen's mechanisms treat a timber fastener as a beam in a bed of crushing wood, bent and pushed sideways and nothing else. A real fastener that tilts across the joint is also pulled along its own axis, and if a washer or a thread resists the pull, it clamps the two members together and adds to what the joint can carry. The addition is a quarter of the axial resistance, it goes only to the modes in which the fastener tilts, and it is capped by fastener type — nothing for a dowel, a quarter for a bolt, all of it for a screw — which is how a screw keeps gaining strength past the thickness at which Johansen's capacity stops.

6 figures

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