Structural Engineering & Statics
where the load goes

Where the load goes.

Every load applied to a structure reaches the ground by some route, and choosing that route is most of what design is. This is a collection of essays about tracing it — one idea at a time, illustrated to the point where the argument becomes visible, with every figure solved rather than drawn to look convincing.

A Pratt truss of 6 panels. A Pratt truss under equal panel-point loads. The joint equilibrium equations were assembled and solved; 10 members came out in tension, 9 in compression and 2 carrying nothing.
Fig. 1 A Pratt truss under load. The colour of each member is not a convention applied by hand: the joint equilibrium equations were assembled and solved, and the members shown in tension came out with a positive force. Note which diagonals pull and which verticals push — that pattern is the whole reason this arrangement is named after somebody.

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451 essays across nine fields, built around 313 ideas with a ladder of their own and 1209 named objects threaded through them. Browse every essay, or by thread, or by the figure family that drew it, or by search. There is also a list of what this site refutes, which is the shortest route in for anybody who has been taught the subject already.

Capacity against load direction, layout by layout. The same capacities as a function of direction on straight axes, for a load through (150, 150) mm. Three rows of two is weakest at 138°, 135.8 kN. Two rows of three is weakest at 312°, 135.8 kN. A ring of six, r = 70.4 mm is weakest at 132°, 150.3 kN. The curves cross: no layout is strongest in every direction, and the one to choose is the one whose lowest point is highest. Connections

The bracket pushed from the wrong side

A six-bolt bracket checked for a load straight down carries 198.5 kN. Push the same load through the same point at 138 degrees and it carries 135.8 — the direction changes the torque as well as the shear, and the worst direction is one no drawing shows. The capacity of a group whose load can turn is a closed curve, the bolt that governs it changes as the load turns, and what the curve rewards is not a larger polar moment but a smaller distance to the furthest bolt.

6 figures Settling down or walking away, cycle by cycle. The total plastic hinge rotation of the beam after each cycle of loading — span 1, both, span 2, neither — with the midspan load at 0.98, 1.02, 1.05, 1.10 times the shakedown load of 126.3 kN. At 0.98 it stops at 0.59 mrad. At 1.02 it grows 4.57 mrad a cycle. At 1.05 it grows 11.43 mrad a cycle. At 1.10 it grows 22.86 mrad a cycle. Nothing collapses in any single cycle; above the shakedown load the beam walks. Materials

The load it can carry once

A two-span beam whose loads come and go span by span collapses at 150 kN under any one arrangement, and walks at 127. Between the two it can carry every arrangement once and none of them forever: each cycle leaves a few more milliradians of rotation at the support and a midspan fifteen millimetres lower. Melan's theorem finds the limit as the last residual moment line that fits, Koiter's as a mechanism no single load state can drive, and a cycle-by-cycle calculation walks exactly where both say it will.

5 figures Three routes to one deflection, and the one that is wrong. Two 2500 mm bars of 250 MPa proof stress meeting at a loaded apex, the drop of the apex against the load. The line is geometry: each bar's extension from its own stress, divided by the sine of its slope. The dots are the derivative of the total complementary energy with respect to the load, and lie on it. The dashed line is the derivative of the strain energy — Castigliano's theorem applied to a material that is not linear — which leaves the truth by ten per cent at 99 kN and at 170 kN gives 308 mm for a deflection of 46.0. Deflection

The other area under the curve

Castigliano's theorem says a deflection is the derivative of the strain energy with respect to the load, and it is true only while the material is linear. Past that, the right energy is the area on the other side of the stress–strain curve. On two aluminium bars at their proof stress the strain energy gives a deflection four times too large, and on a redundant truss minimising it picks a set of forces in perfect equilibrium that no deformed shape can produce.

5 figures The section, once the joint has supplied the fourth equation. Everything above a cut through panel 3. Four severed members; three equations. Moments about the mid-joint of the horizontal remove both diagonals and leave both legs. Moments about the point where the legs' lines meet, 36.0 m up, remove both legs and leave both diagonals. With the joint's result that the diagonals are equal and opposite, that second equation has one unknown: 20.83 kN in each diagonal. The legs follow: 26.1 T and 86.3 C. Equilibrium

The cut that needs a joint first

The method of sections works because a cut through three members leaves three unknowns and a point about which two of them have no moment. A K-braced tower has no such cut anywhere: every section severs two legs and two diagonals. One joint in the middle of a horizontal supplies the missing equation, and only in that order does each step have one unknown — after which the diagonals turn out to be carrying not the shear but the moment about the point where the legs would meet.

6 figures 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. Equilibrium

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.

6 figures The damping that stops helping. The storey drift's white-noise root-mean-square as the bearings' damping is raised from 2 to 50 per cent, each divided by the exact value at 2 per cent. The classical analysis promises that every increment helps, and at 50 per cent predicts 0.20 of the lightly damped drift. The exact analysis flattens: 0.34 at 20 per cent and 0.29 at 50, because the damping force at the bearings is transmitted into the superstructure's own mode, which the classical analysis has decoupled from it. Dynamics

The damping that belongs to no mode

Give every mode its own damping ratio and throw the rest of the damping matrix away, and an isolated building's periods and damping come out right to within a fifth of a per cent. Its storey drift at the superstructure's frequency comes out six times too small. The bearings' dashpot pushes on both modes at once, and the more of it there is the less extra damping buys: the classical analysis promises the drift keeps falling, and it stops.

6 figures

Everything added recently

The fields

Six of them follow the order a load travels — it is applied, resisted by a form, carried as an internal force, met by a section, and then two things can go wrong. The other three are not steps on that route at all: each removes an assumption the first six are resting on. All nine.

Equilibrium

Nothing moves, so everything adds to nothing — and the free body decides what everything means.

47 essays

Structural form

Trusses, cables and arches: the shapes that carry load by geometry rather than by bulk.

54 essays

Internal forces

What a cut reveals — shear, moment and axial force, and the diagrams that track them along a member.

56 essays

Sections and stress

How a cross-section resists a moment, and why where the material sits matters more than how much there is.

53 essays

Stability

Strong enough and still falling over: buckling, slenderness, and loads that make themselves worse.

51 essays

Deflection

Stiffness is not strength. What moves, how far, and why the answer goes as the fourth power of the span.

47 essays

Materials

The assumption every other field rests on: that stress is the modulus times the strain, without limit and in both directions. It is not, and here is what happens instead.

48 essays

Connections

Structures do not fail in the middle of a member. They fail where two of them meet — which is the one place the theory behind every other field explicitly does not hold.

48 essays

Dynamics

Every other field assumes the load arrives slowly and stays. When it does not, the same structure gives a different answer — twice as large for a load put down suddenly, twenty-five times for one applied at the rate the structure likes, and unbounded for one the structure's own motion creates.

47 essays

Threads running through

themes, not chapters

The load must go somewhere

Every force applied to a structure reaches the ground by some route. Choosing that route is most of what design is, and tracing it is most of what analysis is.

267 essays

Geometry beats material

Moving the same steel further from the neutral axis, or making the truss deeper, buys more than making the steel stronger. Shape is the cheap variable.

209 essays

The statics of things that do not move

Every result here is obtained by imagining a motion that does not happen and insisting the sums cancel. Nothing in the subject is measured directly.

66 essays

One support too many

Indeterminacy: more restraints than equations. It buys robustness, costs a stiffness calculation, and makes a structure sensitive to things statics cannot see.

117 essays

Which failure arrives first

A member can yield, buckle, deflect too far or shear through. The governing limit state is rarely the one being thought about.

278 essays

Drawing as calculation

Force polygons, funicular shapes and Cremona diagrams solved real structures for a century. The drawing was not an illustration of the answer — it was the answer.

129 essays

The load that will not hold still

Statics assumes a load arrives slowly and stays. Almost none of them do, and the same structure answers differently when they do not — bounded, if at all, by a damping ratio nobody designed.

74 essays

Scale changes everything

Weight grows as the cube and strength as the square. A large structure is not a small one enlarged, and the difference is why bridges and beetles are built differently.

106 essays