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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728 essays across nine fields, built around 313 ideas with a ladder of their own and 1295 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.

A floor that shears lowers the storey's buckling load before the core does. The buckling factor of a storey of eleven lines of leaning columns, ten bays, each column carrying 3,000 kN over 3.6 m and built a three-hundredth out of plumb, held by a floor to a core sized for a buckling factor of 4.0 were the floor rigid, against the floor's shear stiffness per bay as a multiple of a column's sway softening P/h, logarithmic, with the core at one end, in the middle, or split between both ends; dotted, the rigid floor's 4.0. At 100 times the factor is 1.52, 2.80 and 3.02; at 1,000 times, 3.49, 3.84 and 3.88; a concrete slab is several thousand times. Stability

The floor the columns lean on

A storey of pin-ended columns, every one a little out of plumb, leans on its floor, and the floor leans on a core. Count the floor as rigid and the core needs the sum of the columns' demands. A floor that shears in its own plane is a brace in series with the core, and the far columns lean on it before their push arrives. With a floor a hundred times stiffer per bay than a column's own sway softening, the storey's buckling factor falls from 4.0 to 1.5 and the core is asked for twice as much.

6 figures A passage opposite a corner, and opposite a flat. The section of a 50 mm shaft with a hexagonal bore whose flats are 8 mm from the axis and whose corners are rounded to 1.5 mm, with a passage 5 mm across centred 13 mm from the axis, opposite a corner of the bore (left) and opposite a flat (right), and the largest shear stress each leaves round the passage and round the bore, as a multiple of the plain round shaft's surface stress under the same torque. Opposite a corner the passage peaks at 1.25 and the bore at 1.15; opposite a flat, 1.19 and 0.54. With no passage the bore's corners carry 0.54. Sections and stress

The bore with corners and the passage beside it

A hollow shaft's bore is not always round: a hexagonal socket or a square drive turns the island in Prandtl's soap film into a polygon, and every corner of the bore is a corner of the steel that the film cannot finish. Its corners carry twice the stress of its flats. A passage drilled through the wall beside it can sit opposite a corner, where it doubles that stress again, or opposite a flat, where the bore does not notice it at all.

6 figures Where along its front a surface crack starts to cleave. Where along the front of a surface crack 8 mm deep in a 40 mm plate at −60 °C, 22 degrees below the steel's reference temperature cleavage starts, as a probability per millimetre of front, at the stress that gives a 5 per cent chance of cleavage, against the position on the front, for cracks 80 mm long, 40 mm long and 16 mm long at the surface. The middle third of the front, 60° to 120°, holds 76 per cent, 69 per cent and 28 per cent of the chance. Materials

The front that counts for half its length

A ferritic steel's cleavage toughness depends on how much crack front it has to sample, and for a crack through a plate the front is the plate's thickness. The cracks inspections find are not through cracks; they are thumbnails at weld toes, whose curved front sees a stress intensity that varies round it. Weighted by the weakest link, the front counts for a little over half its length — so counting all of it is safe by 8 per cent, and counting none of it, as the 25 mm toughness does, is unsafe by up to a fifth for a long shallow crack.

6 figures Where the shelter is decides whether the leading row works harder. The share of the cap's shear the leading row and the last row of three rows carry, as a multiple of the average pile's, against the thickness of soft ground over stiffer, in characteristic lengths, with the row factors applied at every depth (solid) or in the soft layer only (dashed). With the factors at every depth the leading row carries 1.24 and the last 0.83 at 2.0 lengths; in the soft layer only, 1.07 and 0.96. In uniform ground the usual reading gives 1.35 and 0.75. Internal forces

The shelter that stops at the interface

Piles in a group shelter one another, and practice counts the shelter as a factor on each row's soil, calibrated on groups in uniform ground. Put the same group in soft ground over stiff and the calibration has to be read one of two ways — the shelter at every depth, or only where the ground is soft — and the two readings disagree about which row works hardest. The first gives the leading pile 17 per cent more moment at the cap; the second gives the last pile 24 per cent more in the ground.

6 figures The silt hands the head back over decades, a season at a time. The basement is under five storeys, its water table 2.0 m above the design head and swinging ±0.5 m over the year, highest in April; pumped drains under its slab start 30 times as conductive as the ground. The head under the slab above the design head, against the years since the drains were new, for drains silting with time constants of 4, 8 and 16 years; dots, a standpipe read every October on the 8-year drains. With those drains the head reaches flotation in the wet season of year 24.1; the reading first exceeds 0.25 m in year 14.8, 9.4 years before. Equilibrium

The drain that fails a season at a time

A basement held at its design head by drains and a pump can fail suddenly, when the pump stops, or slowly, when fines settle in the drains and the head under the slab creeps up while the pump keeps running. The slow failure gives a standpipe read once a year a decade of warning, in its own time constants — if it is read in the wet season. And long before the silt floats the building on its own, it has made every stop of the pump more likely to.

6 figures Taking hold of a swinging mass throws it. The largest stroke of a tuned mass of 2 per cent, as a share of its limit, after the free-mass law takes hold of it while it swings on its own spring and damper in a steady storm at the structure's frequency, against the instant in the structure's period at which the law takes hold; dashed, the law waiting for the stroke to cross zero, and for the mass to come to rest relative to the structure. Depending on the instant the stroke reaches 1.39 to 3.12 times the limit; waiting for zero stroke, 2.81; waiting for rest, 2.27. Dynamics

Taking hold is harder than letting go

A tuned mass driven by a law that cancels its spring is safe to let go of: the transient is the same whenever the switch falls, and the passive mass takes over. Switching the law back on is the reverse operation and behaves nothing like it. The mass the spring was holding is thrown up to three times past its stroke limit, by an amount that depends on the instant, and a switch that is allowed to come back in a storm near its threshold comes back hundreds of times.

6 figures

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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.

81 essays

Structural form

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

80 essays

Internal forces

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

81 essays

Sections and stress

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

81 essays

Stability

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

81 essays

Deflection

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

81 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.

81 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.

81 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.

81 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.

450 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.

329 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.

97 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.

186 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.

495 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.

162 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.

150 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.

164 essays