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.

Start anywhere

461 essays across nine fields, built around 313 ideas with a ladder of their own and 1212 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.

The two knees the environment takes away. The design S-N curve for a category 71 detail and the two shapes a corrosive environment leaves. In air the slope is three to the constant-amplitude limit at 52.3 N/mm², five to the cut-off at 28.7, and nothing below it. With the cut-off removed the second branch continues. Under free corrosion there is one slope of three all the way down and no knee at all. The faint lines are the five bands of the traffic: three of them sit below the cut-off — an ordinary lorry, a van and a car, 90 per cent of the crossings — which is why they do nothing to a detail in air and something to every other curve on the figure. Materials

The cut-off belongs to the water

An S-N curve's cut-off is the most consequential thing on it: on a category 71 detail under ordinary bridge traffic it deletes ninety per cent of the crossings and leaves two bands doing all the damage. It is a property of steel in air. A crack tip that seawater or de-icing salt can reach has no threshold, no endurance limit and no knee, and the same bridge's life runs from 300 years to 45 depending on which of six defensible calculations is asked.

7 figures The magnification a crack sees is a ratio, not a depth. The stress magnification at a weld toe against the crack's depth as a fraction of the plate's thickness, on BS 7910's two-branch fit. It is a function of a/t alone, because a weld's own size scales with the plate it is on, so the elevated field is geometrically similar. The dots are the same absolute starting flaw of 0.20 mm in plates of 12, 16, 25, 40, 60, 80, 100 mm: the flaw does not move and its magnification runs from 1.81 to 3.50. A fixed flaw in a thicker plate is a smaller fraction of it, which puts it deeper inside the raised field rather than nearer the edge of it. Materials

The rule that points sideways

Every fatigue code puts the same detail in a thicker plate into a lower category, by a factor of (25/t) to the power 0.2, and explains nothing. It is a strange rule: a detail's strength made to depend on a dimension at right angles to the crack. Integrate a crack through a weld toe's own stress field and the rule falls out — same form, same sign, and an exponent of 0.13 against the design code's 0.2. Remove the toe's magnification and the effect reverses.

5 figures The same cycles, and two different lives. Two sequences made of exactly the same cycles — blocks of 6.00M at 40 N/mm² alternating with blocks of 0.22M at 90, on a detail starting with a 0.50 mm flaw. Taking the small cycles first, the crack reaches its critical 87.7 mm after 12.36M cycles; taking the large ones first, after 6.36M. Miner's sum at the moment of failure is 0.99 for the first and 0.68 for the second, so a rule that predicts failure at a sum of one is right to within a per cent about the first and 48 per cent unconservative about the second. The mechanism is on the axes: ΔK rises with the crack, so a large block met late finds a longer crack and does more with it. Materials

The record played backwards

Miner's rule adds damage, and a sum has no order. A crack does, twice over: a large block met late finds a longer crack and does more with it, and an overload leaves a plastic zone that slows everything after it. The same cycles rearranged fail at 12.4 million or at 6.4, and a Miner sum that is right to one per cent about the first is out by half about the second. One cycle in four million can add fifty-four per cent to a life.

5 figures Half the life is spent growing the first half-millimetre. The crack length against time for a detail starting with a 0.50 mm flaw under 800 cycles a day of the same five-band traffic the S-N calculation used, integrated by Paris's law. It reaches 1 mm after 91.2 years, 2 mm after 131.3, 10 mm after 172.9 and its critical length of 125.7 mm after 195.8. The curve is nearly flat and then nearly vertical, because the rate goes as the cube of ΔK and ΔK goes as the square root of the crack: the crack spends most of its life being too small to find and the rest being too large to ignore. The dashed lines are the crack lengths at which each band of traffic starts doing anything at all. Materials

The loops a crack grows on

A Miner sum reduces a hundred years of traffic to a number and throws away everything below the cut-off — on this bridge, a third of the vehicles doing exactly none of the damage. Integrate the same spectrum as a crack instead and that third grows thirty-seven per cent of the crack, because a cut-off is a statement about a constant-amplitude test and a crack's threshold is a length rather than a stress. The two calculations disagree about the life by a third and about which vehicles matter entirely.

5 figures Every shape the analogy reaches, which is every single ring. Four frames of 10.0 kN/m on an 8 m span, each solved by the column analogy and each checked against a stiffness solution of the same frame: a portal, largest moment 40.6 kN·m, agreeing to 0.24 per cent; a pitched portal, largest moment 35.6 kN·m, agreeing to 0.15 per cent; a stepped frame, largest moment 46.6 kN·m, agreeing to 0.16 per cent; splayed legs, largest moment 16.9 kN·m, agreeing to 0.09 per cent. The diagrams are drawn normal to each member. Nothing in the method asks what shape the frame is: it needs an area, a centroid and three second moments of the centreline, and a polyline has all five however it bends. Deflection

Three, and what three is a property of

The column analogy works because a closed ring cut once has three redundants and a plane section has three stress resultants. That match is the whole method, and it is topological rather than geometric — a portal, a pitch, a step, a splay and a polygonised arch are all one ring and all exact. Add a second bay and the frame has six redundants with nothing to be a drawing of, and the outer ring on its own is out by 190 per cent.

5 figures A frame bent by nothing at all. A portal of 8.0 m span and 5.0 m columns at EI = 20000.0 kN·m², with its right foot settled 10.0 mm and no load on it anywhere. The moment diagram is drawn on the members: −3.95 kN·m at the left foot, −3.95 at the left knee, −0.00 at the crown, 3.95 and 3.95 on the right. The settlement is drawn hugely magnified; at true scale it is 10.0 mm on an 8.0 m frame. The diagram is antisymmetric, the vertical force the settlement develops is 0.99 kN, and every one of those numbers is proportional to EI. Deflection

The load that is not a load

Settle one foot of a portal frame by ten millimetres and the frame develops moments with nothing applied to it anywhere. In the analogous column the case is simpler than a load case, not harder — the section carries no direct stress at all, and the whole answer is one bending stress. And it scales the wrong way: the moments are proportional to EI, so the stiffer the frame, the more a settlement costs it.

5 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.

51 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.

52 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.

50 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.

270 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.

214 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.

67 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.

122 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.

286 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.

131 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.

77 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.

109 essays