Structural Engineering & Statics
where the load goes
Concept

Second moment of area

6 essays name this object, across 3 fields. What follows is each of them, and the objects they name alongside it.
0.511.52050100150200250300depth of the truss2501671251007150the same moment, resisted by a longer lever arm

Depth is the cheapest strength there is

Doubling the depth of a truss halves its chord forces without adding a gram of material to the chords. Nothing else in structural design is that cheap, and almost every structure has already spent it.

structures · span to depth neutral axiscontribution of each striptotal I = 79.86 × 10⁶the outer strips do almost all of the work

The material far from the middle does nearly all the work

A strip of steel contributes to bending stiffness in proportion to the square of its distance from the centre. Move the same steel outward and the section gets stiffer for nothing.

sections · second moment the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale

The same steel in a different shape, and a factor of forty

Four sections of identical area, identical weight and identical cost. The stiffest is dozens of times the stiffest of the flattest, and the only thing that changed was the arrangement.

sections · section shape x = 1371200 mm² of steel, n = 7.5b = 30018.0 N/mm²371 kN in the steelz = 404C = T = 371 kN · C·z = 150.0 kNm = the applied momentcracked I 1139×10⁶ mm⁴ against uncracked 3422×10⁶ — 67% of the stiffness gone

When half the section has given up

Bending theory puts the neutral axis through the centroid. That is a consequence, not a rule — and when the tension side cracks, the same reasoning moves the axis somewhere else entirely.

sections · bending stress neutral axispeak 82.8stressflow, q = VQ ÷ Imean stress 38.46 — the value a shear divided by an area would givepeak 2.15× that, and in the place bending ignoresthe flow is continuous; the stress jumps wherever the width doesq = 820.0connectors of 20000 each, at 24 centres, are what make this one section

The section made of pieces

Two plates bolted together are twice as strong as one plate. Two plates made to act as one section are eight times as stiff. The difference is entirely in the joint, and the joint has a calculable spacing.

sections · second moment 11.522.533.54050100150200250300span, relative to the first1×5×16×81×256×moment: the squareload: the first powerdeflection: the fourth

Span to the fourth, which is why spans are short

Doubling a span multiplies its deflection by sixteen. No other relationship in ordinary structural work is that steep, and it is the reason long spans are always a different kind of structure.

deflection · span scaling

Named alongside it

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

Neutral axisFirst moment of areaI-sectionLever armParallel axis theoremSelf weightBucklingBuilt up sectionCentroidChord forceCracked sectionDelamination

All concepts