Every essay — page 32
Sections and stress
The section stronger for being cut
Take a square bar standing on its corner, and saw a ninth off its top and bottom tips. It has lost material from the places where material does most for bending, and its first-yield moment has gone up by 5.35 per cent. Nothing has gone wrong with the arithmetic. The bar's plastic moment has gone down in the same cut, and at every curvature the whole bar still carries more than the cut one. What the cut exposes is that a first-yield moment was never a measure of how much a section can hold. It is a measure of how far its furthest fibre stands out.
Stability
The beam that is a different beam upside down
Share the same flange steel unequally between the top and the bottom of a girder and it has two buckling moments, depending on which way the moment bends it: 1,018 kN·m one way up and 283 the other, where the equal-flanged beam from the same steel had 534 both ways. The term responsible was found by Herbert Wagner and is usually left out of beam formulas, on the grounds that most beams are symmetric. It has one simple physical reading. Every fibre leans as the section twists, and it matters whether the compressed ones are near the centre of twist or far from it.
Structural form
The roof that carries everything as shear
A hyperbolic paraboloid is a curved surface made entirely of straight lines, and it carries its load in a way no other roof does. Under a uniform load there is no stress along those straight lines at all. There is only one constant shear, the same everywhere on the surface, which is the same thing as an arch and a hanging cable crossing at right angles at every point, each carrying half the load. It then arrives at the edges with nowhere to go but along them. Membrane theory settles the stress with one line of algebra, and it says nothing about how far the roof will sag, and that has been the trouble with these roofs ever since they were first built.