Concept

I-section — where it appears

A profile with most of its material in two flanges at the extremes of its depth, which a second moment of area rewards and a column's weak axis punishes. Its whole economy is the parallel-axis term: the flanges sit at the extremes of the depth and the web only holds them apart and carries the shear.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

Every strip counts by the square of its distance. A rectangular section divided into equal strips, with each strip's contribution to the second moment of area drawn beside it. The strips are identical in size; only their distance from the neutral axis differs.

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 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

Shear stress across a section. The distribution of shear stress over an I-section, computed as VQ/It by accumulating the first moment of the area above every height. The peak is 0.41 against a mean of 0.19 — a ratio of 2.15 — and it falls at the neutral axis, where the bending stress is zero.

The shear nobody draws

A stack of loose planks slides at its ends when it is loaded. Glue them and the sliding stops — and whatever the glue is now carrying is a stress that no bending calculation contains.

sections · Shear flow

The shear centre of a channel. A channel of 80 by 200, with the shear flow in its flanges drawn. Those flows form a couple, so the load has to be applied 31.7 outside the web to leave the section untwisted — a point in the air, outside the material entirely.

The point that is not in the section

A channel loaded down its web twists. To stop it, the load must be applied through a point outside the steel entirely — in the air beside the section, where nothing can be attached.

sections · Shear centre

The flanges go opposite ways, and the pair of them is the bimoment. A 305 by 165 mm I-section held against warping and twisted by 0.5 kN·m, with the section on the left and the two flanges seen in plan on the right. At the built-in end each flange bends in its own plane, one way at the top and the other at the bottom, through 29.3 mm at the free end — drawn 20 times its true size against the 6 m length. The pair of flange shears is 1.69 kN each, and 1.69 × 295 mm is 0.500 kN·m — the whole torque at that section, carried by two forces neither of which is a torque. The pair of flange moments is 3.04 kN·m each, and 3.04 × 295 mm is 0.897 kN·m², which is the bimoment. It puts 67.0 N/mm² into two diagonally opposite flange tips and takes the same out of the other two, so its net force and its net moment about every axis are zero — which is exactly why no member diagram has a place for it.

The section that cannot stay flat

Twist an I-section and its cross-section dishes out of its own plane. Stop that happening at one end and the member finds a second way to resist — the flanges bend in opposite directions — and the stress resultant that describes it has units nothing else in statics has.

sections · Warping

One point, every plane through it, one circle. A point carrying 140 N/mm² across one face, 0 across the other and 45 of shear. As the plane is turned, the pair (σ, τ) runs round a circle of radius 83.2 centred at 70.0 — and it goes round at twice the rate the plane does, which is the part always misremembered and the part that makes the picture work. The principal stresses are 153.2 and -13.2, on planes 16.4° from the face the 140 acts on; the largest shear on any plane is 83.2, exactly the radius, and it sits 45° from those — which is 90° round the circle. The von Mises stress that ranks this state against any other is 160.2.

The worst stress is not where the worst bending is

Every stress this collection has quoted is a stress on a particular plane, and neither the bending stress nor the shear stress is a property of the point. Turn the plane and both change; one pair of numbers does not, and on a short beam it peaks where neither of them does.

sections · Principal stress

Named alongside it

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

Neutral axis Shear flow Torsion First moment of area Open section Second moment of area Shear centre Shear stress Stress resultant Warping Bending stress Bimoment

All concepts