Second moment of area — where it appears
Named by 26 essays across 6 fields — each of them below, with the objects they name alongside it.
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.
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.
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.
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.
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.
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.
The column that had yielded before it was loaded
A real column sits below both of the two straight answers over the whole middle of the slenderness range, and the usual explanation — that it was not straight — is only half of it. The other half is that the flange tips had already yielded when it left the rolling mill.
Two beams, or one beam four times as stiff
Stack two planks and they bend as two beams whose faces slide past one another. Bond the faces and the pair has one neutral axis, four times the second moment and half the stress. Nothing was added but a restraint on slip.
A section made of two materials, one of them pretended away
Multiplying a material's width by the ratio of the moduli produces a fictitious section of one material with the right neutral axis and the right forces. It is not a trick — it is compatibility and Hooke's law written down — and it says a stiff material takes what its modulus asks for.
The hole that costs nothing, and everything
A service opening removes 30% of a beam's second moment and 0.7% of its deflection. What it costs is not that. Across the opening the shear has nowhere to go but through the two tees, and a tee carrying shear over a length bends.
Folded until it spans
A flat sheet has a second moment of area of B·t³/12 and will not span anything. Folded, the same material has B·t·h²/12, and the gain is exactly the fold depth over the thickness, squared — a ratio with no material in it and no width in it.
Two strengths, depending which way up
A symmetric section has one section modulus. A tee has two, differing by a factor of three, so the same member has two bending strengths and which applies is decided by the sign of the moment. Turn it over and it is a different beam.
The one length a section takes into a column
A section has an area, a second moment, two section moduli, a shear centre and a torsion constant. A column has heard of exactly one of them, and it is none of those — it is the length √(I/A), which is where the whole area would have to sit to give the section the stiffness it has.
The section that changes along the span
A prismatic beam is checked where the moment is largest, and everyone knows where that is. A tapered one is not, because the capacity is moving too — and for a cantilever with a load at its tip the governing station is exactly where the depth has doubled, with no length, no load and no material in the answer.
Half the studs, and most of the beam
Bonding two layers together quadruples the stiffness of the pair. A real connection is a row of studs that deform, so the layers slide a little and the beam sits between the two answers — but not halfway between, and the number that decides where is a single dimensionless group.
Two skins and the space between them
A sandwich panel is a section made of a material that carries the bending and a material that carries none of it. The parallel-axis term is not a correction here — it is 98.8 per cent of the second moment — and the shear deflection is not a correction either.
The section that will not keep its shape
A box girder is closed, so torsion costs it almost nothing. What an eccentric load actually does to it is something a torsion calculation contains no term for — the rectangle becomes a parallelogram, in its own plane, along the whole length of the span.
The web that carries no bending
A corrugated web needs no stiffeners, because the folds give it in one direction a depth it does not have in its thickness. In the other direction the same folds make it an accordion — and a web that cannot be stretched cannot carry a bending stress at all.
Two moments and a neutral axis that obeys neither
Tilt the load on a rolled beam by five degrees and the neutral axis swings by seventy. The section is doubly symmetric, its product of inertia is exactly zero, and none of that helps — because what decides the axis is the moment ratio multiplied by a stiffness ratio of thirty.
The corner columns take more than their share
A framed tube is a hollow cantilever, and a hollow cantilever's flange ought to be uniformly stressed. It is not, and the reason is that the only route the axial force has into a column in the middle of a face is the in-plane shear of the frame — one bay at a time, from the corner inwards.
Stiffer than its cracked section says
At a crack the concrete below the neutral axis has gone and the steel carries the tension alone. Between the cracks it has not gone — bond drags it back into tension, the steel strain drops, and the curvature averaged over a length of beam is neither section's.
Where a deflection comes from
The unit-load method gives a deflection as an integral, and this collection has treated that integral as a number to evaluate. It is not a number. It is a density, and it says which millimetres of the member produced the answer — which is not the same map as where the moment is largest.
The weight that has to be known before it can be found
Every other load arrives from outside and can be looked up. A structure's own weight depends on how big it is, and how big it is depends on the load — so the first calculation on any project is a fixed point, and the fraction of a member spent carrying itself turns out to be the square of its span as a fraction of a span it can never reach.
Where the steel is, not how much of it
A section in bending resists a moment with a couple, and a couple is a force times a distance. The force is bought — it is an area of steel at a stress. The distance is free, decided by where the bars were put, and it is the variable almost nobody optimises because it does not appear on an order.
Halving the panel buys a shorter strut
Subdividing a truss into more panels of the same span and depth barely changes the chord forces, because the couple that carries the moment has not moved. What it changes is the length of every compression member, and a buckling capacity goes as the inverse square of a length.
The axis a column buckles about
A strut buckles about the axis with the smallest second moment of area, and for a section with no axis of symmetry that axis is neither of the two on the drawing. An angle used as a strut is 2.45 times weaker than the number a designer reads off its own dimensions.
Named alongside it
The objects these essays reach for when they reach for this one.
Neutral axisStiffnessSection modulusDeflectionBucklingLever armPlane sectionsSelf-weightServiceabilityShear flowComposite actionCracked section