The section made of pieces

Stack two planks and bend them. They slide against each other at the interface, each bends about its own middle, and the pair is exactly twice as stiff as one plank. Glue them and bend them again, and the assembly is eight times as stiff as one plank — because the pair now bends about a single axis through the middle of the whole thing, and the material far from the middle is suddenly doing something.

The factor of four between those two outcomes is the entire subject of built-up sections, and it is bought by one thing: a connection at the interface capable of carrying the shear that flows there. The parallel-axis theorem is arithmetic about a section that only exists if the connection exists.

What the theorem actually claims #

The parallel-axis theorem says that the second moment of an area about any axis is its second moment about its own centroid, plus its area times the square of the distance between the axes:

$$I = I_{\text{own}} + A d^2$$

Applied to a built-up section, it says that a flange plate far from the neutral axis contributes almost entirely through the $Ad^2$ term — its own $I_{\text{own}}$ is negligible — so moving material outward is enormously effective. It is the same observation that makes a deep truss cheap and one shape beat another at equal weight, and the built-up section is simply the version of it that has to be assembled rather than rolled.

But look at what the theorem quietly assumes. It treats the whole area as bending about one axis, which means every part of it must experience the strain appropriate to its distance from that axis. A flange 250 mm above the neutral axis has to stretch or compress far more than it would if it were bending about its own centre. Something has to force it to.

That something is the shear at the interface. The flange’s stress varies along the beam wherever the moment varies, so the force in the flange varies along the beam, and the difference has to be delivered into it through the joint. Per unit length, that difference is the shear flow.

The flow, and where the spacing comes from #

The shear flow at any level of a section is

$$q = \frac{V Q}{I}$$

where $Q$ is the first moment, about the neutral axis, of all the area beyond the level in question. At the interface between a flange and a web, $Q$ is the whole flange’s area times its distance to the axis — which is precisely the quantity that made the flange so effective in the first place.

That is the trade at the heart of built-up construction, stated as an identity rather than a proverb: the same $Ad$ that buys the stiffness is the $Q$ that has to be delivered through the joint. Push the flange further out and both grow together.

Connector spacing follows immediately. If each connector can carry a force $F$ and they are spaced $s$ apart, they deliver $F/s$ per unit length, and that must be at least $q$:

$$s \le \frac{F}{q} = \frac{F I}{V Q}$$

That last observation is worth stating plainly because it inverts an intuition. The most heavily stressed part of a simply supported beam in bending is midspan, and it is exactly there that the connectors are least needed — the flange force is largest there but it is not changing, and only the change has to be delivered through the joint. The shear diagram is the derivative of the moment, and the connectors are working on the derivative.

Which free body produced the number #

The section in the hero figure is an I-section of 5200 mm² total area and 200 mm depth: flanges 110 wide by 12 thick, a web 14.5 thick, carrying a vertical shear of 200 kN.

The free body is the flange alone, cut from the web along the interface and cut across the beam at two stations one millimetre apart. Three things act on it: the direct stresses on the two cut faces, which differ because the moment differs; and the shear on the interface, which is what makes up the difference.

The flange area is $110 \times 12 = 1320$ mm², its centroid sits 94 mm from the neutral axis, so $Q = 124{,}080$ mm³. The section’s second moment works out at about $3.0 \times 10^7$ mm⁴. Then $q = VQ/I = 200{,}000 \times 124{,}080 / 3.0 \times 10^7 = 820$ N/mm, which is the number the figure prints, and 20 kN connectors at that flow may be no further apart than $20{,}000/820 = 24$ mm. Halve the shear and the same connectors go to 49 mm, which is the linearity of the elastic result made visible.

The assumption the figure rests on is that the connection is stiff enough that no slip occurs at the interface. Real bolted connections slip a little, real welds are flexible, and real shear studs in concrete are deliberately ductile. Partial interaction — a section somewhere between fully composite and not composite at all — is the realistic case, and it has its own theory in which the effective second moment lies between $I_{\text{composite}}$ and $\Sigma I_{\text{own}}$.

The two-plank calculation, in full #

The factor of four deserves to be done rather than quoted, because it is the cleanest demonstration in the subject of what the theorem is worth.

Take two planks, each of breadth $b$ and depth $h$. Separately, each has $I = bh^3/12$ about its own centre, so the pair has $2 \times bh^3/12 = bh^3/6$.

Glued, the assembly is one section of depth $2h$, so $I = b(2h)^3/12 = 8bh^3/12 = 2bh^3/3$.

The ratio is exactly four. Reached the other way, through the theorem: each plank’s centroid is now $h/2$ from the combined axis, so each contributes $bh^3/12 + bh \cdot (h/2)^2 = bh^3/12 + bh^3/4 = bh^3/3$, and two of those is $2bh^3/3$. Same answer, and the second route shows where the gain lives — three-quarters of each plank’s contribution is now the transfer term, and none of that term existed before the glue.

Strength gains by two rather than by four, because the section modulus is $I/c$ and the distance to the extreme fibre has doubled along with everything else. Stiffness goes as the cube of the depth and strength as its square, which is why a built-up section is more dramatically a deflection intervention than a strength one, and why the argument for it is usually made in the wrong currency.

What failure of the joint looks like #

The reversion is the striking part. A built-up section whose joint fails does not break; it reverts, continuously, to being a stack of separate pieces.

The bending capacity falls by the ratio of the two second moments, which for two equal plates is a factor of four, and the deflection rises by the same factor. Nothing snaps. The member simply becomes much more flexible and much more highly stressed, and the pieces begin to slide relative to each other at the ends where the shear is largest — which is the visible symptom, and the reason the ends of a timber flitch beam are where to look.

This is the same mechanism as the delamination of a timber beam, where the connector is the wood’s own longitudinal shear strength and the interface is a plane of weakness the grain provides for free. It is also, exactly, what happens when the shear studs on a composite deck are under-provided: the slab and the steel go on carrying load, separately, at four times the deflection.

The joint that only has to work at the ends #

There is a case where the whole argument relaxes, and knowing which case it is prevents a great deal of unnecessary connection.

A member in pure bending — constant moment, no shear — has no flow at the interface at all, because the flange force does not change along its length. Two plates in a region of constant moment need no connection between them beyond whatever holds them in contact. This is not an exotic case: the middle region of a beam under two symmetric point loads is exactly it, and so is the central portion of many transfer structures.

What such a member does need is a connection at the ends, where the flange force has to be got into the plate in the first place. The total force to be transferred is the same either way — the integral of the flow along the shear span — and the choice is only whether it is delivered continuously or in one lump. A bolted splice is the lump version, and it is why splices are so much heavier than the connectors they replace over the same length.

Where the model stops #

Elastic distribution. $q = VQ/I$ is an elastic result. Once the connectors begin to yield they redistribute — a heavily loaded connector sheds force to its neighbours — so the ultimate capacity of a group is closer to the sum of the individual capacities than the elastic distribution suggests. Codes exploit this by permitting uniform spacing of ductile shear studs across a whole shear span, which is a plastic argument and would be indefensible elastically.

Constant section. The derivation assumes $I$ and $Q$ do not vary along the beam. A haunched girder or a plated beam with a curtailed flange has a longitudinal variation in the flange force that comes partly from the changing moment and partly from the changing section, and the second part is often forgotten. Curtailment points are where built-up sections have trouble.

One interface. A section with several interfaces — a box girder with four corners, a sandwich panel with two faces — has a flow at each, and they are not equal unless the section is symmetric.

Stability of the pieces. A thin web between two heavy flanges is a plate in shear, and it can ripple long before anything yields. Building a section out of separated pieces creates slender pieces, and their local stability becomes a separate check that the solid section never needed.

Fatigue. A connector under cycling shear is a fatigue detail, and the governing check is usually fatigue rather than static strength, because a weld at an interface is exactly the geometry crack growth likes.

The figure has a limitation it cannot escape. The shear flow is drawn as a distribution across the depth of the section, at one station along the beam, which is the correct way to show its magnitude and the wrong way to show what it is. The flow is a force per unit length running along the beam, out of the plane of the drawing, and the picture necessarily shows it as an ordinate on a cross-section. A reader who takes the plotted curve as a spatial distribution rather than an intensity has the mental image the drawing invites, and it is not the one the equation describes.

The generalisation #

The rule generalises past sections to any assembly meant to act as a unit: the connection has to carry the derivative of what the assembly is carrying.

A composite deck’s studs carry the derivative of the slab’s compressive force. A glued laminated beam’s glue lines carry the derivative of each lamination’s stress. A bolted splice carries the whole flange force, which is the extreme case where the derivative is delivered in one place. A diaphragm holding a building’s floors together carries the derivative of the shear distributed between its stability elements. Even the free body itself is the same idea used as a method rather than as a structure: cut something out, and whatever crossed the cut has to be drawn on the piece. In all of them, the assembly’s efficiency and the connection’s demand grow together, because they are the same geometric quantity read two ways.

The corollary is a good design instinct: whenever a section’s stiffness comes from separation, ask what holds the separated parts together. A deep truss, a box girder, a sandwich panel and a composite deck are all the same idea — put the material far apart and connect it — and all of them fail the same way, which is that the connection is the part nobody drew.

A last consequence, and the one that most often decides a scheme. Because the connection demand grows with the same $Ad$ that buys the stiffness, there is a depth past which the connection becomes the difficult part of the design rather than the sections. A very deep plate girder is not limited by its flanges; it is limited by the weld that has to run the whole length of it, twice, carrying a flow proportional to the depth it is exploiting. Efficiency in one currency is expenditure in another, and the built-up section is the clearest place in the subject to watch the exchange rate.

The idea is old and the analysis is not. Built-up wrought-iron girders were riveted in the 1840s, and the rivet spacing was set by rule and by experience: closer at the ends, open in the middle, which is right, and known long before $VQ/I$ was available to say why. Jourawski derived the shear-flow formula in 1844 while investigating exactly this question on the timber trestle bridges of the St Petersburg–Moscow railway, whose built-up beams were failing by delamination rather than by bending. The formula arrived because a joint failed.

The ladder from here #

Later rungs on this anchor: partial interaction, and the effective stiffness of a connection that slips. Shear lag, where a wide flange does not carry uniform stress and the effective width is less than the real one. Curtailment, and the flange force that changes because the section does. The plastic distribution of connector forces and why uniform spacing is permitted. Sandwich construction, where the core is a connector with almost no material in it. And the built-up compression member, where the connection has a second job entirely — holding the pieces at their spacing while they try to buckle apart, which is a stiffness requirement rather than a strength one and is the subject of the brace that need not be strong.