The eccentricity a purlin cannot avoid

The shear centre of a channel is a point that is not in the section — a place in the air, outside the material, through which a transverse load has to act if the member is not to twist.

That is a curiosity when the section is on a page. On a roof it is a design case, because a purlin is a channel or a zed, its load arrives from sheeting fixed to its top flange, and there is nowhere on the member that the load could be applied to miss the shear centre by less than the section’s own proportions dictate.

Which free body produced the number #

The free body is a short length of the channel, cut across the member, with the shear flow on the cut face resolved into its parts.

The web carries almost all of the vertical shear, and its resultant acts along the web. The flanges carry a horizontal shear flow which builds from zero at the toe to a maximum at the web — and the two flange flows point in opposite directions, one inward and one outward, because the flow is continuous round the section.

Equal, opposite and separated by the depth: that is a couple. It contributes no net force in any direction, so it is invisible to a check on forces, and it contributes a moment $q_f h$ that has to be balanced by something.

The only thing available is the applied load, which must therefore act at an offset $e$ such that its own moment about the web cancels the flange couple. That offset is the shear centre, and

$$e = \frac{b^2 h^2 t}{4 I}$$

for a channel of flange width $b$, depth $h$ and uniform thickness $t$. Nothing in that expression is a strength and nothing in it is a load — it is a property of the shape, and the section has it whether or not anybody uses it.

What the shape decides #

Because $e$ is fixed by geometry, the only way to move it is to change the section, and the changes are not free.

The eccentricity goes roughly as $b^2$ and the second moment roughly as $b$, so widening a flange costs torsion faster than it buys bending. That is the trade, and it is the reason purlin sections have narrow flanges relative to their depth — narrower than a beam of the same depth would have, and narrow enough that the flange is close to being an outstand rather than a flange.

Depth is the only dimension that improves both, which is a familiar answer arriving from an unfamiliar direction: the material far from the middle does the work in bending, and here it also moves the shear centre back toward the web.

The torque is not a tolerance #

It is worth being precise about what kind of quantity this is, because it is routinely filed under the wrong heading.

An eccentricity from tolerance is what happens when a load lands 10 mm from where the drawing says. It is small, it has a distribution, and it is covered by a factor somewhere.

This is not that. The load is applied exactly where the drawing says — on the top flange, over the web, or wherever the sheeting fixings are — and it misses the shear centre by 38 mm because the shear centre is 38 mm outside the section. Getting the construction perfect changes nothing.

So a channel purlin under a uniform load $w$ carries a distributed torque of $we$ along its whole length, as a matter of geometry, and the only questions are what resists it and what it costs.

How large the torque actually is #

Numbers make the case, and they are quickly got.

A 100 × 250 × 8 channel spanning 6 m at 1.8 m centres under a roof load of 1.0 kN/m² carries 1.8 kN/m. Its shear centre is 38.0 mm outside the web, so the distributed torque is 1.8 × 0.038 = 0.068 kNm per metre, and over half the span that accumulates to about 0.21 kNm at each end if the ends can hold a torque.

That is a small number and the section it is applied to is smaller still. The torsional shear stress from 0.21 kNm on a slit shape with $J = 1.14\times10^5$ mm⁴ and $t = 8$ mm is $Tt/J = 15$ N/mm² — modest. The twist, though, is the quantity that governs: $TL/GJ$ over 3 m at $G = 81{,}000$ is 0.068 radians, about 4°, on a member whose sheeting has to stay flat.

The stress is comfortable and the deformation is not, which is the shape of nearly every torsion problem in an open section. That is why the check on a purlin is a serviceability one and why it is answered by restraint rather than by section size.

What an open section does with a torque #

The answer to the first question is: very little, on its own.

A channel is the slit shape. One slit costs a factor of hundreds, and the 93° is not a design case — it is an arithmetic demonstration that St Venant torsion alone cannot carry what a purlin is asked to carry.

Two things save the member and neither is the section’s torsion constant.

Warping torsion. A channel restrained against warping at its ends resists twist by bending its flanges in opposite directions in their own planes, which is a much stiffer mechanism than shearing through the thickness. For a short span this dominates, and the twist is far less than the $GJ$ calculation says.

The cladding. The sheeting is fixed to the top flange at close centres and is very stiff in its own plane. It restrains the flange laterally, which restrains the twist, and it does so continuously.

The second is the real answer for a roof, and it means that the purlin’s torsional design is a calculation about the sheeting rather than about the purlin.

What the restraint actually is #

That has a consequence worth stating plainly: a purlin is a member whose capacity depends on a component supplied by somebody else.

The sheeting provides two distinct restraints and they are not equivalent. It prevents the top flange moving laterally, which is a translational restraint, and through the fixings’ resistance to rotation it partially prevents the section twisting, which is a rotational one. The first is nearly complete and the second is not, since a fixing through a thin sheet is a poor rotational spring.

So the standard treatment gives the purlin a lateral restraint at the top flange and a partial torsional restraint whose stiffness comes from the sheeting’s own bending stiffness and the fixing’s pull-out — a chain of small stiffnesses in series, of which the smallest decides.

Three consequences follow from the restraint being at the top flange only.

Under downward load the top flange is in compression and restrained, which is the favourable case and the one purlin systems are designed around.

Under uplift the bottom flange is in compression and unrestrained, so the member’s lateral-torsional capacity collapses — which is why purlins have anti-sag ties and why the wind uplift case governs almost every light-gauge roof.

And in the construction condition there is no sheeting at all: the purlin is an unrestrained channel carrying a torque of $we$, and the twist is the one the figure above describes.

Where the flange flow comes from #

The couple in the first figure has an origin worth showing, because it makes the shear centre’s position feel less arbitrary.

The flow in a flange is $VQ/I$ with $Q$ the first moment of the flange area outboard of the point in question. It is zero at a free edge, where there is no area outboard, and grows linearly along the flange to its junction with the web.

On an I-section the two halves of each flange carry equal and opposite flows that cancel, so the flange contributes no net horizontal force and no couple — which is why a doubly symmetric section’s shear centre is at its centroid, and why nobody meets this problem on a beam.

On a channel the flange is one-sided. Its flow does not cancel, the top and bottom flanges give opposite forces, and the couple exists. The shear centre’s offset is therefore a direct consequence of the flange being on one side of the web, and any section with that property has it.

The other places the same offset turns up #

A purlin is the case that made this a design problem, and it is not the only one.

A crane runway girder is often a channel or an I-section with a channel capping, and the wheel load lands on the rail at the top. Any lateral surge from the crab acts at rail level, well above the shear centre, and the torque is applied by a moving load at a point that changes — so the member sees a travelling torque as well as a travelling shear.

A lintel over an opening in a cavity wall carries the outer leaf on a shelf projecting from its web. That shelf is deliberately eccentric, and the twist it causes is the reason a lintel is a closed or nearly closed shape rather than an angle.

A stringer on a stair or a walkway is a channel with treads fixed to one flange. The load is eccentric to the shear centre by more than the flange width, and there is no cladding.

And a member in a lattice bolted through one leg carries its axial load eccentric to both the centroid and the shear centre, which is the single-angle strut problem with a torque added to it.

The common feature is that all four are members where the connection to whatever is being carried is on one side. An open section attached along one face is eccentric to its own shear centre by construction, and the only general remedy is to close the section.

The zed, which is worse and is used anyway #

A zed purlin has its flanges pointing opposite ways, which makes it point-symmetric about its centroid — so its shear centre is at its centroid, and a load through the centroid produces no torque at all.

That sounds like a solution and is not, because a zed has no axis of symmetry, so a vertical load produces bending about a principal axis at an angle and the section deflects sideways. The product of inertia that causes it is large for a zed, and the sideways movement is often larger than the vertical one.

So the two common purlin sections divide the problem rather than solving it. A channel twists and does not move sideways; a zed moves sideways and does not twist. Both are restrained by the same sheeting, both are governed by the same uplift case, and the choice between them is made on how they lap at supports rather than on either effect.

What the eccentricity does to the bending check #

The torque is not the only consequence of the offset, and the second one is often the larger.

A load acting eccentric to the shear centre is statically equivalent to the same load through the shear centre plus a torque. The load through the shear centre bends the member and the torque twists it — and once it has twisted, the load’s line of action has moved relative to the section, so the bending is no longer about the axis the calculation assumed.

For a channel restrained by sheeting the twist is small and this second-order coupling is negligible. For an unrestrained channel — the construction case, or a member in a wall where nothing is fixed to it yet — it is not, and the section is in biaxial bending with a torque, at an angle that depends on how far it has already twisted.

That is the mechanism by which a lightly restrained open section fails, and it is why the failure looks like a lateral movement rather than like a twist: the section rolls over, and the observer records a beam that has fallen sideways rather than one that was in torsion.

Where the model stops #

The section is thin-walled and the thickness is uniform. A rolled channel has tapered flanges and a root radius, and its shear centre is a few per cent from the thin-walled value.

St Venant torsion is treated alone. For a purlin of ordinary proportions the warping term dominates, and the 93° above is an upper bound on the twist rather than a prediction of it.

The restraint is treated as continuous. Sheeting fixings are at 300 mm or so, so the restraint is discrete, and the purlin bulges between them in a mode nothing here computes.

Nothing here is a stability calculation. The twist described is a first-order response to a torque; the failure that actually governs a purlin under uplift is lateral-torsional buckling of an unrestrained bottom flange, which is a different question with the same section properties in it.

The two flanges are assumed to carry the same flow. They do under pure bending; under a load that has already twisted the section they do not, and the shear centre computed from the undeformed geometry is not quite the point the deformed section wants.

And the load is assumed to act on the top flange. A purlin also carries the sheeting’s own dead weight, a construction load applied anywhere, and a point load from a service hanger fixed to the bottom flange — each at a different eccentricity from the shear centre, and each producing a torque of a different sign.

What to carry away #

The shear centre’s position is a property of the shape and cannot be detailed around. For a channel it is outside the material, so every point on the section is eccentric to it, and the offset is $b^2h^2t/4I$.

The trade is flange width against depth. Narrow flanges reduce the eccentricity faster than they reduce the second moment; depth improves both. That is the whole of why a purlin looks the way it does.

And the resistance comes from outside the member. An open section’s own torsional stiffness is two orders of magnitude below a closed one’s, so what actually stops a purlin twisting is the sheeting screwed to it — a component chosen for weathering, supplied by a different trade, and load-bearing in a way its own specification does not mention. The internal force with no diagram is carried here by something that is not on the structural drawing at all.

Two other essays follow the same section property into different trouble. A section loaded straight down that moves sideways is the principal-axis half of the same geometry, and a section that cannot stay flat is what the torsion the eccentricity produces actually does to it.

The ladder from here #

Later rungs on this anchor: the shear centre of a closed and of a multi-cell section, where the shear flow has a circuit to run round and the offset collapses. Warping torsion computed properly, with the warping constant and the two-term differential equation. The purlin’s restraint modelled as a rotational spring rather than as a restraint, and the stiffness chain that supplies it. Lateral-torsional buckling of a monosymmetric section, where the shear centre’s offset from the centroid changes the answer through the Wagner effect. Cold-formed lipped sections, where lips exist partly to move the shear centre. And the general theory of thin-walled beams, in which the centroid, the shear centre and the warping function are three aspects of one formulation.

The shear centre was identified independently by Maillart and by Timoshenko in the early 1920s, and the delay is instructive: shear flow had been understood since Jourawski in 1855, and it took another seventy years for anybody to add up the flange flows and notice they made a couple. The result then sat as a curiosity for another thirty, until cold-formed sections made thin open shapes cheap and the couple started twisting roofs.