Two moments and a neutral axis that obeys neither

A rolled beam is a machine for being strong in one direction. A 305 by 102 universal beam has a second moment of $5.91 \times 10^7\,\mathrm{mm^4}$ about its strong axis and $1.92 \times 10^6$ about its weak one — a factor of thirty-one — and that ratio is the entire reason the shape exists.

It is also the reason the section is so easy to ruin. Turn the load five degrees out of the plane of the web and the stress in the worst corner goes up by ninety per cent.

Which free body produced the number #

Cut the section and impose the assumption everything here rests on: plane sections stay plane. The strain at a point is then a linear function of position, so the stress is too, and the general linear function of two coordinates is

$$\sigma = \frac{M_y\,y}{I_y} + \frac{M_z\,z}{I_z}$$

for a section whose product of inertia is zero — which for a doubly symmetric shape it is, exactly, by symmetry. The two bending problems superpose and nothing couples them.

The neutral axis is where that expression vanishes, which gives $z/y = -(M_y I_z)/(M_z I_y)$, or

$$\tan\alpha = \frac{M_z}{M_y}\cdot\frac{I_y}{I_z}$$

measured from the strong axis. Two ratios multiply: how much of the moment went the weak way, and how much less stiff the section is that way. The first is small and the second is thirty-one, so the product is not small.

At one degree of tilt the axis is already at 28°. At five it is at 70. At ten it is at 80, and past that it barely moves, because the section has effectively given up bending about its strong axis at all.

The difference from the section next door #

This is a different argument from the one about unsymmetric sections, and the two are worth separating carefully, because they look alike and share a page in most textbooks.

There, the load is straight down, the section is an angle or a zed, its product of inertia $I_{yz}$ is not zero, and the consequence is that a vertical load produces a horizontal movement. The coupling is a property of the section.

Here, $I_{yz}$ is exactly zero and there is no coupling at all. What produces the answer is that the load is not straight down — and the section’s response to the part of it that is not is enormously more compliant than its response to the part that is. The coupling is a property of the load.

The two can happen together, and an angle carrying a load off the vertical is genuinely both. The reason for keeping them apart is that the fixes are different: the first is fixed by rotating the section or restraining it laterally, and the second is fixed by making sure the load is where it was said to be.

Where the weak axis takes over #

The stress at the worst corner is

$$\sigma = M\left(\frac{\cos\theta}{Z_y} + \frac{\sin\theta}{Z_z}\right)$$

and the two terms are equal when $\tan\theta = Z_z/Z_y$. For this section that is 5.5 degrees.

Past five and a half degrees the beam is a weak-axis member with a strong-axis correction, which is an odd sentence about a section chosen entirely for its strong axis, and it is why the tilt matters so much more than the geometry suggests.

The purlin, which is the case this is really about #

Nothing on a building site is five degrees out of plane by accident. What is out of plane by design is a purlin.

A purlin sits on a sloping rafter and carries a vertical load — snow, wind, its own roof — through a section whose web is perpendicular to the roof rather than to the ground. The angle between the load and the web is therefore the roof pitch, and roof pitches are not five degrees. At twenty degrees the corner stress on this section is 4.5 times what a strong-axis calculation gives.

That is why purlins have sag bars. A sag bar is a rod running up the slope between the purlins at mid-span, and its only job is to hold the purlin against moving down the slope — which halves the effective span for the weak-axis component while leaving the strong-axis span alone. It is the cheapest possible response to the arithmetic above: it does not make the section stronger, it removes the term.

And it explains a detail that looks arbitrary from outside. Sag bars are put in at mid-span for one row and at the third points for two, in a pattern decided by the weak-axis moment and having nothing to do with the vertical load at all.

The beam moves the way the axis says, not the way the load does #

The deflection follows the same arithmetic and gets less attention.

A member deflects perpendicular to its neutral axis, so a load five degrees off the web moves this beam at 70 degrees to the vertical — sideways nearly as much as down. The magnitudes are the two components computed separately, $\delta_y \propto \cos\theta/I_y$ and $\delta_z \propto \sin\theta/I_z$, and the ratio of the second to the first is again the tilt times the stiffness ratio.

For a purlin that is the sag: a purlin without sag bars does not deflect downwards, it slides down the slope, and the visible symptom is a roof whose lines are wrong rather than a roof that is sagging. On a twenty-degree pitch the down-slope movement of this section is eleven times the movement normal to the roof.

And the serviceability check is nearly always written about the wrong one. A deflection limit of span over two hundred applied to the vertical component passes a member whose real movement is several times that, in a direction nobody measured.

Four corners, and only one of them is the answer #

Superposing two bending stresses on a doubly symmetric section gives four corner values, and they are not symmetric in any useful way: two corners have both terms of the same sign and two have them opposed.

The corner where they add is the one that decides the section, and the corner diagonally opposite it is its mirror in tension. The other two carry the difference of the two terms, which for a small tilt is nearly the strong-axis stress alone and for a large one is nearly the weak-axis stress with the sign the other way.

That is worth stating because it decides where a strain gauge goes and where a crack starts, and because it is the reason a biaxially bent section that is asymmetric in one direction — a channel, a zed, a section with unequal flanges — has no single governing corner at all. It has one for each combination of signs, and the check is four checks.

The elastic check is a plane and the plastic one is a curve #

Set the corner stress to the yield stress and the capacity is

$$\frac{M_y}{M_{y,el}} + \frac{M_z}{M_{z,el}} = 1$$

which is a straight line in the two moments — a diamond, if both signs are drawn. That is the elastic answer and it is exact, because the two stresses genuinely add at the corner.

The plastic answer is not a straight line, and the reason is worth stating precisely: the plastic neutral axes for the two cases are not the same line. Bending about the strong axis puts the whole top half in compression; bending about the weak axis puts the whole of one side in compression. Combine them and the yielded region is a wedge whose boundary is neither — so the section carries more of the combination than the sum of the parts suggests, and the interaction bulges outward with an exponent of about two on the strong axis.

For the section here the elastic capacity at five degrees is 73 kNm and the plastic one is 114, a ratio of 1.58 — against a strong-axis shape factor of only 1.13. Biaxial bending is where the plastic reserve is largest, precisely because it is where the elastic corner stress is most pessimistic about what the rest of the section is doing.

The shape factors are not the same either #

A doubly symmetric I-section has a strong-axis shape factor of about 1.13 — the plastic modulus is thirteen per cent more than the elastic one, because most of the material is already near the extreme fibre and there is little left to recruit.

About the weak axis the same section has a shape factor of 1.54, which is close to a rectangle’s 1.5, because about that axis the section is essentially two rectangles laid flat. The flanges, which are the reason the strong-axis factor is small, are the reason the weak-axis factor is large.

So the two directions differ not only in capacity but in kind: the strong axis is an elastic section with a little reserve and the weak axis is a plastic one. A designer who reaches for a plastic check to recover margin recovers more of it in the direction that was not supposed to be carrying anything.

The crane girder, where the tilt is a load rather than a geometry #

A purlin is out of plane because the roof is. A crane girder is in plane and gets its second moment from the crane.

A travelling crane applies a horizontal surge along the rail — from accelerating the crab, from skewing on the rails, from the load swinging — and that surge acts at the top flange, several hundred millimetres above the section’s centroid. So the girder carries a vertical wheel load, a horizontal load at the top flange, and the torsion the offset between them produces, all at once and all at their maxima at the same instant.

The arithmetic on this page prices the first two. A horizontal surge of a tenth of the vertical wheel load is a five- or six-degree tilt of the resultant, and this section’s curve says that costs it most of a factor of two. Which is why crane girders are almost never plain I-sections: they are I-sections with a channel or a plate on the top flange, and the extra piece is there to carry the horizontal component about an axis the plain section is bad at.

Where the model stops #

Nothing here is about torsion, and a load out of plane usually brings some. A vertical load on a sloping purlin acts through a line that misses the shear centre, so the section twists as well as bending both ways, and the twist adds a warping stress the superposition above has no term for. The check is honest only while the member is torsionally restrained — by the sheeting, by the sag bars, by the cleats — which for a purlin it largely is and for a crane girder taking a lateral surge it is not.

Lateral-torsional buckling is left out entirely. A member bent about its strong axis and pushed sideways at the same time is being asked the lateral-torsional question in its most direct form, and the resistance to that is not the section’s weak-axis modulus but a curve of its own.

The section is taken as doubly symmetric. A channel is not, and a channel bent about both axes is both essays at once, with $I_{yz}$ non-zero and a shear centre outside the section.

And the corner is treated as a point. A rolled section has a root radius, and the fibre furthest from a neutral axis at 70 degrees is the tip of a flange rather than the corner of a rectangle, which is a per cent or two rather than a mechanism.

What the pictures cannot show #

The first figure draws the neutral axis as a line through the section. That line is a construction: nothing along it is unstressed in any measurable sense, because the section is also carrying shear, and often axial force, and the axis of zero bending stress is not the axis of zero anything else.

Nor can any of these figures show what actually happens on site, which is that the load is rarely exactly at the pitch either. A purlin picks up a component from the sheeting’s own restraint, from the wind blowing up the slope, and from the fact that snow does not lie normal to a roof. The five degrees this essay opens with is not an error bar; it is a design case that is usually specified and often exceeded.

The assumption the figure rests on #

The moment is assumed to be applied about a known direction, and the section to be free to bend about whichever axis the arithmetic gives it.

Neither is quite true when a member is sheeted. Trapezoidal roof sheeting screwed to a purlin restrains it in the plane of the roof, which does not remove the weak-axis moment so much as change what carries it — part goes into the sheeting as a diaphragm and part stays in the purlin, in a proportion nobody computes and everybody assumes. The consequence is that the honest range of answers for a real purlin runs from the number in this essay down to something like a strong-axis calculation, and which end of that range a particular roof sits at depends on fixings, on sheet thickness, and on whether the sheeting has been replaced since.