Unsymmetrical bending and shear centre

Simple bending theory assumes the moment acts about a principal axis of the section. Often it does not. A purlin on a sloping roof carries vertical gravity load, but its own axes are tilted with the roof, so the load has components about both axes. An angle section has principal axes inclined to its legs, so even a vertical load about a leg axis produces bending about two principal axes at once. This is unsymmetrical, or biaxial, bending.

The working method is to resolve the moment into components about the two principal axes, find the stress from each with the ordinary bending formula, and add them with the right signs at each corner. The neutral axis is no longer perpendicular to the plane of loading; it rotates towards the weaker axis, and the maximum stress usually occurs at a corner far from that neutral axis. For a rectangle the corner stress can be nearly double the value you would get by ignoring the tilt.

Shear stress raises a second issue. In an open thin-walled section such as a channel, the shear flow in the flanges forms a couple. Unless the load passes through a particular point, the shear centre, the section twists as it bends. For doubly symmetric sections such as I-beams the shear centre is at the centroid. For a channel it lies outside the web, on the side away from the flanges, which is why channel purlins are often fixed with the flanges facing a particular way or braced against twist.

These effects matter most in cold-formed steel, angles used as beams, crane rails and curved or skew members. Design codes handle them through biaxial bending interaction checks and through lateral-torsional buckling rules, which are themselves sensitive to where the load acts relative to the shear centre.