Steel Angles: Principal Axes u–v and the Angle α Explained

A rolled steel angle is two rectangular legs joined at a heel. It is natural to describe it by the geometric axes y–y and z–z running parallel to the legs. That is how the dimensions and the values I_y and I_z are tabulated. But unlike an I- or H-section, an angle has no axis of symmetry parallel to a leg. In the y–z system the product of inertia I_yz (the term that couples bending about one axis with curvature about the other) is not zero.

The consequence: y–y and z–z are not the directions of extreme stiffness. If you rotate the reference axes, there is exactly one orientation in which the product of inertia vanishes. These are the principal axes u–u and v–v, inclined at the angle α to the geometric axes. About u–u the second moment of area reaches its maximum I_u, about v–v its minimum I_v. Every cross-section has such a pair of axes: in doubly symmetric shapes they simply coincide with y–y and z–z, which is why the question only becomes visible with angles and other asymmetric profiles.

An equal-leg angle does have one axis of symmetry: the bisector through the heel. An axis of symmetry is always a principal axis, so u–u lies on that bisector and v–v runs perpendicular to it. That fixes α = 45° exactly, for every equal angle regardless of leg size or thickness. Take L 50×50×5 as a typical example.

Note a subtlety: symmetry makes I_y = I_z, but neither of them is an extreme value. The section is markedly stiffer about the heel bisector (I_u) and markedly softer about the axis perpendicular to it (I_v). An equal angle therefore still needs its principal-axis values whenever stiffness or buckling is assessed: the convenient-looking leg-parallel axes tell only part of the story.

For an unequal-leg angle the symmetry is gone and α is no longer 45°. The strong axis u–u rotates toward the longer leg, and the more unequal the legs, the further α drops below 45°. There is no simple geometric shortcut, which is why section tables list tan α for each profile alongside the moments of inertia:

Reading a row such as L 100×65×7 shows the pattern clearly: I_u exceeds both leg-parallel values, I_v is smaller than either of them, and tan α encodes how far the principal directions are rotated away from the legs. In hand calculations the same angle also converts a bending moment applied about a geometric axis into its components about u–u and v–v.

Rotating the reference axes redistributes the second moments of area but never changes their sum. The polar moment about the centroid is fixed, so the invariant I_u + I_v = I_y + I_z holds exactly for every section: it is the moments-of-inertia analogue of Mohr's circle for stress. As the axes rotate towards the principal orientation, one moment grows at the expense of the other until they reach the extremes I_u (maximum) and I_v (minimum), at which point the product of inertia is zero.

This gives you a free sanity check: for any table row or any hand calculation, the sum of the principal moments must equal the sum of the geometric ones, with I_u at least as large as the larger of I_y and I_z, and I_v no larger than the smaller of the two. If a computed α or a transformed pair breaks this identity, the arithmetic is wrong, not the theory.

A pin-ended compression member buckles about the axis with the smallest radius of gyration. For a single angle that is always the minor principal axis v–v with its radius i_v, regardless of how the legs happen to be oriented in the structure. Checking flexural buckling of an angle strut about y–y or z–z alone therefore overestimates its capacity; the member will find the weak diagonal direction on its own.

In practice single-angle diagonals are bolted through one leg to a gusset plate, so the axial force is eccentric to the centroid. Design codes deal with this by modified (equivalent) slenderness rules for angles connected through one leg rather than by explicit biaxial bending, but the slenderness in those rules is still built on i_v, which is why the tables highlight it.

Load an unrestrained angle in a plane parallel to one leg and you are bending it about a non-principal axis. Because I_yz ≠ 0, the result is unsymmetric (skew) bending: the beam deflects out of the plane of loading, the neutral axis is inclined, and the extreme stresses appear at the heel and the leg toes rather than where a symmetric-beam formula would put them. The clean way to compute it is to resolve the moment into components about u–u and v–v, superpose the two stress fields, and check the deflection components in both principal directions, or to load the section in a principal plane in the first place.

This explains where angles are actually used. In trusses and bracing they carry essentially axial force, where only the area and i_v matter. As masonry lintels, unequal angles work with the long leg vertical; the wall above restrains the horizontal deflection component, so the awkward part of skew bending is suppressed and the simple geometric-axis check becomes defensible. An unrestrained angle used as a laterally free beam, however, deserves a full principal-axis analysis.