Unequal angle — why it bends the way it does, not the way you pushed it
Load an angle straight down and it moves sideways. That is not a defect — it is what a section with no axis of symmetry does, and the two numbers that explain it are and .
- Section
- 150 × 90 × 12 UA
- Long leg
- 150 mm
- Short leg
- 90 mm
- Thickness
- 12 mm
- Symmetry
- nonethe whole point
Part 1 — Two legs meeting at a heel
A 150 × 90 × 12 unequal angle, drawn sharp.
Two rectangles sharing a corner:
REC 0,0 90,12 the 90 mm horizontal leg · REC 0,12 12,150 the 150 mm vertical leg
Drawn from the heel at the origin, so the whole section lives in the positive quadrant. That placement is deliberate: an angle has no axis of symmetry at all, and putting the heel at the origin makes the centroid’s offset in both directions easy to read.
Area comes to 2736 mm², with the centroid at (21.39, 51.39) mm — offset from the heel in both directions, and inside neither leg.
Part 2, step 1 — The product of inertia, and what it means
The term that vanishes on every symmetric section.
Every other section in this series had , because each had at least one axis of symmetry. The angle has none, and here -1.912 × 10⁶ mm⁴.
It is negative because most of the material sits where and have opposite signs relative to the centroid — the vertical leg is left of center and above it, the horizontal leg is right of center and below it.
A non-zero has one immediate consequence: the x and y axes you drew are not the principal axes. Bending applied about x produces curvature about y as well. The section refuses to bend the way you pushed it.
Part 2, step 2 — Rotate 20° and the coupling disappears
And the weak axis is much weaker than Iy suggested.
The principal axes are the pair about which vanishes. Mohr’s circle gives both the angle and the two moments:
Here 19.95°, and the principal moments are:
| About | Geometric axes | Principal axes |
|---|---|---|
| Strong | Ix = 6.318 × 10⁶ | Imax = 7.012 × 10⁶ |
| Weak | Iy = 1.743 × 10⁶ | Imin = 1.049 × 10⁶ |
That is the number to take away. The true weak axis carries 1.049 × 10⁶ mm⁴ — only 60% of what suggests. Design an angle strut on and you have overstated its buckling stiffness by 66%.
The invariant holds as a check: — 8.061 = 8.061 × 10⁶ mm⁴.
Hand calculation vs the tool
| Quantity | By hand | StructureCalcs | |
|---|---|---|---|
| Area | 90×12 + 12×138 = 2 736 mm² | 2736 mm² | |
| Centroid x̄ | weighted: (1080×45 + 1656×6)/2736 = 21.39 | 21.39 mm | |
| Centroid ȳ | (1080×6 + 1656×81)/2736 = 51.39 | 51.39 mm | |
| Ixy | NON-ZERO — no axis of symmetry | -1.912 × 10⁶ mm⁴ | |
| Principal angle | ½·atan(−2Ixy/(Ix−Iy)) | 19.95° | |
| Mohr invariant | Imax + Imin = Ix + Iy = 8.061 × 10⁶ | 8.061 × 10⁶ mm⁴ | |
| Imin vs Iy | Imin MUST be ≤ Iy | 1.049 vs 1.743 × 10⁶ — 60% |
Every value was worked by hand with the classical method, then checked against this site’s solver — the same engine the Try it button opens. This agreement is re-run automatically on every build.
Now make it yours
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