Principal axes of a cross-section

The idea

The second moments of area about the centroidal x and y axes form a symmetric matrix, [Ixx, Ixy; Ixy, Iyy]. Its eigenvalues are the principal second moments: the largest (the strong axis) and the smallest (the weak axis) values over every rotation of the axes. Its eigenvectors point along those axes. It is the same mathematics as principal stresses, which is why the two sheets look alike.

Worked example

The sample takes Ixx = 8.2 x 10^6 mm^4, Iyy = 3.1 x 10^6 mm^4 and a product of inertia Ixy = -2.4 x 10^6 mm^4. Results from the engine:

The closed form is the mean plus and minus R, with R = sqrt(((Ixx - Iyy) / 2)^2 + Ixy^2), and it agrees with the eigenvalues. A second check: the sum I1 + I2 is 11300000 mm⁴, equal to Ixx + Iyy, 11300000 mm⁴, because the trace of the matrix does not change when the axes rotate.

Signs and conventions

The sign of the product of inertia depends on the quadrants the section occupies and on the convention your textbook uses for it, and the matrix can be written with +Ixy or -Ixy. The principal values are the same either way; only the sign of the angle changes. Check the direction of the strong axis on a sketch of the section before you use it.

Using the result

Resolve a bending moment into components about the two principal axes: each component bends the section about its own axis only, and the stresses from the two add. For buckling, the weak axis governs: a column check should use I2, not the smaller of Ixx and Iyy, which for an unsymmetric section is larger than I2. The column buckling calculator uses the smaller second moment of a rectangle; for another shape, give it I2.

The sheet source

The sample as it ships (this page adds two lines for the trace check):

# Principal axes of a section

The second moments of area about any pair of axes form a symmetric matrix. Its eigenvalues
are the principal second moments and its eigenvectors are the principal axes, so the same
`eigenvals` and `eigenvecs` calls that give principal stresses give the strong and weak axes
of an unsymmetric section.

## Section properties about the centroid
Ixx: 8.2e6 mm^4 | second moment about x
Iyy: 3.1e6 mm^4 | second moment about y
Ixy: -2.4e6 mm^4 | product of inertia

I: [Ixx, Ixy; Ixy, Iyy]

## Principal values and axes
I_p: eigenvals(I)
I_2: I_p.1
I_1: I_p.2

The larger value is the strong axis. Each row of `eigenvecs` is a unit vector along an axis:

V: eigenvecs(I)
theta_p: atan(V.(2, 2)/V.(2, 1))

## Check with the closed form
I_avg: (Ixx + Iyy)/2
R: sqrt(((Ixx - Iyy)/2)^2 + Ixy^2) -> mm^4
I_1c: I_avg + R
I_2c: I_avg - R

I_sum: I_1 + I_2 -> mm^4 | the trace after rotation
I_xy_sum: Ixx + Iyy -> mm^4 | the trace before rotation