Principal Axes and Principal Moments of Inertia Calculator

What are principal axes?

The second moment of area of a section depends on the axis it is taken about. As the axes are rotated within the section, I_x and I_y change. At one angle the moment of inertia reaches its maximum, and about the perpendicular axis its minimum. These two axes are the principal axes and the values are the principal moments of inertia. The product of inertia is zero for the principal axes.

Sections with an axis of symmetry (I sections, boxes, channels) already have their principal axes along the symmetry axes. For unsymmetric sections such as angles and Z sections the principal axes are inclined and have to be calculated.

Formulas

• Principal moments: I₁, I₂ = (I_x + I_y) / 2 ± √(((I_x − I_y) / 2)² + I_xy²)
• Principal axis angle: tan 2θ = −2 × I_xy / (I_x − I_y)
• Invariant sum: I₁ + I₂ = I_x + I_y

The angle is measured from the x axis, positive counterclockwise. These relations have the same form as Mohr's circle for stress: the centre of the circle is (I_x + I_y) / 2 and its radius is the square root term.

Example

A 100 × 100 × 10 mm angle without corner radii has I_x = I_y = 180.0 cm⁴ and I_xy = −106.6 cm⁴ about its centroidal axes. With the two moments equal, the radius is simply 106.6. So I₁ = 180.0 + 106.6 = 286.6 cm⁴ and I₂ = 180.0 − 106.6 = 73.4 cm⁴. The axis of maximum inertia is at 45 degrees, which is the angle's axis of symmetry.

Why it matters

A compression member buckles about its weakest axis: for an angle that is I₂ and the matching minimum radius of gyration. An unsymmetric beam loaded off its principal axes bends obliquely, and the moment then has to be resolved along the principal axes.

Limits

The tool only performs the axis transformation. All three inputs must refer to axes through the same point. For rolled sections the corner radii make table values differ from the sharp cornered calculation by a few percent.