EngineerStudio AI logoEngineerStudio AI

Parallel Axis Theorem Calculator

Moment of inertia about a shifted axis: I = Ic + A·d².

INPUTS
×10⁶ mm⁴
mm²
mm
RESULTS
Transfer term A·d²
144.4 ×10⁶ mm⁴
Total I = Ic + A·d²
144.533 ×10⁶ mm⁴
A·d² share of total
99.9 %
Rule
I is always minimum about the centroidal axis — shifting the axis only ever adds inertia
0501001502002503003500100200300400500600
Inertia grows with the square of the shiftaxis shift d [mm]I [×10⁶ mm⁴]
How it works

The parallel axis theorem is how composite section properties are built: each part contributes its own centroidal inertia plus area times distance squared to the common axis. Because d enters squared, placing area far from the neutral axis is overwhelmingly effective — a flange's own Ic is usually negligible next to its A·d². It's the mathematical reason I-beams, box girders and sandwich panels exist.

Worked example

A 200×20 mm flange plate (Ic = 0.133×10⁶ mm⁴, A = 4000 mm²) located 190 mm from a girder's neutral axis contributes I = 0.133 + 4000·190²/10⁶ = 144.5×10⁶ mm⁴ — the A·d² term is over 1000× its own Ic.

Frequently asked questions

Related calculators
Need the full picture — SFD, BMD, deflected shape and a report?

Open EngineerStudio to build this as an interactive project with a verified solver and AI tutor.