Parallel Axis Theorem (I = I_c + Ad²)

A section's second moment of area is only ever quoted about one particular axis, and the moment you build a shape out of parts, most of those parts are nowhere near the axis the whole thing bends about. The transfer term fixes that in one line: move an area A a distance d away from its own centroid and its contribution grows by Ad². Take a 200 × 20 mm flange plate sitting 240 mm above the neutral axis of a girder. Its own $I_c = bh^3/12 = 133\,333$ mm⁴; the transfer term is $4000 \times 240^2 = 230{,}400{,}000$ mm⁴. The plate's own stiffness is 0.06% of what it actually delivers.

That ratio is the whole argument for the I-beam. Material only earns its keep by being far from the neutral axis, and the transfer term is quadratic while the local term is not. It also explains a fact that surprises people: you can nearly always ignore $I_c$ for a thin flange and lose almost nothing, but you can never ignore it for the web, whose centroid is right on the axis and whose entire contribution is the local term.

Two traps. The theorem only works from the part's own centroidal axis outward, never between two arbitrary parallel axes, so if you already transferred once you must come back to the centroid before transferring again. And d is measured to the centroid of the composite section, which you have to locate first by taking area moments. Getting the composite centroid wrong is the most common failure in a built-up section calculation, and it is silent: the arithmetic all works, the answer is just wrong.