Moment of Inertia, Section Modulus & Polar Moment - Theory

What is the moment of inertia of a section?

The area moment of inertia (also called the second moment of area) is a geometric property of a cross-section that quantifies how its area is distributed about an axis. It governs how stiff and how strong a beam or column is in bending: the further the material lies from the bending axis, the larger the moment of inertia and the more bending the section resists. It has units of length to the fourth power (mm⁴, cm⁴, in⁴). For an elemental area dA at distance z from the axis,

Properties are always reported about the centroidal axes - the axes through the area\'s centre of mass. Following Eurocode 3 (EN 1993) the major (strong) axis is labelled $y\text{-}y$ and the minor (weak) axis $z\text{-}z$. Bending about the major axis produces the smallest deflection for a given load.

Section modulus and bending stress

The elastic section modulus $W$ turns the moment of inertia into a strength measure. It is the moment of inertia divided by the distance $c$ from the centroid to the extreme fibre:

The bending stress $\sigma$ is largest at the fibre furthest from the centroid. For an asymmetric section (a tee, a channel) the top and bottom fibres are at different distances, so the section has a different modulus - and a different peak stress - top and bottom. The calculator reports the modulus for all four extreme fibres.

Polar moment of inertia

The polar second moment of area $I_p$ measures resistance to twisting about the longitudinal axis. By the perpendicular-axis theorem it is simply the sum of the two in-plane values:

For a solid circle this gives $I_p = \pi d^{4}/32$. Note that for non-circular sections the true St. Venant torsion constant $J$ (which appears in the torsional stiffness $GJ$) is not equal to$\,I_p$; the polar second moment of area reported here is exact only for circular sections.

Radius of gyration

The radius of gyration is the distance from the axis at which the whole area could be concentrated to give the same moment of inertia. It is central to column buckling - the slenderness is the buckling length over the radius of gyration, and buckling occurs about the weaker axis (smaller $i$).

The parallel-axis theorem

Built-up sections (I-sections, channels, tees) are split into simple rectangles. Each rectangle\'s moment of inertia is first found about its own centroid, then transferred to the overall section centroid with the parallel-axis theorem:

where $I_c$ is the moment of inertia about the part\'s own centroid, $A$ its area and$\,d$ the distance between the two parallel axes. Summing the shifted contributions of every part gives the section\'s moment of inertia. The same procedure locates the centroid first, as the area-weighted average of the part centroids.

Standard formulas

For the common solid shapes the closed-form formulas (about the major centroidal axis) are:

I-section (wide-flange beam)

The I-section is the workhorse of steel construction. Because it is doubly symmetric, its centroid is at mid-height and the simplest way to find the major-axis moment of inertia is the "full rectangle minus the two side voids"method: take the solid $b \times h$ rectangle and subtract the two rectangular gaps either side of the web, each $(b-t_w)/2$ wide and $h_w = h - 2t_f$ tall:

The minor-axis value $I_z$ is dominated by the two flanges (their $b^{3}$ term). The section moduli follow as $W_y = 2I_y/h$ and $W_z = 2I_z/b$. A real rolled section also has root radii (fillets) at the web–flange junctions; the calculator adds that material when you enter a non-zero radius $r$, which slightly increases the area and inertia.

T-section

A tee is singly symmetric, so - unlike the I-section - its centroid is not at mid-height: it sits closer to the flange. You must locate the centroid first as the area-weighted average of the flange and stem, then apply the parallel-axis theorem to each part:

Because the centroid is off-centre, the top and bottom fibres are at different distances, so the tee has twodifferent section moduli - $W_{y,top} = I_y/c_t$ and $W_{y,bot} = I_y/c_b$ - and bends with different peak stress top and bottom. The calculator reports both.

Enter your dimensions in the tool to see every property - area, centroid, $I_y$, $I_z$,$\,I_p$, the section moduli, the radii of gyration and the plastic modulus - with a live diagram.