Moment of Inertia Calculator

About Moment of Inertia Calculation

The calculator determines the moment of inertia and other geometric properties of a cross-section: area A, second moments of area I, section moduli W, and radii of gyration i. These values are used for strength and deflection checks of beams, columns, and similar members, and for selecting or comparing section shapes.

The calculation is performed for typical section shapes defined by linear dimensions. The results refer to axes passing through the centroid of the section unless stated otherwise.

Guidelines and recommendations

What is calculated and in which units

Section area A is computed from the linear dimensions and represents the amount of material in the cross-section. Units: mm² (or equivalent when converted).

Second moments of area I_x and I_y describe how the area is distributed about the x and y axes and are used in deflection and buckling calculations. Units: mm⁴.

Section moduli W_x and W_y are used to relate bending moment to the maximum bending stress. Units: mm³.

Radii of gyration i_x and i_y are used in stability checks of members (for example, slenderness). Units: mm.

Core relations used in the calculation

From second moment of area to radius of gyration the calculator uses the definition of radius of gyration.

ix = √( Ix / A ), iy = √( Iy / A )

Section modulus is computed as the ratio of second moment of area to the distance from the neutral axis to the most distant fibre.

Wx = Ix / ymax, Wy = Iy / xmax

Here y_max and x_max are the maximum distances from the axis to the extreme points of the section in the corresponding direction. For symmetric shapes this is typically half the height or width. For non-symmetric shapes the distances to the top and bottom edges may differ, and the corresponding extreme distances are used.

How second moments of area are computed for built-up sections

Built-up section (for example, an I-section, channel, angle, or hollow section) is represented as a sum and difference of simple shapes. For each part the calculator first finds its area A_k, the location of its centroid, and its own second moments of area about axes through that part’s centroid.

Transfer to the common axes is done using the parallel axis theorem.

Ix = Σ( Ix,k + Ak·Δyk2 ), Iy = Σ( Iy,k + Ak·Δxk2 )

Here Δx_k and Δy_k are the offsets of the part centroid relative to the overall section centroid. Holes and cut-outs are treated as “negative” parts, meaning their areas and moments are subtracted.

Typical formulas for basic shapes

Rectangle (width b, height h) is calculated using standard expressions about centroidal axes.

A = b·h, Ix = b·h3/12, Iy = h·b3/12

Circle (diameter d) is calculated using standard expressions about any diameter through the centre.

A = π·d2/4, Ix = Iy = π·d4/64

Circular hollow section (outer diameter D, wall thickness t) is calculated as the difference of two circles. The inner diameter d is taken as d = D − 2t (provided that D > 2t).

A = π·(D2 − d2)/4, Ix = Iy = π·(D4 − d4)/64

How the final value is chosen when there are multiple axes and extreme distances

Separate axes are evaluated separately: I_x, W_x, i_x relate to bending and stability about the x axis, and I_y, W_y, i_y relate to the y axis.

Minimum values may be shown as the “worst-case” for stability, especially for non-symmetric sections. A common practice is to use i_min = min(i_x, i_y) and I_min = min(I_x, I_y).

Practical guidance for using the results

Bending and stress are commonly assessed using the section modulus. For the same bending moment, a larger W gives lower bending stress.

Deflection is sensitive to the second moment of area. For the same material and span conditions, deflection is inversely proportional to I. Increasing the section depth usually increases I_x much more than increasing the width.

Stability of members is often linked to radius of gyration and slenderness. A common relation uses the ratio L/i, where a smaller i gives a more critical case.

EU standards reference

EN 1993-1-1 (Eurocode 3) uses geometric properties (area, second moments of area, radii of gyration, section moduli) in the design of steel members.

EN 1995-1-1 (Eurocode 5) uses the same properties for timber members in strength and serviceability checks.

EN 1999-1-1 (Eurocode 9) similarly uses section properties for aluminium structures.

In these documents the geometry formulas are generally treated as standard background, and the calculated properties are then used in material strength and stability checks.

FAQs

What is the difference between second moment of area and section modulus

Second moment of area I describes how the area is distributed and directly affects stiffness and deflection. Section modulus W also accounts for the distance to the extreme fibres, so it is convenient for estimating bending stress. For a non-symmetric section, W can differ for different sides.

Why does one section have different values about the x and y axes

A section can be stiff in one direction and flexible in the other. For example, a tall narrow shape usually has a large I_x and a small I_y. That is why bending and buckling checks are done separately for each axis.

How are voids in a hollow section or holes in a section handled

A void is handled by subtracting the inner shape from the outer one. The inner area and second moments of area are taken with a negative sign because there is no material there. For a circular hollow section, the condition D > 2t must be satisfied.

Can different section shapes be compared using area only

Area indicates mass for the same material density, but it does not represent stiffness or bending resistance. Two sections with the same area can have very different I and W. For beams in bending it is common to compare both W for stress and I for deflection.

Which results are most often used for column stability

It is common to look at the radii of gyration and take the minimum i, since it produces the most critical slenderness. This is especially important for built-up and non-symmetric sections. The design check then accounts for effective length, end restraints, and the relevant Eurocode rules.