Engineering · Mechanical Engineering · Machine Elements
Moment of Inertia Calculator
Calculates the second moment of area (area moment of inertia) for common cross-sectional shapes used in structural and mechanical analysis.
Calculator
Formula
I is the second moment of area (mm⁴ or m⁴). For a rectangle: b is the width and h is the height parallel to the bending axis. For a solid circle: d is the diameter. For a hollow circle: d_o is the outer diameter and d_i is the inner diameter. For an I-beam: b_f is the flange width, h is the total height, t_w is the web thickness, and h_w is the web height.
Source: Hibbeler, R.C. — Mechanics of Materials, 10th Edition; Roark's Formulas for Stress and Strain, 8th Edition.
How it works
The second moment of area, commonly called the moment of inertia in engineering contexts, quantifies how a cross-section's area is distributed about a neutral bending axis. The further the material is located from the centroidal axis, the greater its contribution — which is why I-beams and hollow tubes are far more efficient than solid bars of equivalent area. A higher value of I means the section is stiffer and will deflect less under the same applied bending moment.
Each cross-sectional shape has its own closed-form formula. For a rectangle of width b and height h, the formula is I = bh³/12, where h is measured parallel to the bending axis. For a solid circle of diameter d, I = πd⁴/64. A hollow circular tube uses I = π(d_o⁴ − d_i⁴)/64, which is simply the solid outer circle minus the missing inner area. A symmetric I-beam is treated as a large rectangle minus two rectangular cutouts on either side of the web: I = (bh³/12) − ((b − t_w)(h − 2t_f)³/12). The elastic section modulus S = I/c, where c is the distance from the neutral axis to the extreme fiber, determines the maximum bending stress. The radius of gyration r = √(I/A) is used in column buckling calculations.
These values are foundational inputs for beam deflection formulas, Euler column buckling analysis, bending stress calculations (σ = Mc/I), and finite element pre-processing. Structural engineers use them when selecting steel sections from AISC tables, mechanical engineers apply them when designing shafts and housings, and aerospace engineers rely on them when sizing thin-walled airframe members.
Worked example
Problem: A simply supported steel beam spans 4 m and must carry a central point load. The designer selects a rectangular section with b = 100 mm width and h = 200 mm height. Calculate the second moment of area, section modulus, and radius of gyration.
Step 1 — Second moment of area:
I = bh³ / 12 = (100 × 200³) / 12 = (100 × 8,000,000) / 12 = 66,666,667 mm⁴ ≈ 6.667 × 10⁷ mm⁴
Step 2 — Elastic section modulus:
c = h/2 = 200/2 = 100 mm
S = I / c = 66,666,667 / 100 = 666,667 mm³
Step 3 — Radius of gyration:
A = b × h = 100 × 200 = 20,000 mm²
r = √(I / A) = √(66,666,667 / 20,000) = √3,333.3 = 57.74 mm
If this same beam were replaced by an I-beam with b = 100 mm, h = 200 mm, t_w = 10 mm, and t_f = 15 mm, the web height is h − 2t_f = 170 mm, and the calculation becomes:
I = (100 × 200³)/12 − (90 × 170³)/12 = 66,666,667 − 36,741,250 = 29,925,417 mm⁴ — but the cross-sectional area is only 4,700 mm², giving a much higher specific stiffness per unit mass than the solid rectangle.
Limitations & notes
This calculator computes the second moment of area about the centroidal horizontal axis only (the x-axis through the centroid). It does not compute the moment of inertia about an arbitrary or offset axis — use the parallel axis theorem (I = I_c + Ad²) for that purpose. The I-beam formula assumes a symmetric section with equal flanges; unsymmetric or tapered flanges require numerical integration. Results are purely geometric and do not account for material properties, residual stresses, local buckling, or torsional effects. For very thin-walled sections, shear-lag and local plate buckling considerations may make classical beam theory unreliable. Always verify critical structural members against relevant design codes such as AISC 360, Eurocode 3, or ASME standards.
Frequently asked questions
What is the difference between moment of inertia and second moment of area?
In physics, 'moment of inertia' refers to a body's resistance to rotational acceleration about an axis (units: kg·m²). In structural and mechanical engineering, the term is commonly used interchangeably with 'second moment of area' (units: mm⁴ or m⁴), which is a purely geometric property of a cross-section. This calculator computes the geometric second moment of area used in beam bending and column buckling analysis.
Why is an I-beam more efficient than a solid rectangular beam?
An I-beam concentrates material at the flanges, far from the neutral axis, where it contributes most to bending stiffness (since I ∝ distance⁴ from the axis). The web carries shear. This geometry achieves a high second moment of area with relatively little material, giving a superior stiffness-to-weight ratio compared to a solid rectangular section of the same area.
How do I use the section modulus to find maximum bending stress?
The maximum bending stress in a beam is given by σ = M / S, where M is the applied bending moment and S is the elastic section modulus (I/c). For example, if M = 10 kN·m = 10 × 10⁶ N·mm and S = 666,667 mm³, the extreme-fiber stress is σ = 10,000,000 / 666,667 ≈ 15 MPa. This must be kept below the material's allowable stress.
What is the radius of gyration used for?
The radius of gyration r = √(I/A) represents the equivalent distance from the axis at which the entire cross-sectional area could be concentrated to give the same second moment of area. It is primarily used in Euler column buckling calculations — specifically in computing the slenderness ratio (L/r), where a higher slenderness ratio indicates greater susceptibility to buckling under compressive loads.
Can I calculate the moment of inertia about an axis that is not at the centroid?
Yes — use the Parallel Axis Theorem: I_x = I_c + A·d², where I_c is the centroidal second moment of area (calculated here), A is the cross-sectional area, and d is the perpendicular distance between the centroidal axis and the new axis. This is essential when compositing multiple shapes, such as a T-section made from two rectangles, where each piece has a different centroid offset from the combined section's centroid.
Last updated: 2025-01-15 · Formula verified against primary sources.