Area moments of inertia
A beam's resistance to bending depends on how its cross-section's area is spread from the centroid. The area moment of inertia measures exactly that.
Definition
The area moment of inertia, or second moment of area, of a section about an axis is \(I = \int y^2\,dA\), where y is the distance from the axis. Because distance is squared, area far from the axis contributes far more. Its units are length to the fourth power, such as mm to the 4.
Rectangles
For a rectangle of width b and height h about its centroidal horizontal axis, \(I = \frac{bh^3}{12}\). Height is cubed, so doubling depth makes a beam eight times stiffer in bending, while doubling width only doubles it.
On edge versus flat
A 50 by 200 mm plank stood on edge has \(I = \frac{50 \times 200^3}{12} \approx 33.3 \times 10^6\) mm⁴; laid flat, \(I = \frac{200 \times 50^3}{12} \approx 2.08 \times 10^6\) mm⁴, sixteen times less. That is why floor joists are installed on edge.
The parallel axis theorem
The moment of inertia about any axis parallel to a centroidal axis is \(I = \bar{I} + Ad^2\), where d is the distance between the axes. It allows composite sections to be built up: each part's own \(\bar{I}\) plus its area times its distance from the overall centroid squared.
Why I-beams are I-shaped
An I-beam places most of its material in the top and bottom flanges, far from the centroid, where it contributes most to I, with a thin web joining them. It achieves high bending stiffness with little weight. Steel I-beams became standard in buildings and bridges in the late nineteenth century.
Circular sections
A solid circular section of radius r has \(I = \frac{\pi r^4}{4}\). A hollow tube puts material far from the centre, so it is stiffer for its weight than a solid rod; bicycle frames, scaffolding, and bones use this principle.
Radius of gyration
The radius of gyration \(k = \sqrt{I/A}\) is the distance at which the whole area could be concentrated to give the same I. It is used in column buckling calculations: slender columns with small k buckle at lower loads.
Worked example
Two 100 by 20 mm flanges are joined by a 20 by 160 mm web to form an I-section 200 mm deep. About the centroid: the web gives \(\frac{20 \times 160^3}{12} \approx 6.83 \times 10^6\) mm⁴; each flange gives \(\frac{100 \times 20^3}{12} + 2{,}000 \times 90^2 \approx 16.27 \times 10^6\) mm⁴. The total is about \(39.4 \times 10^6\) mm⁴. A solid 100 by 200 mm rectangle has \(66.7 \times 10^6\) mm⁴ but uses 20,000 mm² of material; the I-section reaches about 59% of that stiffness with only 7,200 mm², 36% of the material.
Key takeaways
The area moment of inertia I is the integral of y squared dA, in length to the fourth power, measuring how a cross-section resists bending. A rectangle has I = bh cubed over 12, so depth matters far more than width: a plank on edge is 16 times stiffer than flat. The parallel axis theorem, I = I bar + Ad squared, builds up composite sections. I-beams and hollow tubes place material far from the centroid for stiffness with low weight. The radius of gyration, root of I over A, governs column buckling.
Flashcard
Integral of y^2 dA
Check your understanding
How many times stiffer in bending is a 50 by 200 mm plank on edge than laid flat?