Engineering Mechanics – Units and Definitions What is the correct SI unit for the second moment of area (also called the area moment of inertia) of a plane area?

Introduction / Context:
In strength of materials and structural analysis, we frequently use the area moment of inertia (also called the second moment of area) to quantify a section’s resistance to bending and deflection. This is not the same as the mass moment of inertia used in dynamics. Knowing the correct SI unit prevents dimensional mistakes in formulas for beam bending and column buckling.

Given Data / Assumptions:

• We are referring to a geometric property of an area, not a mass distribution.
• Standard SI base units are used (m, kg, s).
• Common symbols: I, Ix, Iy for area moments about reference axes.

Concept / Approach:
The area moment of inertia is defined by integrals such as Ix = ∫ y^2 dA and Iy = ∫ x^2 dA. The integrand multiplies an area element by the square of a length. Therefore, its unit must be area (m^2) times length squared (m^2), giving m^4. This is purely geometric and independent of material density.

Step-by-Step Solution:
Start from definition: I = ∫ (distance)^2 dA. Unit of distance^2 = m^2. Unit of area element dA = m^2. Multiply: m^2 * m^2 = m^4 ⇒ unit of I is m^4.

Verification / Alternative check:
In beam deflection formulas like δ = (W L^3) / (3 E I) for a cantilever with end load, W has unit N, L^3 has m^3, E has N/m^2, so I must supply m^4 to keep δ in metres. The dimensional balance confirms I in m^4.

Why Other Options Are Wrong:
kg·m^2: unit of mass moment of inertia (rotational dynamics), not area property. kg·m·s^2 and kg/m^2: unrelated to second moment of area. N·m·s: not relevant; mixes force, length, and time.

Common Pitfalls:
Confusing area moment (m^4) with mass moment (kg·m^2). Using cm^4 or mm^4 without converting properly to SI.

Final Answer:
m^4