- 4.1 Centroid — Definition and Formulae
- 4.2 Moment of Inertia — Definition and Derivations (Rectangle, Triangle)
- 4.3 Product of Inertia
- 4.4 Parallel Axis Theorem
- 4.5 Perpendicular Axis Theorem (Polar MOI)
- 4.6 Complete Standard Sections — ALL with Diagrams, Centroid and MOI Formulae
- 4.7 Principal Axes and Principal Moments of Inertia
- 4.8 Rotation of Axes
4.1 Centroid
The centre of area for plane cross-sectional areas (circle, rectangle, triangle etc.) is called the centroid. Represented by C or G. Distance of centroid from origin:
If a plane area has an axis of symmetry about any axis → centroid lies on that axis of symmetry.
If area is symmetrical about two axes → centroid lies at the point of intersection of those two axes of symmetry (e.g., rectangle, circle).
4.2 Moment of Inertia (Second Moment of Area)
MOI of a plane area = second moment of area. If axis x_c, y_c are centroidal axes → I_xc and I_yc are called centroidal moments of inertia or self moments of inertia.
Derivation — Rectangle (b × d) from source:
I_x = ∫y²·dA = ∫₋d/2^(d/2) y²·b·dy
= 2b∫₀^(d/2) y²·dy = 2b·[y³/3]₀^(d/2)
∴ I_x = bd³/12 (centroidal)
Strip at dist y from BASE A-A: dA = b·dy
I_BB = ∫₀^d y²·b·dy = b[y³/3]₀^d
∴ I_BB = bd³/3 (about base)
Similarly: I_y = db³/12 (centroidal)
Derivation — Triangle (base b, height h) from source:
dA = b_y·dy = (b/h)(h−y)dy
I_AA = ∫₀^h y²·dA = (b/h)∫₀^h y²(h−y)dy
∴ I_AA = bh³/12 (about base)
By parallel axis theorem (d = h/3 from base):
I_cc = I_AA − A·(h/3)² = bh³/12 − (bh/2)·(h²/9)
∴ I_cc = bh³/36 (centroidal)
4.3 Product of Inertia
- If x_c, y_c are centroidal axes → I_xcyc = self product of inertia
- Product of inertia depends on coordinate system — can be positive, negative, or zero
- If area has any axis of symmetry → I_xy = 0 about that axis
- Sections with I_xy = 0: rectangle, circle, T-section, I-section, isosceles triangle (see shapes below)
Product of inertia is zero for any section with at least one axis of symmetry
I_xy = ∫₀^b x·(d/2)·d·dx = (h²/2b²)∫₀^b x(b−x)²dx = b²h²/24
4.4 Parallel Axis Theorem
Let x_c, y_c = centroidal axes. Let x, y = other axes parallel to centroidal axes at perpendicular distances d_x and d_y respectively. Let I_xc and I_yc are centroidal MOI. Then:
I_y = I_yc + A·d_y²
I_xy = I_xcyc + A·d_x·d_y
4.5 Perpendicular Axis Theorem + 4.6 Notation
If I_x and I_y are MOI about two axes x and y in the plane of area, then MOI about longitudinal axis z (perpendicular to the plane) = sum of I_x and I_y. I_z is also called Polar Moment of Inertia.
A = area | (x̄, ȳ) = coordinate of centroid C | I_x, I_y = MOI about x and y axis
I_p = I_x + I_y = polar MOI | I_BB = MOI about base axis B-B
4.6 Standard Sections — Complete Centroid & MOI with Diagrams
bh³/12
hb³/12
bh³/3
bh(h²+b²)/12
0 (two axes of symmetry)
bh³/36
hb³/36
bh³/12
bh²(b−2c)/72
bh(h²+b²−bc+c²)/36
πr⁴/4 = πd⁴/64
5πr⁴/4 = 5πd⁴/64
πr⁴/2 = πd⁴/32
0 (two axes of symmetry)
πab³/4
πba³/4
πab(b²+a²)/4
0 (two axes symmetry)
h³(a²+4ab+b²) / 36(a+b)
h³(3a+b) / 12
16bh³/105
2bh³/15
b²h²/12
| Section | Area | I_x (centroidal) | I_y (centroidal) | I_BB (base) | I_xy |
|---|---|---|---|---|---|
| Rectangle (b×h) | bh | bh³/12 | hb³/12 | bh³/3 | 0 |
| Triangle (b×h) | bh/2 | bh³/36 | hb³/36 | bh³/12 | bh²(b-2c)/72 |
| Circle (r) | πr² | πr⁴/4 = πd⁴/64 | πr⁴/4 | 5πr⁴/4 | 0 |
| Semicircle (r) | πr²/2 | ≈0.1098r⁴ | πd⁴/8 | πr⁴/8 | 0 |
| Quarter Circle (r) | πr²/4 | πr⁴/16 | πr⁴/16 | πr⁴/8 | — |
| Circ. Ring (r,t) | 2πrt | πr³t | πr³t | — | 0 |
| Ellipse (a,b) | πab | πab³/4 | πba³/4 | — | 0 |
| Trapezoid (a,b,h) | h(a+b)/2 | h³(a²+4ab+b²)/36(a+b) | — | h³(3a+b)/12 | — |
4.7 Principal Axes and Principal Moments of Inertia
Principal moments of inertia are the maximum and minimum values of MOI at the axis of zero product of inertia. The corresponding axes are called principal axes. Major principal MOI = I₁ (maximum); Minor = I₂ (minimum).
I₂ = I_min = (I_x+I_y)/2 − √[{(I_x−I_y)/2}² + I_xy²]
- Product of inertia about principal axes = ZERO
- Principal axes always pass through centroid
- At least one of principal axes is always a symmetrical axis
- Symmetrical axes are always principal axes, but reverse is not true
θ_p₁ and θ_p₂ = θ_p₁ + 90°; substituting gives I₁ and I₂.
Note: I_x + I_y = I₁ + I₂ = constant
4.8 Rotation of Axes
When axes are rotated by angle θ (anticlockwise positive) from x-y to x’-y’ (same centroid or origin), the MOI and product of inertia about new axes (from source, equations i, ii, iii):
I_y’ = (I_x+I_y)/2 − (I_x−I_y)/2 · cos2θ + I_xy · sin2θ …(ii)
I_x’y’ = (I_x−I_y)/2 · sin2θ + I_xy · cos2θ …(iii)
I_x’ + I_y’ = I_x + I_y (sum is CONSTANT regardless of rotation)
Chapter 4: Centroids and Moments of Inertia — Strength of Materials · MADE EASY Postal Study Package 2019
All sections with individual SVG diagrams showing shape, centroid (CG), axes, and dimensions