Engineer's Notebook §2.5

Second moment of area

The second moment of area measures how a section's material is spread about a bending axis. The further the material sits from the axis, the stiffer the beam. For a solid rectangle about its centroid:

I =b·h³12
I
second moment of area
m⁴
b
width, parallel to the axis
m
h
depth, across the axis
m
A
area
m²
Z
elastic section modulus
m³
y
distance from the axis
m
d
diameter, or inner diameter
m
D
outer diameter
m
s
distance between parallel axes
m
A rectangular cross-section with its horizontal x–x and vertical y–y centroidal axes. Beside it, two bars compare the second moment of area about each axis. y x b = 50 mm h = 150 mm Ix Iy
Fig. 2.5-1Section drawn to scale. Bars compare I about each axis.
A = bh7,500 mm²
Z = bh²/61.88 × 10⁵ mm³
Iy = hb³/121.56 × 10⁶ mm⁴
Ix = bh³/121.41 × 10⁷ mm⁴

Worked example

Uses the values in Fig. 2.5-1. Change them and the working updates.

Given
  • bwidth50 mm
  • hdepth150 mm
Find

The second moment of area about the horizontal centroidal axis, Ix.

Solution
Ix = bh³/12 = 50 mm × (150 mm)³ / 12 = 1.406 × 10⁷ mm⁴ = 1.406 × 10⁻⁵ m⁴
Answer
Ix = 1.41 × 10⁷ mm⁴

About the vertical axis, hb³/12 = 1.56 × 10⁶ mm⁴.

Other sections

About a centroidal axis unless stated.

  • Solid circle

    I = πd⁴64

  • Tube

    I = π(D⁴ − d⁴)64

  • Hollow rectangle

    I = BH³ − bh³12

  • Parallel axis, distance s away

    I = Ic + As²

Assumptions
  • I is taken about an axis through the centroid, for a section of one material.
  • For a rectangle the centroidal axes are also the principal axes, so bending about x–x causes no sideways bending.
  • Dimensions are those of the real section. Rounded corners and holes reduce I.

Derivation

By definition I = ∫ y² dA, where y is the distance of each element of area dA from the axis.

  1. For a rectangle, take strips of width b and thickness dy: dA = b dy.
  2. I = ∫ by² dy, taken from y = −h/2 to y = h/2.
  3. I = b[y³/3] between those limits, which is b(h³/24 + h³/24) = bh³/12.

Depth enters cubed and width only once. That is why joists stand on edge: doubling h makes a beam eight times stiffer, while doubling b only doubles it.