Area Moment of Inertia – Typical Cross Sections I

Moment of Inertia by Integration: Understanding Problem and Solution: Strengths of Materials
Moment of Inertia by Integration: Understanding Problem and Solution: Strengths of Materials

Area Moment of Inertia – Typical Cross Sections I

Typical cross sections and their Area Moment of Inertia.

Area Moment of Inertia or Moment of Inertia for an Area – also known as Second Moment of Area – I, is a property of shape that is used to predict deflection, bending and stress in beams.

Area Moment of Inertia – Imperial units

  • inches4

Area Moment of Inertia – Metric units

  • mm4
  • cm4
  • m4

Converting between Units

  • 1 cm4 = 10-8 m4 = 104 mm4
  • 1 in4 = 4.16×105 mm4 = 41.6 cm4

Example – Convert between Area Moment of Inertia Units

9240 cm4 can be converted to mm4 by multiplying with 104

(9240 cm4) 104 = 9.24 107 mm4

Area Moment of Inertia (Moment of Inertia for an Area or Second Moment of Area)

for bending around the x axis can be expressed as

Ix = ∫ y2 dA (1)

where

Ix = Area Moment of Inertia related to the x axis (m4, mm4, inches4)

y = the perpendicular distance from axis x to the element dA (m, mm, inches)

dA = an elemental area (m2, mm2, inches2)

The Moment of Inertia for bending around the y axis can be expressed as

Iy = ∫ x2 dA (2)

where

Iy = Area Moment of Inertia related to the y axis (m4, mm4, inches4)

x = the perpendicular distance from axis y to the element dA (m, mm, inches)

Area Moment of Inertia for typical Cross Sections I

Solid Square Cross Section

The Area Moment of Inertia for a solid square section can be calculated as

Ix = a4 / 12 (2)

where

a = side (mm, m, in..)

Iy = a4 / 12 (2b)

Solid Rectangular Cross Section

The Area Moment of Ineria for a rectangular section can be calculated as

Ix = b h3 / 12 (3)

where

b = width

h = height

Iy = b3 h / 12 (3b)

Solid Circular Cross Section

The Area Moment of Inertia for a solid cylindrical section can be calculated as

Ix = π r4 / 4

= π d4 / 64 (4)

where

r = radius

d = diameter

Iy = π r4 / 4

= π d4 / 64 (4b)

Hollow Cylindrical Cross Section

The Area Moment of Inertia for a hollow cylindrical section can be calculated as

Ix = π (do4 – di4) / 64 (5)

where

do = cylinder outside diameter

di = cylinder inside diameter

Iy = π (do4 – di4) / 64 (5b)

Square Section – Diagonal Moments

The diagonal Area Moments of Inertia for a square section can be calculated as

Ix = Iy = a4 / 12 (6)

Rectangular Section – Area Moments on any line through Center of Gravity

Rectangular section and Area of Moment on line through Center of Gravity can be calculated as

Ix = (b h / 12) (h2 cos2 a + b2 sin2 a) (7)

Symmetrical Shape

Area Moment of Inertia for a symmetrical shaped section can be calculated as

Ix = (a h3 / 12) + (b / 12) (H3 – h3) (8)

Iy = (a3 h / 12) + (b3 / 12) (H – h) (8b)

Nonsymmetrical Shape

Area Moment of Inertia for a non symmetrical shaped section can be calculated as

Ix = (1 / 3) (B yb3 – B1 hb3 + b yt3 – b1 ht3) (9)

Area Moment of Inertia vs. Polar Moment of Inertia vs. Moment of Inertia

  • “Area Moment of Inertia” is a property of shape that is used to predict deflection, bending and stress in beams
  • “Polar Moment of Inertia” as a measure of a beam’s ability to resist torsion – which is required to calculate the twist of a beam subjected to torque
  • “Moment of Inertia” is a measure of an object’s resistance to change in rotation direction.

Section Modulus

  • the “Section Modulus” is defined as W = I / y, where I is Area Moment of Inertia and y is the distance from the neutral axis to any given fiber

Related Topics

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