Area Moment of Inertia

Second Moment of Inertia or Area Moment of Inertia

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

Imperial units

  • inches4

Metric units

  • mm4
  • cm4
  • m4

Converting between units

  • 1 cm4 = 10-8 m = 104 mm
  • 1 in4 = 4.16x105 mm4 = 41.6 cm4
  • 1 cm3 = 10-6 m = 103 mm

Definition of Second Moment of Inertia

Ix = ∫ y2 dA         (1)

where

Ix = second moment of inertia

y = the perpendicular distance from axis x to the element dA

dA = an elemental area

Area Moment of Inertia for some common Cross Sections

Solid Square Cross Section

area moment of inertia square section

Ix = b4 / 12         (2)

where

b = side


Iy = b4 / 12         (2b)

Solid Rectangular Cross Section

area moment of inertia rectangular section

Ix = b h3 / 12         (3)

where

b = width

h = height


Iy = b3 h / 12         (3b)

Solid Circular Cross Section

area moment of inertia circular section

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

area moment of inertia cylindrical section

Ix = π (do4 - di4) / 64         (5)

where

do = cylinder outside diameter

di = cylinder inside diameter


Iy = π (do4 - di4) / 64         (5b)

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