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Area Moment of Inertia - Typical Cross Sections I

Area Moment of Inertia, Moment of Inertia for an Area or Second Moment of Area for typical cross section profiles

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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 = 4.16x105 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

Ix = 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
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