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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)

Area Moment of Inertia - Moment of Inertia for an Area - 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

Area moment of inertia - Square 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

Area moment of inertia - rectangular 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

Area moment of inertia - circular 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

Area moment of inertia - cylindrical 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

Square - Diagonal Area Moments of Inertia

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

Area Moment of Inertia through Center of Gravity for Rectangular Section

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

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

Symmetrical Shape

Area Moment of Inertia - Symetrical 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 - Nonsymetrical 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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