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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

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.

• inches4

• 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

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

• Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more
• Beams and Columns - Deflection and stress, moment of inertia, section modulus and technical information of beams and columns

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