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 Daisy powerline 990 repair manual.

(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

I_x = ∫ y² dA (1)

where

I_x = 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

I_y = ∫ x² dA (2)

where

I_y = 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

I_x = a⁴ / 12 (2)

where

a = side (mm, m, in.)

I_y = a⁴ / 12 (2b)

Solid Rectangular Cross Section

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

I_x = b h³ / 12 (3)

where

b = width

h = height

I_y = b³ h / 12 (3b)

Solid Circular Cross Section

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

I_x = π r⁴ / 4

= π d⁴ / 64 (4)

where

r = radius

d = diameter

I_y = 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

I_x = π (d_o⁴ - d_i⁴) / 64 (5)

where

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d_o = cylinder outside diameter

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d_i = cylinder inside diameter

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I_y = π (d_o⁴ - d_i⁴) / 64 (5b)

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Square Section - Diagonal Moments

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

I_x = I_y = 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

I_x = (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

I_x = (a h³/ 12) + (b / 12) (H³ - h³) (8)

I_y = (a³ h / 12) + (b³ / 12) (H - h) (8b)

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Nonsymmetrical Shape

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

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I_x = (1 / 3) (B y_b³- B₁ h_b³ + b y_t³ - b1 h_t³) (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
'PolarMoment 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

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

Area Moment of Inertia - Imperial units

Area Moment of Inertia - Metric units

Converting between Units

Example - Convert between Area Moment of Inertia Units

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

Area Moment of Inertia for typical Cross Sections I

Solid Square Cross Section

Solid Rectangular Cross Section

Solid Circular Cross Section

Hollow Cylindrical Cross Section

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Square Section - Diagonal Moments

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

Symmetrical Shape

Nonsymmetrical Shape

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

Section Modulus

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