# Area Moment of Inertia

## Area Moment of Inertia, Moment of Inertia of an Area or Second Moment of Area

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**Area Moment of Inertia **or** Moment of Inertia of 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

*inches*^{4}

**Area Moment of Inertia** - Metric units

*mm*^{4}*cm*^{4}*m*^{4}

### Converting between Units

*1 cm*^{4}= 10^{-8}m = 10^{4}mm*1 in*^{4}= 4.16x10^{5}mm^{4}= 41.6 cm^{4}*1 cm*^{3}= 10^{-6}m^{3}= 10^{3}mm^{3}

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

can be expressed as

*I _{x} = ∫ y^{2} dA (1)*

*where *

*I _{x} = area moment of inertia (m^{4}, mm^{4}, inches^{4})*

*y = the perpendicular distance from axis x to the element dA (m, mm, inches^{})*

*dA = an elemental area ( m^{2}, mm^{2}, inches^{2})*

### Area Moment of Inertia for some typical Cross Sections

#### Solid Square Cross Section

*I _{x} = b^{4} / 12 (2)*

*where*

*b = side*

* *

*I _{y} = b^{4} / 12 (2b)*

#### Solid Rectangular Cross Section

*I _{x} = b h^{3} / 12 (3)*

*where*

*b = width *

*h = height*

* *

*I _{y} = b^{3} h / 12 (3b)*

#### Solid Circular Cross Section

*I _{x} = π r^{4} / 4 *

* = π d ^{4} / 64 (4)*

*where *

*r = radius*

*d = diameter*

* *

*I _{y} = π r^{4} / 4*

* = π d ^{4} / 64 (4b)*

#### Hollow Cylindrical Cross Section

*I _{x} = π (d_{o}^{4} - d_{i}^{4}) / 64 (5)*

*where *

*d _{o} = cylinder outside diameter*

*d _{i} = cylinder inside diameter*

* *

*I _{y} = π (d_{o}^{4} - d_{i}^{4}) / 64 (5b)*

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

- "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
- "Area Moment of Inertia" is a property of shape that is used to predict deflection, bending and stress in beams

### 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 fibre

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