# 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.

**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^{4}= 10^{4}mm^{4}*1 in*^{4}= 4.16x10^{5}mm^{4}= 41.6 cm^{4}

**Example - Convert between Area Moment of Inertia Units**

*9240 cm ^{4}* can be converted to

*mm*by multiplying with

^{4}*10*

^{4}*(9240 cm ^{4}) 10^{4} = 9.24 10^{7} mm^{4}*

**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} = ∫ y^{2} dA (1)*

*where *

*I _{x} = Area Moment of Inertia related to the x axis (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})*

The Moment of Inertia for bending around the y axis can be expressed as

*I _{y} = ∫ x^{2} dA (2)*

*where *

*I _{y} = Area Moment of Inertia related to the y axis (m^{4}, mm^{4}, inches^{4})*

*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} = a^{4} / 12 (2)*

*where*

*a = side (mm, m, in..)*

* *

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

#### Solid Rectangular Cross Section

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

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

*where*

*b = width *

*h = height*

* *

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

#### Solid Circular Cross Section

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

*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

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

*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)*

#### Square Section - Diagonal Moments

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

*I _{x} = I_{y} = a^{4} / 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) (h^{2} cos^{2} a + b^{2} sin^{2} a) (7)*

#### Symmetrical Shape

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

*I _{x} = (a h^{3 }/ 12) + (b / 12) (H^{3} - h^{3}) (8)*

*I _{y} = (a^{3} h / 12) + (b^{3} / 12) (H - h) (8b)*

#### Nonsymmetrical Shape

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

*I _{x} = (1 / 3) (B y_{b}^{3 }- B_{1} h_{b}^{3} + b y_{t}^{3} - b1 h_{t}^{3}) (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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