# Area Moment of Inertia – Typical Cross Sections I

Moment of Inertia by Integration: Understanding Problem and Solution: Strengths of Materials
Moment of Inertia by Integration: Understanding Problem and Solution: Strengths of Materials

Area Moment of Inertia – Typical Cross Sections I

## Typical cross sections and their Area Moment of Inertia.

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 = 4.16×105 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

Iy = 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

• Beams and Columns

Deflection and stress, moment of inertia, section modulus and technical information of beams and columns.

• Mechanics

Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

## Related Documents

• American Standard Beams – S Beam

American Standard Beams ASTM A6 – Imperial units.

• American Standard Steel C Channels.

Dimensions and static parameters of American Standard Steel C Channels

• American Wide Flange Beams

American Wide Flange Beams ASTM A6 in metric units.

• Area Moment of Inertia – Typical Cross Sections II

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

• Area Moment of Inertia Converter

Convert between Area Moment of Inertia units.

• Beams – Fixed at Both Ends – Continuous and Point Loads

• Beams – Fixed at One End and Supported at the Other – Continuous and Point Loads

• Beams – Supported at Both Ends – Continuous and Point Loads

• British Universal Columns and Beams

Properties of British Universal Steel Columns and Beams.

• Cantilever Beams – Moments and Deflections

Maximum reaction forces, deflections and moments – single and uniform loads.

• Center Mass

Calculate position of center mass.

• Center of Gravity

A body and the center of gravity.

• Euler’s Column Formula

Calculate buckling of columns.

• HE-A Steel Beams

Properties of HE-A profiled steel beams.

• HE-B Steel Beams

Properties of HE-B profiled steel beams.

• HE-M Steel Beams

Properties of HE-M profile steel beams.

• Mass Moment of Inertia

The Mass Moment of Inertia vs. mass of object, it’s shape and relative point of rotation – the Radius of Gyration.

• Mild Steel – Square Bars

Typical weights of mild steel square bars.

• Normal Flange I-Beams

Properties of normal flange I profile steel beams.

• Pipe Formulas

Pipe and Tube Equations – moment of inertia, section modulus, traverse metal area, external pipe surface and traverse internal area – imperial units

• Radius of Gyration in Structural Engineering

Radius of gyration describes the distribution of cross sectional area in columns around their centroidal axis.

• Section Modulus – Unit Converter

Convert between Elastic Section Modulus units.

• Steel Angles – Equal Legs

Dimensions and static parameters of steel angles with equal legs – metric units.

• Steel Angles – Unequal Legs

Dimensions and static parameters of steel angles with unequal legs – imperial units.

• Steel Angles – Unequal Legs

Dimensions and static parameters of steel angles with unequal legs – metric units.

• Structural Lumber – Section Sizes

Basic size, area, moments of inertia and section modulus for timber – metric units.

• Structural Lumber – Properties

Properties of structural lumber.

• Three-Hinged Arches – Continuous and Point Loads

Support reactions and bending moments.

• W-Beams – American Wide Flange Beams

Dimensions of American Wide Flange Beams ASTM A6 (or W-Beams) – Imperial units.

• Weight of Beams – Stress and Strain

Stress and deformation of vertical beams due to own weight.

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