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
- 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
- 1 cm3 = 10-6 m3 = 103 mm3
Area Moment of Inertia (Moment of Inertia of an Area or Second Moment of Area)
can be expressed as
Ix = ∫ y2 dA (1)
where
Ix = area moment of inertia (m4, mm4, inches4)
y = the perpendicular distance from axis x to the element dA (m, mm, inches)
dA = an elemental area (m2, mm2, inches2)
Area Moment of Inertia for some typical Cross Sections
Solid Square Cross Section
Ix = b4 / 12 (2)
where
b = side
Iy = b4 / 12 (2b)
Solid Rectangular Cross Section
Ix = b h3 / 12 (3)
where
b = width
h = height
Iy = b3 h / 12 (3b)
Solid Circular Cross Section
Ix = π r4 / 4
= π d4 / 64 (4)
where
r = radius
d = diameter
Iy = π r4 / 4
= π d4 / 64 (4b)
Hollow Cylindrical Cross Section
Ix = π (do4 - di4) / 64 (5)
where
do = cylinder outside diameter
di = cylinder inside diameter
Iy = π (do4 - di4) / 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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