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.
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^{4} by multiplying with 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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