Area Moment of Inertia

Area Moment of Inertia 

Area Moment of Inertia or Moment of Inertia for an Area - also known as Second Momentof Area - I, is a property of shape that is used to predict deflection, bending and stress inbeams.Area Moment of Inertia - Imperial units- inches⁴Area Moment of Inertia - Metric units- mm⁴- cm⁴- m⁴Converting between Units-    1 cm⁴ = 10⁻⁸ m⁴= 10⁴mm⁴-    1 in⁴= 4.16x10⁵ mm⁴ = 41.6 cm⁴Area 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
Ix = 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

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 = b³h / 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
r = radius
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 cos 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 yb ³ - B1 hb ³ + b yt ³ - b1 ht ³)      (9)
--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