The area moment of inertia, also known as the second moment of area, is a 2D geometric property of the cross-sectional shape. It predicts bending, deflection and stresses in beams. It indicates how the points in the cross section are dispersed around an arbitrary axis. The units of the area moment of inertia are metres to the fourth power (m4).
The area moment of inertia is usually denoted by the letter I. The convention is that the X-axis is horizontal in the cross-sectional plane and is denoted as Ix. The Y-axis is vertical in the cross-sectional plane and is denoted by Iy. The usual assumption is that the “width” of any shape is the length of the side along the X-axis and the “height” is the length of the side in the Y-axis.
The Moment of Inertia calculator uses the following formulas:
[Note: the term names in the formulas below relate to the terms in the pictures under the “Shape” section of the calculator]
Triangle
I_x = \frac {b \times h^3} {36}
I_y = \frac {(b^3 \times h) - (b^2 \times h \times a) + (b \times h \times a^2)} {36}
Rectangle
I_x = \frac {a \times b^3} {12}
I_y = \frac {a^3 \times b} {12}
Semicircle
I_x = (\frac {\pi} {8}-\frac {8} {9 \times \pi}) \times r^4
I_y = \frac {\pi \times r^4} {8}
Circle
I_x = I_y = \frac {\pi \times r^4} {4}
Ellipse
I_x = \frac {\pi \times a^3 \times b} {4}
I_y = \frac {\pi \times a \times b^3} {4}
Regular Hexagon
I_x = I_y = \frac {5 \times \sqrt 3 \times a^4} {16}
Try our Beam Deflection calculator to find out how the area moment of inertia is used.
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