STATICS  THEORY






Integrals of the form
are called first moments of the area. Accordingly, integrals of the form
are called second moments of the area, or area moments of inertia. 


Moments of Inertia for a Plane Area

Moment of Inertia for a Plane Area 

For an area A that lies in the xy plane as shown, the area moments of
inertia about the x and y axes are
Moments of inertia that are calculated about the centroid of the area
are denoted
I_{x'} I_{y'}
The moments of inertia for basic shapes are tabulated in
Sections Appendix. 





Polar Moment of Inertia for a Plane Area

Polar Moment of Inertia for a Plane Area


The moment of inertia for an area that lies in the xy plane can also be calculated about the z axis, which is known as the polar moment of inertia. The polar moment of inertia of the area A is calculated as
If the polar moment of inertia is calculated at the centroid of the area, it is denoted
J_{z'} = I_{x'} + I_{y'}
The polar moment of inertia is commonly used when calculating the torsion of shafts. 





Parallel Axis Theorem for an Area

Parallel Axis Theorem
Parallel Axis Theorem
(Polar Moment of Inertia)


If the area moments of inertia about the centroid are known, then the moments of inertia about any other parallel axis can be found as
I_{x} = I_{x'} + Ad_{y}^{2}
I_{y} = I_{y'} + Ad_{x}^{2}
I_{z} = I_{z'} + Ad^{2}
Parallel Axis Theorem from the Centroid 





Moments of Inertia for Composite Areas

Moment of Inertia  Composite Parts


Just as the centroid of an area can be calculated by breaking the area into simpler composite parts, the moments of inertia of a complicated area can be calculated by breaking the area into simpler composite parts.
For an arbitrary axis, the moments of inertia for an area made of composite parts are given by
This technique also works for polar moment of inertia. 





Subtraction of Material (Holes)



Previously, in the
Centroid: Composite Parts section, it was shown that when the centroid of line, area, or volume are calculated, holes or cutouts can be accounted for by considering them a negative line length, area, or volume respectively. When calculating moments of inertia, we can deal with cutouts and holes in the same manner. Both the moments of inertia about the centroid of the hole as well as the area of the hole are considered negative when we are summing the composite parts. 


