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Area Moment Of Inertia

The Area Moment Of Inertia or Second Moment of Area is a geometrical property of a beam and depends on a reference axis. 
The Moment of Inertia of a beam's cross-sectional area measures the beams ability to resist bending. The larger the Moment of Inertia the less the beam will bend. 
The smallest Moment of Inertia about any axis passes through the centroid.
The following are the mathematical equations to calculate the Moment of Inertia:
Ixequ. (1)
Iyequ. (2)
Where , y is the distance from the x axis to an infinitesimal area dA.
x is the distance from the y axis to an infinitesimal area dA.

Perpendicular axis theorem 


The moment of inertia of a plane area about an axis normal to the plane is equal to the sum of the moments of inertia about any two mutually perpendicular axes lying in the plane and passing through the given axis.
Izz = Ixx + Iyy

Polar Moment of Inertia

The Moment Of Inertia of an area about an axis perpendicular to its plane (ie. Z - axis) is called as Polar Moment of Inertia.
The Polar Area Moment Of Inertia of a beams cross-sectional area measures the beams ability to resist torsion. The larger the Polar Moment of Inertia the less the beam will twist.
Using the Perpendicular axis theorem  yields the following equations for the Moment of Inertia (J) with respect to Z - axis (ie. perpendicular to X and Y axises)

The following are the mathematical equations to calculate the Polar Moment of Inertia:
Jequ. (3)
x is the distance from the y axis to an infinitesimal area dA.
y is the distance from the x axis to an infinitesimal area dA.
J = Iz = Ix +Iy

Parallel Axis Theorem 


The moment of Inertia of an area about any axis, parallel to centroidal axis in the plane of the area is equal to the sum of moment of Inertia about its centroidal axis of the area and the product of area with square of the distance between these two axises. 
Where A - is the cross-sectional area, and 
d -is the perpendicular distance between the centroidal axis(BB') and the parallel axis(AA'). 

Radius of Gyration 

The radius of gyration of a body is referred to as the radial distance from the rotational axis at which, the entire body mass/ area is supposed to be concentrated and have same moment of inertia. 
The Radius of Gyration kx of an Area (A) about an axis (x) is defined as: 
The radius of gyration Kx    

Where Ix is the Moment of Inertia about the axis (x), and A is the area. 
If no axis is specified the centroidal axis is assumed.

Area Moment of Inertia of some standard Sections 

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