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Beam deflection and torsion problems are the basics of traditional (non-FEA approach) structural mechanical design calculations. When solving such problems you have to deal with the two terms area moment of inertia and polar moment of inertia.
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Area Moment of Inertia and Its Application
The area moment of inertia is represented by “I" in calculations. Interestingly, mass moment of inertia also is represented by “I" though some difference between the mass moment of inertia the area moment of inertia exist. The area moment of inertia is the property of a shape and is used in the beam deflection equation:
M/I = σ/Y = E/R
From the equation you can see the stress value (σ) decreases with the increase of the area moment of inertia (I) or in other words it resists beam bending or deflections. So, if you are solving a beam problem, or treating any other problem as a beam problem, then you have to use the area moment of inertia (I). Also for calculating the maximum deflection angle and distance of different types of beam, you will be required to use the area moment of inertia.
There are different formulas for calculating different cross sections. I have an article on How to calculate area moment of inertia for different sections.
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Polar Moment of Inertia and Its Application
The polar moment of inertia is represented by “J". It is used in the torsion equation of shaft:
T/J = τ/r = Gθ/L
As J increase in the above equation, the torque produced in shaft is reduced. So, the polar moment of inertia (J) is used to predict the resistance of a cross section against torsion. The value of the polar moment of inertia of a circular cross section shaft is given by:
Since most of the time you will deal with the shafts with circular cross sections, so the above formula is the only one which you have to remember practically. Although, formulas are available for the polar moment of inertia for other cross sections also.
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The area moment of inertia and the polar moment of inertia are used for mechanical design calculations. The area moment of inertia comes into the picture for beam deflection and bending related problems, and the polar moment of inertia for the torsion of shaft problems.