Find out moment of inertia, Mechanical Engineering

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Find out Moment of Inertia:

Find out Moment of Inertia of an I shaped area around its centroidal axis as illustrated in Figure.

82_Find out Moment of Inertia.jpg

 Solution

 (i)        The area of I section may be divided into three parts namely, A1, A2 and A3.

                  A =  A1 +  A2  +  A3  = (400 × 150 × 2 + 300 × 200) = 18 × 104

The centroid G is at mid-depth as illustrated in Figure (a).

Now, I A ( x) = I A ( x)  +  I A  ( x)  +  I A ( x)

where,

 I A ( x)  = I A ( x)  = (1 /12) × 400 × 1503 + 400 × 150 × (225)2

or,

I A 3 ( x)  = I A 1 ( x)  = 106 [112.5 + 3037.5] = 3150 × 106  mm4

I A2 ( x)  = (1/12) × 200 × 3003 = 450 × 106   mm4

∴ I A ( x)  = [(2 × 3150) + 450] × 106

= 6750 × 106  mm4

 (ii)       On the other hand, the M. I. of I-section about x axis may be attained by subtracting the M. I. of area [2 × ( A2′ )] from the M. I. of area ( A1′ ) as illustrated in Figure (b).

Here,

A1′ = 400 × 600 mm2      and  A2′ = 100 × 300 mm2

∴ A =  A1′ - 2 A2′

= 24 × 104  - 6 × 104  = 18 × 104  mm2

Then,

 I A ( x)  = I( A1′ )  - I ( A2′ ) × 2

=400 × 6003/12 - 2 × 100 × 3003 / 12

=   (1/12)  × 106 [86400 - 5400] = 6750 ×106  mm4


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