Find out the axial moment of inertia, Mechanical Engineering

Find out the axial moment of inertia:

Find out the axial moment of inertia of a rectangular area of base b and height d around centroidal axis GX and the base B1B2.

Solution

With Reference to Figure, where centroidal axis GX divides the area at mid-depth, i.e. d/2.

2295_Find out the axial moment of inertia.png

Figure

For a thin strip, illustrated shaded, of width b and thickness (very small) dy, all of the points on it are at a constant distance y from axis GX,

                                                          ∴ dA = bdy

Letting y as positively upward from centroidal axis GX: for elements below

GX, y shall be treated as negative.

                                                        ∴ ∫   A dy = 0

1152_Find out the axial moment of inertia1.png

                                                                         =bd3 /12

Likewise, referring to y axis through centroid G,

                                                       IGY  =   db3 /12

Moment of inertia around base B1 B2 may be computed either directly or by utilizing parallel axis theorem.

Direct approach is as. Referring to Figure,

I B1 B2   = ∫ dA × ( y′)2

1459_Find out the axial moment of inertia2.png

On the other hand, using theorem of parallel axis, we have

I B1 B2 = IGX  + A ( y1 )2

Where y1  =  d/2 = perpendicular distance between GX and B1B2.

= b d 3/12 + b d (d/2)2           ?

= b d 3/ 3

Posted Date: 1/29/2013 1:18:17 AM | Location : United States







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