Find increase in the length of bar, Mechanical Engineering

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Find increase in the length of bar:

Q: Find increase in the length of bar of uniform section because of self weight.

Sol.: Take a bar of cross sectional area A and length l hanging freely under its own weight as shown in figure

Let attention be focused on a small element of length dy at distance x from the lower end. If ' ω ' is specific weight (weight per unit volume) of bar material, then total tension at section m-n equals weight of bar for length y and can be given by

P =     Ay

As a result of which load, the elemental length dx elongates by small amount Ax, and
147_Find increase in the length of bar.png

The total change in length of bar because of self weight is worked out by integrating above expression between limits y = a and y = l. Thus,

1247_Find increase in the length of bar1.png

The total change in length of bar because of self weight is worked out by

integrating the above expression between limits y = a and y = l. Thus,
702_Find increase in the length of bar2.png 
If W is total weight of bar (W= wAl), then w = W/Al. In that case, the total extension of the bar
2488_Find increase in the length of bar3.png 

344_Find increase in the length of bar4.png

Figure

Thus total extension of bar because of self weight is equal to the extension that would be produced if ½ of the weight of bar is applied at its end.


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