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

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,

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,
 
If W is total weight of bar (W= wAl), then w = W/Al. In that case, the total extension of the bar
 

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.