Find out change in length of circular bar, Mechanical Engineering

Find out change in length of circular bar of uniform taper

Sol.: The stress at any cross section can be found by dividing load by area of cross section and extension can be found by integrating extensions of small length over whole length of the bar. We shall consider following cases of variable cross section:

Consider circular bar which tapers uniformly from diameter d1 at bigger end to diameter d2 at smaller end, and subjected to axial tensile load P as shown in figure given below.

Let us consider small strip of length dx at distance x from the bigger end.

Diameter of elementary strip:

dx = d1  - [(d1  - d2)x]/L

= d1  - kx; where k = (d1  - d2)/L

 

551_Find out change in length of circular bar.png

The cross sectional area of strip,

1463_Find out change in length of circular bar1.png

Stress in the strip,

242_Find out change in length of circular bar2.png 
Strain in strip

862_Find out change in length of circular bar3.png 
Elongation of strip

887_Find out change in length of circular bar4.png 


The total elongation of tapering bar can be worked out by integrating above expression between limits  x = 0 to  x = L

1013_Find out change in length of circular bar5.png 
BY putting the value of  k = (d1 -  d2) /l in above expression, we obtain
208_Find out change in length of circular bar6.png 

  If the bar is of uniform diameter d throughout the length of it, then δL = 4.P.L/(Π.E.d2)

= P.L/[(Πd2/4).E] = P.l/A.E.

Posted Date: 10/20/2012 8:42:57 AM | Location : United States







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