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:

d_x = d₁  - [(d₁  - d₂)x]/L

= d₁  - kx; where k = (d₁  - d₂)/L

The cross sectional area of strip,

Stress in the strip,

 
Strain in strip

 
Elongation of strip

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

 
BY putting the value of  k = (d₁ -  d₂) /l in above expression, we obtain
 

  If the bar is of uniform diameter d throughout the length of it, then δL = 4.P.L/(Π.E.d^₂)

= P.L/[(Πd_²/4).E] = P.l/A.E.