### Example: Change in length for a bar with continuously varying load or dimensions

A tapered bar AB of solid circular cross-section and length L (Fig. 1-a) is supported at end B and subjected to a tensile load P at the free end A. The diameters of the bar at ends A and B are dA and dB, respectively. Determine the elongation of the bar due to the load P, assuming that the angle of taper is small.

Figure 1

in this example force is constant through the whole length of the tapered, to obtain the elongation we should derive an expression for area, therefore we should set an origin for our coordination x. to simplify the problem we will extend the side of the bar until it intersect at point O, we set the origin of our coordination system at point O.

From triangle similarity

dA/LA=dB/LB

For d(X)

d(x)/X=dA/LA

d(X)=dA*X/LA

A(X)=π*(diameter^2)/4
diameter for the bar=dx
A(X)=π*((dA*X/LA)^2)/4
A(X)=π*((dA^2*X^2)/LA^2)/4

dδ=(N(X)*dx)/(A(X)*EA)

dδ=(4*P*LA^2dx)/(EA*π*dA^2*X^2)

δ

δ

δ
for simplification

substitute  this to δ

we substitute LA/LB=dA/dB see previous equation

δ