The stress at any cross section can be found by dividing the load by the area of cross section and extension can be found by integrating extensions of a small length over whole of the length of the bar. We shall consider the following cases of variable cross section:
Consider a circular bar that tapers uniformly from diameter d1 at the ** end to diameter d2 at the smaller end, and subjected to axial tensile load P as shown in fig. Let us consider a small strip of length dx at a distance x from the ** end. Diameter of the elementary strip:
Cross-sectional area of the strip,
Elongation of the strip
The total elongation of this tapering bar can be worked out by integrating the above expression between the limits x = 0 to x = L
Putting the value of k = (d1 – d2) /l in the above expression, we obtain
If the bar is of uniform diameter d throughout its length, then
δL = 4.P.L/(Π.E.d2)
= P.L/[(Πd2/4).E] = P.l/A.E.; Which is same as last article
