Question

A bar of steel of length "l" and is of uniform thickness ‘"t" The width of the bar varies uniformly from a at one end to b (a > b) at the other end. Find the extension of the rod when it carries an axial pull p.

Answer 1

Consider any section X-X distance x from the bigger end.

Width of section

= a - \(\frac{a-b}L\) x

= a - kx where k = \(\frac{a-b}2\)

Thickness of section = t 

Area of section = t(a – kx)

Stress on the section

 \(\delta\)x = \(\frac{p}{t(a-kx)}\) 

Extension of an elemental length

\(\delta\)x = \(\frac{pdx}{t(a-kx)E}\)

Total extension of the rod