What is the ratio of elongation of a rectangular bar due to self-weight to the elongation if total weight of the bar is applied at its end?

Explanation:

Elongation of the prismatic bar due to self-weight:

\({{\bf{\delta }}_1} = \frac{{{\bf{\gamma }}{{\bf{l}}^2}}}{{2{\bf{E}}}} = \frac{{{\bf{W }}{{\bf{l}}}}}{{2{\bf{AE}}}}\) 

Elongation of bar if total weight of bar is applied at the end is 

Elongation of the conical bar due to self-weight:

\({{\bf{\delta }}_2} = \frac{{{\bf{W }}{{\bf{l}}}}}{{A{\bf{E}}}}\)

Where,

γ = unit weight of the member, l = length of the member and E = young modulus of elasticity

The ratio of the above 2 is given by

\(\frac{{{{\bf{\delta }}_2}}}{{{{\bf{\delta }}_1}}} = {{\frac{{{\bf{W }}{{\bf{l}}}}}{{2A{\bf{E}}}}}\over\frac{{{\bf{W }}{{\bf{l}}}}}{{A{\bf{E}}}}}= {1 \over 2}\)