The strain energy stored in a body due to suddenly applied load compared to when it is applied gradually is

Concept:

In gradual loading, the loading starts from zero and increases gradually till the body is fully loaded, while in sudden loading, the load is suddenly applied on the body.

\({σ _{gradual}} = \frac{P}{A}\)

\({σ _{sudden}} = \frac{{2P}}{A}\)

\({\rm{Strain\;energy}}\left( {\rm{U}} \right) = \frac{{{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}\)

Calculation:

Given:

σ_gradual = σ, σ_sudden = 2σ.

\({\rm{Strain\;energy}}\left( {\rm{U}} \right) = \frac{{{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}\)

\({{\rm{U}}_{{\rm{gradual}}}} = \frac{{{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}{\rm{\;}},\)

\({{\rm{U}}_{{\rm{suddenly}}}} = \frac{{{{\left( {2{\rm{σ }}} \right)}^2}{\rm{V}}}}{{2{\rm{E}}}} = \frac{{4{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}}\)

\(\frac{{{{\rm{U}}_{{\rm{sudden}}}}}}{{{{\rm{U}}_{{\rm{gradual}}}}}} = \frac{{4{{\rm{σ }}^2}{\rm{V}}}}{{2{\rm{E}}}} \times \frac{{2{\rm{E}}}}{{{{\rm{σ }}^2}{\rm{V}}}} = 4\)

The strain energy stored in a body due to suddenly applied load compared to when it is applied gradually is four times.