What is the ratio of the strain energy in bar X to that in bar Y
What is the ratio of the strain energy in bar X to that in bar Y when the material of the two bars is the same? The cross-sectional areas are as indicated over the indicated length.
A. 1/3
B. 2/3
C. 4/3
D. 1/6
Please scroll down to see the correct answer and solution guide.
Right Answer is: B
SOLUTION
Concept:
The strain energy stored in a bar due to axial loading P having uniform length L and uniform cross-sectional area A and modulus of elasticity E is given by, \({\rm{U}} = \frac{{{{\rm{P}}^2}{\rm{L}}}}{{2{\rm{EA}}}}\)
If the bar consists of different geometry connected in a series configuration, then the strain energy stored in the bar,
\({\rm{U}} = \sum \frac{{{{\rm{P}}^2}{{\rm{L}}_{\rm{i}}}}}{{2{\rm{E}}{{\rm{A}}_{\rm{i}}}}}\)
Calculations:
For bar X,
\({{\rm{U}}_{\rm{x}}} = \frac{{{{\rm{P}}^2}{\rm{L}}}}{{2{\rm{EA}}}}\)
For bar Y,
\({{\rm{U}}_{\rm{y}}} = \frac{{{{\rm{P}}^2}\frac{{\rm{L}}}{2}{\rm{\;}}}}{{2{\rm{EA}}}} + \frac{{{{\rm{P}}^2}\frac{{\rm{L}}}{2}{\rm{\;}}}}{{2{\rm{E}}\frac{{\rm{A}}}{2}}} = \frac{{3{{\rm{P}}^2}{\rm{L}}}}{{4{\rm{EA}}}}\)
∴ So the ratio of strain energies, \(\frac{{{{\rm{U}}_{\rm{x}}}}}{{{{\rm{U}}_{\rm{y}}}}} = \frac{{\frac{{{{\rm{P}}^2}{\rm{L}}}}{{2{\rm{EA}}}}}}{{\frac{{3{{\rm{P}}^2}{\rm{L}}}}{{4{\rm{EA}}}}}} = \frac{2}{3}\)
∴ The ratio of the strain energy in bar X to that in bar Y to is 2/3.