Strain Energy — Worked Examples

A gradually applied force reaches $P=100\text{ kN}$ when the elastic displacement is $\delta=1.00\text{ mm}$. Determine the stored strain energy.

A prismatic bar carries $P=100\text{ kN}$, with $L=2.00\text{ m}$, $A=500\text{ mm}^2$, and $E=200\text{ GPa}$. Determine axial strain energy.

A linear-elastic material is stressed to $100\text{ MPa}$ in uniaxial tension. Take $E=200\text{ GPa}$. Determine strain-energy density.

A steel has $\sigma_y=250\text{ MPa}$ and $E=200\text{ GPa}$. Determine its linear-elastic modulus of resilience.

A prismatic member has constant bending moment $M=10\text{ kN}\cdot\text{m}$ over $L=2.0\text{ m}$. Take $E=200\text{ GPa}$ and $I=50\times10^6\text{ mm}^4$. Determine the bending strain energy.

A shaft carries constant torque $T=1.00\text{ kN}\cdot\text{m}$ over $L=2.00\text{ m}$. Take $J=1.00\times10^6\text{ mm}^4$ and $G=80\text{ GPa}$. Determine torsional strain energy.

A linear elastic member would develop $50\text{ MPa}$ stress if a load were applied gradually. Under the ideal sudden-load model, the same load is applied instantaneously from zero drop height. Determine the ideal peak stress.

A weight $W=500\text{ N}$ falls $h=200\text{ mm}$ onto a collar attached to a steel rod with $L=3000\text{ mm}$, $A=100\text{ mm}^2$, and $E=200000\text{ MPa}$. Use the ideal linear axial-impact model to determine the peak stress.

Show that Castigliano's theorem reproduces the familiar deformation of a prismatic axial bar with strain energy $U=P^2L/(2AE)$.

A beam has no real vertical load at point $C$, but its vertical displacement at $C$ is required using Castigliano's theorem. What is the correct setup?

A unit vertical load at point $B$ causes a $2.4\text{ mm}$ vertical displacement at point $A$ in a linear elastic structure. Under the corresponding reciprocal loading and direction, what displacement at $B$ is caused by a unit load at $A$?