Elasticity — Question 9

Question 9

The effect of added area depends on what the loading device holds fixed

A uniform elastic rod has stiffness $k=EA/L$ and stored energy $U=\tfrac12k\delta^2$. Compare two otherwise identical rods, one with area $A$ and one with area $2A$. Treat each as loaded quasistatically from its own unloaded state.

1. At the same final force $F$, compare their extensions and stored energies.

2. At the same final extension $\delta$, compare forces and energies. Explain why the two energy comparisons differ.

Question 9 - Solution

The effect of added area depends on what the loading device holds fixed

(a) Derivation and calculation

Doubling area doubles stiffness: $k'=E(2A)/L=2k$. At a common final force, $$\delta_A=F/k,\qquad\boxed{\delta_{2A}=F/(2k)=\delta_A/2}.$$ Since $U=F^2/(2k)$, $$U_A=F^2/(2k),\qquad\boxed{U_{2A}=F^2/(4k)=U_A/2}.$$

(b) Result and interpretation

At a common final extension, $$F_A=k\delta,\qquad\boxed{F_{2A}=2k\delta=2F_A},$$ $$U_A=\tfrac12k\delta^2,\qquad\boxed{U_{2A}=\tfrac12(2k)\delta^2=2U_A}.$$ The work is $\int_0^\delta kx\,dx$, the area under the respective loading line. Fixing force selects different endpoints from fixing displacement, reversing the energy comparison.

Physical check

The identity $U=F\delta/2$ gives both ratios directly from the corresponding force and extension ratios.