Thermal Expansion — Question 9

Question 9

Preventing expansion creates compressive stress

A straight uniform rod is initially stress-free between rigid supports. It has Young modulus $E=200\,\mathrm{GPa}$, expansion coefficient $\alpha=12.0\times10^{-6}\,\mathrm{K^{-1}}$ and area $A=2.00\times10^{-4}\,\mathrm{m^2}$. Use the supplied small-strain relation $\epsilon=\sigma/E+\alpha\Delta T$, with tensile stress positive. Assume the rod remains straight and elastic.

1. Find stress and support-force magnitude for a $50.0\,\mathrm K$ rise.

2. If allowable compressive stress is $150\,\mathrm{MPa}$, find the maximum rise in this model. Explain why this is not also a buckling guarantee.

Question 9 - Solution

Preventing expansion creates compressive stress

(a) Method and calculation

Rigid supports impose zero total strain:$$0=\sigma/E+\alpha\Delta T\ \Longrightarrow\ \sigma=-E\alpha\Delta T.$$$$\sigma=-120\,\mathrm{MPa},\qquad |F|=|\sigma|A=24000\,\mathrm N.$$

(b) Interpretation and further calculation

$$E\alpha\Delta T\le150\times10^6\,\mathrm{Pa}.$$$$\Delta T_{\max}=\frac{150\times10^6}{200\times10^9(12\times10^{-6})}=62.5\,\mathrm K.$$A slender rod can buckle before reaching that material-stress limit; geometric stability requires a separate model.

Physical check

Heating gives compression with the stated tensile-positive convention. Cooling would reverse the stress if both supports can sustain tension.