Chapter 17, Problem 17.79P

(a) Equation (17.12) gives the stress required to keep the length of a rod constant as its temperature changes. Show that if the length is permitted to change by an amount ΔL when its temperature changes by ΔT, the stress is equal to

$\frac{F}{A}=Y\left(\frac{\Delta L}{L_0}-\alpha \Delta T\right)$

where F is the tension on the rod, L₀ is the original length of the rod, A its cross-sectional area, α its coefficient of linear expansion, and Y its Young’s modulus. (b) A heavy brass bar has projections at its ends (Fig. P17.79). Two fine steel wires, fastened between the projections, are just taut (zero tension) when the whole system is at 20°C. What is the tensile stress in the steel wires when the temperature of the system is raised to 140°C? Make any simplifying assumptions you think are justified, but state them.

Figure P17.79