Find out the stresses set up in the brass:

A compound bar is built from three bars 50 mm broad by 12 mm thick fastened together to compose a bar 50 mm broad by 36 mm thick. The middle bar is of aluminum alloy for which E = 70 GN/m² and the outside bars are of brass with E = 100 GN/m². If the bars are at first fastened at 18°C and the temperature of entire assembly is then grow up to 50°C, find out the stresses set up in the brass & the aluminium.

For computation reason, take following values:

α_brass = 18 × 10^-6 K^-1 , and α _aluminium = 22 × 10^-6 K^-1

Solution

Assume the stress in aluminium bar is σ_a and that in each of brass bar be σ_b. Then for equilibrium

Force in brass = Force in aluminium

σ_b × 50 × 12 × 2 = σ_a × 12 × 50

or         2 σ_b = σ_a

and from extension considerations, we obtain

(σ_a/E_a) L + (σ_b / E_b )L = L(α_a - α _b) ΔT

∴          2σ_b/(70 ×10⁹)   + σ_b/ (100 ×10⁹ )= (22 -18) ×10^-6 × (50 - 18)

 or        27σ_b = 4 × 10^-6 × 32 × 700 ×  10⁹

or         σ_b = 3.32 N/mm²    (T)

and      σ_a = 6.64 N/mm²    (C)