Determine the stresses set up in the brass:
A compound bar is constructed from three bars 50 mm wide by 12 mm thick fastened together to form a bar 50 mm wide 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 initially fastened at 18°C and the temperature of whole assembly is then raised to 50°C, determine the stresses set up in the brass and the aluminium.
For computation purposes, take following values:
αbrass = 18 × 10-6 K-1 , and αaluminium = 22 × 10-6 K-1
Solution
Let the stress in aluminium bar be σa and that in each brass bar be σb. Then for equilibrium
Force in brass = Force in aluminium
σb × 50 × 12 × 2 = σa × 12 × 50
or 2 σb = σa
and from extension considerations, we get
(σa/Eα) L + (σb/Eb) L = L(αa - αb) ΔT
∴ 2σb/70 ×109 + σb/100 ×109 = (22 -18) ×10-6 × (50 - 18)
or 27σb = 4 × 10-6 × 32 × 700 × 109
or σb = 3.32 N/mm2 (T)
and σa = 6.64 N/mm2 (C)