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:

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

Solution

Let the stress in aluminium bar be σ_a and that in each brass bar be σb. 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 get

(σ_a/E_α) 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)