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Compound tube: stresses, load share and shortening (750 kN)

Short answer

Equal strain gives σ_s = 2σ_b. With total load 750 kN and areas A_s=4398.2 mm², A_b=5340.7 mm², solving yields σ_s=106.10 MPa, σ_b=53.05 MPa, loads P_s=466.67 kN, P_b=283.33 kN, shortening 0.1326 mm.

A compound tube consists of steel tube 130mm internal diameter and 10mm thickness and outer brass tube 160mm internal diameter and 10mm thickness. The two tubes are of same length. The combined tube carries a compressive load of 750 kN. Find the stresses and load carried by each of the steel tube and the brass tube and the amount it shortens. Length of each tube is 250mm. Esteel=2×105 MPaE_{steel} = 2\times10^5\ \text{MPa} Ebrass=1×105 MPaE_{brass} = 1\times10^5\ \text{MPa}.

Principle: both tubes have the same length and are loaded together, so they shorten equally: εs=εb\varepsilon_s = \varepsilon_b. The total load is shared: Ps+Pb=750P_s + P_b = 750 kN.

Areas

Steel: inner d=130d = 130 mm, outer D=150D = 150 mm. Brass: inner d=160d = 160 mm, outer D=180D = 180 mm.

As=π4(1502−1302)=4398.2 mm2,Ab=π4(1802−1602)=5340.7 mm2A_s = \frac{\pi}{4}(150^2 - 130^2) = 4398.2\ \text{mm}^2, \qquad A_b = \frac{\pi}{4}(180^2 - 160^2) = 5340.7\ \text{mm}^2

Stresses

Equal strain gives

σsEs=σbEb⇒σs=EsEbσb=2σb\frac{\sigma_s}{E_s} = \frac{\sigma_b}{E_b} \Rightarrow \sigma_s = \frac{E_s}{E_b}\sigma_b = 2\sigma_b P=σsAs+σbAb=σb(2As+Ab)⇒750 000=σb(2×4398.2+5340.7)P = \sigma_sA_s + \sigma_bA_b = \sigma_b(2A_s + A_b) \Rightarrow 750\,000 = \sigma_b(2\times4398.2 + 5340.7) σb=53.05 MPa,σs=106.10 MPa\sigma_b = 53.05\ \text{MPa}, \qquad \sigma_s = 106.10\ \text{MPa}

Loads carried

Ps=σsAs=466.67 kN,Pb=σbAb=283.33 kNP_s = \sigma_sA_s = 466.67\ \text{kN}, \qquad P_b = \sigma_bA_b = 283.33\ \text{kN}

Check: 466.67+283.33=750.00466.67 + 283.33 = 750.00 kN.

Shortening

δ=σsLEs=106.10×2502×105=0.1326 mm\delta = \frac{\sigma_s L}{E_s} = \frac{106.10\times250}{2\times10^5} = 0.1326\ \text{mm}

(Brass gives the same: 53.05×250/105=0.132653.05\times250/10^5 = 0.1326 mm.)

Answer: σs=106.10\sigma_s = 106.10 MPa, σb=53.05\sigma_b = 53.05 MPa; Ps=466.67P_s = 466.67 kN, Pb=283.33P_b = 283.33 kN; shortening =0.1326= 0.1326 mm.

✓ Answer checked by Claude Opus 5.5 (high effort)

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