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Compound tube: stresses and load share (1000 kN)

Short answer

Equal strain and total load 1000 kN, with A_s=50.27 cm², A_b=56.55 cm², give σ_s = 2σ_b = 12732.4 N/cm² (127.32 MPa) and σ_b = 6366.2 N/cm² (63.66 MPa). Loads: steel 640.0 kN, brass 360.0 kN.

A compound tube consists of a steel tube 15cm internal diameter and 1cm thickness and an outer brass tube of 17cm internal diameter and 1cm thickness. Two tubes are of the same length. The compound tube carries an axial load of 1000 kN. Find the stresses and the load carried by each tube. Take Esteel=2×107 N/cm2E_{steel} = 2\times10^7\ \text{N/cm}^2 and Ebrass=1×107 N/cm2E_{brass} = 1\times10^7\ \text{N/cm}^2

Both tubes carry the load together and have the same length, so their strains are equal and Ps+Pb=1000P_s + P_b = 1000 kN =106= 10^6 N.

Areas

Steel: inner 15 cm, outer 17 cm. Brass: inner 17 cm, outer 19 cm.

As=π4(172−152)=50.27 cm2,Ab=π4(192−172)=56.55 cm2A_s = \frac{\pi}{4}(17^2 - 15^2) = 50.27\ \text{cm}^2, \qquad A_b = \frac{\pi}{4}(19^2 - 17^2) = 56.55\ \text{cm}^2

Stresses

σsEs=σbEb⇒σs=2×107107σb=2σb\frac{\sigma_s}{E_s} = \frac{\sigma_b}{E_b} \Rightarrow \sigma_s = \frac{2\times10^7}{10^7}\sigma_b = 2\sigma_b 106=σb(2As+Ab)=σb(2×50.27+56.55)⇒σb=6366.2 N/cm210^6 = \sigma_b(2A_s + A_b) = \sigma_b(2\times50.27 + 56.55) \Rightarrow \sigma_b = 6366.2\ \text{N/cm}^2 σs=12732.4 N/cm2\sigma_s = 12732.4\ \text{N/cm}^2

Loads

Ps=σsAs=640.00 kN,Pb=σbAb=360.00 kNP_s = \sigma_sA_s = 640.00\ \text{kN}, \qquad P_b = \sigma_bA_b = 360.00\ \text{kN}

Check: 640.00+360.00=1000.00640.00 + 360.00 = 1000.00 kN.

Answer: steel: 12732.4 N/cm212732.4\ \text{N/cm}^2 (127.32127.32 MPa), load 640.0640.0 kN; brass: 6366.2 N/cm26366.2\ \text{N/cm}^2 (63.6663.66 MPa), load 360.0360.0 kN.

✓ Answer checked by Claude Opus 5.5 (high effort)

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