Question 1 of 8: Composite Brass/Steel Column with a Gap

Question 1: Composite Brass/Steel Column with a Gap (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Approach. Because the tube is separated from the cap by a gap, the brass post alone carries the entire load until either (a) the post compresses enough to close the 1 mm gap, engaging the tube, or (b) the post yields first — whichever happens at the smaller load; compare the two.

1. Section properties. Brass post area (solid circle, $d=60$ mm): $$A_{br}=\frac{\pi}{4}(60)^2 = 2827.4\text{ mm}^2$$ Steel tube area (annulus, 60–80 mm): $$A_{st}=\frac{\pi}{4}\left(80^2-60^2\right)=2199.1\text{ mm}^2$$
2. Displacement needed to close the 1 mm gap, if the brass post alone carried the load. The post shortens by $\delta = PL/(A_{br}E_{br})$; equivalently, the compressive strain needed for a 1 mm shortening over 250 mm is $$\varepsilon_{gap}=\frac{\delta_{gap}}{L}=\frac{1}{250}=0.0040$$
3. Strain at which the brass post itself yields. $$\varepsilon_{y,br}=\frac{\sigma_{y,br}}{E_{br}}=\frac{70}{100{,}000}=0.00070$$ Converting to a displacement: $\delta_{y,br}=\varepsilon_{y,br}\,L = 0.00070(250)=0.175\text{ mm}$. Since $0.175\text{ mm} \ll 1\text{ mm}$, the brass post reaches its yield stress while the gap is still $1-0.175=0.825$ mm open — the steel tube never engages; it carries no load at the critical state.
4. Governing load. With the tube inactive, the post alone must not exceed its yield stress: $$\boxed{P_{max}=\sigma_{y,br}A_{br}=70(2827.4)=197{,}920\text{ N}\approx 197.9\text{ kN}}$$