Question 2 of 7: Bridge-Cable Wire Count; Measuring Yield Strength and Poisson’s Ratio

Question 2: Bridge-Cable Wire Count; Measuring Yield Strength and Poisson’s Ratio (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.

Given. $L=20$ ft $=240$ in; total load $P=20{,}000$ lb; wire diameter $d=3/16$ in; $\sigma_y=100{,}000$ psi, allowable stress $=0.7\sigma_y$; $E=30\times10^6$ psi; $\delta_{allow}=0.5$ in.

Find. (a) Minimum number of wires, $n$. (b) Experimental method for yield strength and Poisson’s ratio.

Approach

Two independent constraints govern $n$: the wire stress must not exceed 70% of yield, and the cable’s total elastic stretch must not exceed the allowable elongation. Each wire carries an equal share $P/n$ of the load, so both constraints give a lower bound on $n$; the governing (larger) value wins.

1. Wire cross-sectional area. $$A=\frac{\pi}{4}d^2=\frac{\pi}{4}(0.1875)^2=0.02761\ \text{in}^2$$
2. Constraint 1 — allowable stress. $\sigma_{allow}=0.7(100{,}000)=70{,}000$ psi. $$\frac{P}{nA}\le\sigma_{allow}\ \Rightarrow\ n\ge\frac{P}{\sigma_{allow}A}=\frac{20{,}000}{70{,}000(0.02761)}=10.35$$ so stress alone requires $n\ge11$.
3. Constraint 2 — allowable elongation. Each wire stretches $\delta=(P/n)L/(AE)$: $$n\ge\frac{PL}{AE\,\delta_{allow}}=\frac{20{,}000(240)}{0.02761(30\times10^6)(0.5)}=11.59$$ so elongation alone requires $n\ge12$.
4. Governing constraint and final check. Elongation governs (12 $>$ 11): $$\boxed{n=12\ \text{wires}}$$ Check at $n=12$: stress $=20{,}000/[12(0.02761)]=60{,}361$ psi $<70{,}000$ psi ✓; elongation $=(20{,}000/12)(240)/[0.02761(30\times10^6)]=0.483$ in $<0.5$ in ✓. Eleven wires would satisfy the stress limit but stretch $0.527$ in $>0.5$ in, so 11 is not enough — the elongation limit is the binding constraint.
5. (b) Measuring yield strength and Poisson’s ratio. Machine a standard round tensile coupon of the 1080 steel and pull it in a calibrated tensile-testing machine while recording load continuously. Yield strength: mount an axial extensometer on the gauge length, plot engineering stress vs. engineering strain, and apply the standard 0.2% offset construction (a line of slope $E$ drawn from $e=0.002$, parallel to the elastic region; its intersection with the stress-strain curve is the offset yield strength) — the offset method is used because many steels, 1080 included when quenched/tempered, do not show a sharp, unambiguous yield point. Poisson’s ratio requires measuring both the axial strain and the transverse (diametral) strain simultaneously and elastically: mount a second extensometer (or biaxial strain gauge rosette) across the specimen diameter in addition to the axial extensometer, load only into the elastic region (below yield, so the specimen is undamaged and the strains are recoverable), and compute $$\nu=-\frac{\varepsilon_{transverse}}{\varepsilon_{axial}}$$ from the slope of transverse strain vs. axial strain over several small elastic load increments.