Question 1.P.13: A uniform beam weighing 500 N is held in a horizontal positi......
A uniform beam weighing 500 N is held in a horizontal position by three vertical wires, one attached to each end of the beam, and one at the mid-length. The outer wires are brass of diameter 0.125 cm, and the central wire is of steel of diameter 0.0625 cm. If the beam is rigid and the wires are of the same length, and unstressed before the beam is attached, estimate the stresses in the wires. Young’s modulus for brass is 85 GN/m² and for steel is 200 GN/m².
On considering the two outer brass wires together, we may take the system as a composite one consisting of a single brass member and a steel member. The area of the steel member is
A_s = \frac{\pi}{4}(0.625 \times 10^{-3})^2 = 0.306 \times 10^{-6} m^2The total area of the two brass members is
A_b = 2\left[\frac{\pi}{4}(1.25 \times 10^{-3})^2\right] = 2.45 \times 10^{-6} m^2Equations (1.20) then give, for the steel wire
\sigma_1 = \frac{PE_1}{A_1E_1 + A_2E_2}, \sigma_2 = \frac{PE_2}{A_1E_1 + A_2E_2} (1.20) \\\\ \sigma_s = \frac{500}{(0.306 \times 10^{-6}) + (2.45 \times 10^{-6})(85/200)} = 370 MN/m^2and for the brass wires
\sigma_b = \frac{500}{(0.306 \times 10^{-6})(200/85) + (2.45 \times 10^{-6})} = 158 MN/m^2