Halliday, Resnick & Walker — Equilibrium and Elasticity: Problem Problem_83

Lifting cable danger. A crane is to lift a steel beam at a construction site (Fig. 12.70). The beam has length \(L = 12.0\text{ m}\), a square cross section with edge length \(w = 0.540\text{ m}\), and density \(\rho = 7900\text{ kg/m}^3\). The main cable from the crane is attached to two short steel cables of length \(h = 7.00\text{ m}\) and radius \(r = 1.40\text{ cm}\) symmetrically attached to the beam at distance \(d\) from the midpoint. There are three attachment points at \(d_1 = 1.60\text{ m}\), \(d_2 = 4.24\text{ m}\), and \(d_3 = 5.90\text{ m}\). What is the tension \(T_{\text{short}}\) in each short cable for (a) \(d_1\), (b) \(d_2\), and (c) \(d_3\)? What is the stress \(\sigma\) in each short cable for (d) \(d_1\), (e) \(d_2\), and (f) \(d_3\)? Safety protocol requires that the stress in those cables not exceed 80% of the yield stress of \(415 \times 10^6\text{ N/m}^2\). (g) Which of the attachment points pass that requirement?