Show, by direct calculation, that the conclusion of Exercise E10.18 - that the titanium alloy beam with \(Z \sigma_{f}=10^{4} \mathrm{Nm}\) and \(\phi_{B}^{f}\) of 10 is much lighter than the 6061 aluminium alloy beam of the same strength and \(\phi_{B}^{f}-\) is consistent with the idea that to minimize mass for a given stiffness one should maximize \(\frac{\left(\phi_{\beta}^{f} \sigma_{f}\right)^{2 / 3}}{ho}\).

Data From Exercise E10.18

Use the 4-segment chart for strength-limited design of the text to compare the mass per unit

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length, m/L, of a section with Z = 104 Nm (where Z is the section modulus) made from a. 6061 grade aluminium alloy with a shape factor of 3, strength of =200 MPa and density p= 2700 kg/m, and b. titanium alloy with a shape factor 3 of 10, strength of = 480 MPa and density p=4420 kg/m