A cylindrical rod of diameter \(6.7 \mathrm{~mm}\) fabricated from a \(70 \mathrm{Cu}-30 \mathrm{Zn}\) brass alloy is subjected to rotating-bending load cycling; test results (as \(S-N\) behavior) are shown in Figure 8.20. If the maximum and minimum loads are \(+120 \mathrm{~N}\) and \(-120 \mathrm{~N}\), respectively, determine its fatigue life. Assume that the separation between loadbearing points is \(67.5 \mathrm{~mm}\).

Diameter of the rod is given as \(6.7 \mathrm{~mm}\). We can calculate the cross-sectional area (A) using the formula for the area of a circle: \(A=\pi\left(\frac{d}{2}\right)^{2}=\pi\left(\frac{6.7 \mathrm{~mm}}{2}\right)^{2}\). Calculating the cross-sectional area, we get: \(A \approx 35.29 \mathrm{~mm}^{2}\).

Now we will calculate the maximum and minimum stresses (σ_max and σ_min) using the formula for stress: \(\sigma = \frac{F}{A}\), where F is the force and A is the cross-sectional area. For maximum stress (σ_max), F is the maximum load, \( +120 \mathrm{~N}\): \(\sigma_{max} = \frac{120 \mathrm{~N}}{35.29 \mathrm{~mm}^2} \approx 3.4 \mathrm{MPa}\). For minimum stress (σ_min), F is the minimum load, \( -120 \mathrm{~N}\): \(\sigma_{min} = \frac{-120 \mathrm{~N}}{35.29 \mathrm{~mm}^2} \approx -3.4 \mathrm{MPa}\).

We can now calculate the stress amplitude (σ_a) and mean stress (σ_m) using the following formulas: \(\sigma_{a} = \frac{\sigma_{max} - \sigma_{min}}{2} = \frac{3.4 \mathrm{MPa} - (-3.4 \mathrm{MPa})}{2} = 3.4 \mathrm{MPa}\). \(\sigma_{m} = \frac{\sigma_{max} + \sigma_{min}}{2} = \frac{3.4 \mathrm{MPa} + (-3.4 \mathrm{MPa})}{2} = 0 \mathrm{MPa}\).

Now that we have the stress amplitude and mean stress values, we can use the S-N behavior graph to find the fatigue life of the rod. We can compare the stress amplitude value with the S-N graph, which shows cycles (N) on the x-axis and stress amplitude (S) on the y-axis. From the graph, we can find the approximate number of cycles that correspond to a stress amplitude of \(3.4 \mathrm{MPa}\). This will give us an estimate of the fatigue life (N) of the rod. Using the \(S-N\) behavior graph given in Figure 8.20, we find that the fatigue life for a stress amplitude of \(3.4 \mathrm{MPa}\) is approximately between 2 million and 4 million cycles. Therefore, the estimated fatigue life of the rod falls within this range.