The fatigue data for a steel alloy are given as follows: $$ \begin{array}{|c|c} \hline \text { Stress Amplitude [MPa (ksi)] } & \text { Cycles to Failure } \\\ \hline 470(68.0) & 10^{4} \\ \hline 440(63.4) & 3 \times 10^{4} \\ \hline 390(56.2) & 10^{5} \\ \hline 350(51.0) & 3 \times 10^{5} \\ \hline 310(45.3) & 10^{6} \\ \hline 290(42.2) & 3 \times 10^{6} \\ \hline 290(42.2) & 10^{7} \\ \hline 290(42.2) & 10^{8} \\ \hline \end{array} $$ (a) Make an \(S-N\) plot (stress amplitude versus logarithm of cycles to failure) using these data. (b) What is the fatigue limit for this alloy? (c) Determine fatigue lifetimes at stress amplitudes of \(415 \mathrm{MPa}(60,000 \mathrm{psi})\) and \(275 \mathrm{MPa}(40,000 \mathrm{psi})\). (d) Estimate fatigue strengths at \(2 \times 10^{4}\) and \(6 \times\) \(10^{5}\) cycles.

Step by step solution

01

Create the S-N plot

To create the \(S-N\) plot, we can plot the stress amplitude (S) on the y-axis and the logarithm of cycles to failure (N) on the x-axis. We will be using the given data from the table to make this plot. Using graph paper or software, plot these points and connect them with smooth lines. The data for the plot are: (4, 470) (4.4771, 440) (5, 390) (5.4771, 350) (6, 310) (6.4771, 290) (7, 290) (8, 290)

02

Determine the fatigue limit

The fatigue limit of a material is the stress level below which the material will not fail, even after a large number of loading cycles. In the \(S-N\) plot, the fatigue limit is apparent as the stress amplitude levels off on the y-axis with increasing cycles to failure. From the plotted data, we can observe that the fatigue limit for this alloy is approximately 290 MPa (42.2 ksi).

03

Determine fatigue lifetimes at stress amplitudes of 415 MPa and 275 MPa

To find the fatigue lifetimes at stress amplitudes of 415 MPa (60,000 psi) and 275 MPa (40,000 psi), we can use the \(S-N\) curve to estimate the respective cycles to failure, N. Interpolating between the data points on the curve, we can estimate the cycles to failure for both values: 1. For 415 MPa (60,000 psi): approximately N = 4.25 x 10^4 cycles 2. For 275 MPa (40,000 psi): approximately N = 1.5 x 10^7 cycles

04

Estimate fatigue strengths at 2 x 10^4 and 6 x 10^5 cycles

Now we will use the \(S-N\) curve to estimate the fatigue strengths at 2 x 10^4 and 6 x 10^5 cycles. Interpolating between the data points on the curve, we can estimate the stress amplitude, S, for both values: 1. At 2 x 10^4 cycles: approximately S = 455 MPa (65.9 ksi) 2. At 6 x 10^5 cycles: approximately S = 325 MPa (47.1 ksi)

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