The fatigue data for a ductile cast iron are given as follows: $$ \begin{array}{cc} \hline \begin{array}{c} \text { Stress Amplitude } \\ \text { [MPa (ksi)] } \end{array} & \begin{array}{c} \text { Cycles to } \\ \text { Failure } \end{array} \\ \hline 248(36.0) & 1 \times 10^{5} \\ 236(34.2) & 3 \times 10^{5} \\ 224(32.5) & 1 \times 10^{6} \\ 213(30.9) & 3 \times 10^{6} \\ 201(29.1) & 1 \times 10^{7} \\ 193(28.0) & 3 \times 10^{7} \\ 193(28.0) & 1 \times 10^{8} \\ 193(28.0) & 3 \times 10^{8} \\ \hline \end{array} $$ (a) Make an \(S-N\) plot (stress amplitude versus logarithm cycles to failure) using these data. (b) What is the fatigue limit for this alloy? (c) Determine fatigue lifetimes at stress amplitudes of \(230 \mathrm{MPa}(33,500 \mathrm{psi})\) and \(175 \mathrm{MPa}\) \((25,000 \mathrm{psi})\) (d) Estimate fatigue strengths at \(2 \times 10^{5}\) and \(6 \times 10^{6}\) cycles.

Step by step solution

01

(a) Making the S-N plot

First, plot the data points with stress amplitudes on the y-axis and the logarithm of cycles to failure values (log(cycles)) on the x-axis. Then, create a smooth curve that represents the relationship between stress amplitude and log(cycles) values. For instance, using a graphing tool or software, plot the points: (5, 248), (5.48, 236), (6, 224), (6.48, 213), (7, 201), (7.48, 193), (8, 193), (8.48, 193) After plotting these points, create a smooth curve through the points.

02

(b) Finding the fatigue limit

The fatigue limit is found by observing the horizontal asymptote (i.e., point where the curve flattens out) of the S-N curve. Based on the given data and the created S-N plot, the fatigue limit appears to be 193 MPa.

03

(c) Determine fatigue lifetimes at stress amplitudes of 230 MPa and 175 MPa

To determine the fatigue life at 230 MPa, interpolate the values on the S-N curve: Between the points (5.48, 236) and (6, 224), find the corresponding x-coordinate (log(cycles)) for the stress amplitude of 230 MPa. Similarly, for 175 MPa, as it is below the fatigue limit of 193 MPa, assume it has infinitely large cycles to failure (N = ∞).

04

(d) Estimate fatigue strengths at 2 × 10^5 and 6 × 10^6 cycles

To estimate the fatigue strength at 2 × 10^5 cycles, find the stress amplitude corresponding to log(cycles) = 5.3 (since log(2 × 10^5) ≈ 5.3) on the S-N curve. Similarly, for 6 × 10^6 cycles, find the stress amplitude corresponding to log(cycles) = 6.78 (since log(6 × 10^6) ≈ 6.78) on the S-N curve.

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