The fatigue data for a brass alloy are given as follows: $$ \begin{array}{cc} \hline \begin{array}{c} \text { Stress Amplitude } \\ \text { (MPa) } \end{array} & \begin{array}{c} \text { Cycles to } \\ \text { Failure } \end{array} \\ \hline 310 & 2 \times 10^{5} \\ 223 & 1 \times 10^{6} \\ 191 & 3 \times 10^{6} \\ 168 & 1 \times 10^{7} \\ 153 & 3 \times 10^{7} \\ 143 & 1 \times 10^{8} \\ 134 & 3 \times 10^{8} \\ 127 & 1 \times 10^{9} \\ \hline \end{array} $$ (a) Make an \(S-N\) plot (stress amplitude versus logarithm cycles to failure) using these data. (b) Determine the fatigue strength at \(5 \times 10^{5}\) cycles. (c) Determine the fatigue life for \(200 \mathrm{MPa}\).

Short Answer

Expert verified
Answer: The fatigue strength at 5 x 10^5 cycles can be found by observing the S-N plot and noting the corresponding stress amplitude value at the logarithm of 5.698 cycles. The fatigue life for a stress amplitude of 200 MPa can be determined by locating the point on the S-N plot where the stress amplitude is 200 MPa and noting the corresponding logarithm value of the number of cycles to failure. Then, convert the logarithm value back to the actual number of cycles using 10^x.

Step by step solution

01

Create an S-N plot

Plot the given data points on a graph with the stress amplitude on the y-axis and the logarithm of the number of cycles to failure on the x-axis. For each point, the stress amplitude has already been given in MPa, and we can find the logarithm of the number of cycles to failure. Here is the modified table with the logarithm of the number of cycles to failure: $$ \begin{array}{ccc} \hline \text { Stress Amplitude (MPa) } & \text { Cycles to Failure } & \log_{10}(\text{Cycles to Failure}) \\ \hline 310 & 2 \times 10^{5} & 5.301 \\ 223 & 1 \times 10^{6} & 6 \\ 191 & 3 \times 10^{6} & 6.477 \\ 168 & 1 \times 10^{7} & 7 \\ 153 & 3 \times 10^{7} & 7.477 \\ 143 & 1 \times 10^{8} & 8 \\ 134 & 3 \times 10^{8} & 8.477 \\ 127 & 1 \times 10^{9} & 9 \\ \hline \end{array} $$ Using any graphing software or a graphing calculator, plot the data points: (5.301, 310), (6, 223), (6.477, 191), (7, 168), (7.477, 153), (8, 143), (8.477, 134), and (9, 127).
02

Determine the fatigue strength at 5 x 10^5 cycles

To determine the fatigue strength at \(5 \times 10^{5}\) cycles, we first need to find the logarithm of the number of cycles. \(log_{10}(5\cdot 10^5) = 5.698\). Now that we have the \(x\)-value, we will estimate the value of the stress amplitude (fatigue strength) by observing the graph. Locate the point on the plot where \(x = 5.698\) and note the corresponding stress amplitude value. This value represents the fatigue strength at \(5\times 10^5\) cycles.
03

Determine the fatigue life for 200 MPa

In order to determine the fatigue life for 200 MPa, observe the plot we created in Step 1. Locate the point on the plot where the stress amplitude is 200 MPa and note the corresponding logarithm value of the number of cycles to failure. Convert the logarithm value back to the actual number of cycles by using \(10^x\), where \(x\) is the logarithm value obtained from the plot. The resulting value represents the fatigue life for 200 MPa.

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