Goodman Failure
Consider a material with a known yield strength of 200 MPa is subjected to cyclical tensile loading during a reliability test. If both the mean and alternating stresses are 100 MPa and 50 MPa respectively, does fatigue failure occur according to the modified Goodman theory?
Expand Hint
The modified Goodman theory is a method for predicting the fatigue failure of a material under cyclic loading. It states that fatigue failure will occur whenever
$$$\frac{\sigma_a}{S_e}+\frac{\sigma_m}{S_{ut}}\geq 1\:\:or\:\:\frac{\sigma_{max}}{S_y}\geq 1$$$
if
$$\sigma_m \geq 0$$
where
$$S_e$$
is the endurance limit,
$$S_{ut}$$
is the ultimate strength,
$$S_y$$
is the yield strength,
$$\sigma_a$$
is the alternating stress, and
$$\sigma_m$$
is the mean stress.
Hint 2
$$$\sigma_{max}=\sigma_m+\sigma_a$$$
The modified Goodman theory is a method for predicting the fatigue failure of a material under cyclic loading. It states that fatigue failure will occur whenever
$$$\frac{\sigma_a}{S_e}+\frac{\sigma_m}{S_{ut}}\geq 1\:\:or\:\:\frac{\sigma_{max}}{S_y}\geq 1$$$
if
$$\sigma_m \geq 0$$
where:
- $$S_e$$ is the endurance limit (the max load level where a material can be cycled indefinitely without failure).
- $$S_{ut}$$ is the ultimate strength (max stress a material can withstand from being pulled before yielding).
- $$S_y$$ is the yield strength (the max load a material can withstand without deforming).
- $$\sigma_a$$ is the alternating stress (subtracting the max and min stress levels and dividing by 2)
- $$\sigma_m$$ is the mean stress (adding the max and min stress levels and dividing by 2)
- $$\sigma_{max}=\sigma_m+\sigma_a$$
- $$n$$ is the factor of safety
The modified Goodman theory can also be expressed with a factor of safety:
$$$\frac{\sigma_a}{S_e}+\frac{\sigma_m}{S_{ut}}\geq \frac{1}{n}\:\:or\:\:\frac{\sigma_{max}}{S_y}\geq \frac{1}{n}$$$
if
$$\sigma_m \geq 0$$
.
Solving the for the max stress:
$$$\sigma_{max}=100MPa+50MPa=150\:MPa$$$
Checking if the modified Goodman theory predicts failure:
$$$\frac{150MPa}{200MPa}=0.75< 1$$$
Because the calculated 0.75 is less than 1, the modified Goodman theory states that
failure will not occur
.
Fatigue failure will not occur.
Time Analysis
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