Fatigue Analysis of a Gondola Ski Lift (Part 3)

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You presented the results to your boss, and he/she is satisfied with your results of determining the worst case stress and its location. The next step is to start applying fatigue analysis.

STEP 4: Determine a minimum and maximum stress

Fatigue analysis is based off of a fluctuating load. Thus one must have two loading states to perform fatigue analysis. In our example one has two states: (1) an unloaded state and (2) a loaded state. The states a are illustrate below for this scenario:

For this scenario the unloaded state consists of the weight of the Gondola ( $P_{unloaded}= 2000 \text{ lbf}$ ). The loaded state consists of the weight of the Gondola and the passengers. Recall that the loaded state consists of 25 passengers with an average weight of 200 lbf. Therefore the loaded state is equal to $P_{Loaded} = 2000 + (25)(200) = 7000 \text{ lbf}$ . Next one calculates the minimum and maximum stress at the worst case location using the unloaded and loaded states. The calculation is illustrated below using the equation derived in Part 2 for the stress at the worst case stress location:

$\begin{align*}\label{Step4}\sigma_{min} &= \frac{P_{unloaded}}{A}(1+\frac{24}{r})\\&= \frac{2000}{3.14}(1+\frac{24}{1})\\&=15,915 \; psi\\\\\sigma_{max} &= \frac{P_{Loaded}}{A}(1+\frac{24}{r})\\&= \frac{7000}{3.14}(1+\frac{24}{1})\\&=55,704 \; psi\\\end{align*}$

STEP 5: Determine the mean and alternating stresses i.e. the operating point

After determining the minimum and maximum stress, one must determine a mean and alternating stress because these two quantities are plotted on a fatigue-life diagram. The mean and alternating stresses are calculated below with simple formulas:

$\begin{align*}\label{Step4}\sigma_{mean} &= \frac{\sigma_{min}+\sigma_{max}}{2}\\&= \frac{15,915+55,704}{2}\\&=35,810 \; psi\\\\\sigma_{alt} &= \frac{\sigma_{max}-\sigma_{min}}{2}\\&= \frac{55,704-15,915}{2}\\&=19,894 \; psi\\\end{align*}$

Now that one knows the operating point ( $\sigma_{mean},\sigma_{alt}$ ), one must construct a fatigue-life diagram to determine the life expectancy of the component.

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