Chapter 7.10, Problem 30AAP

A fatigue test is made with a mean stress of 17,500 psi (120 MPa) and a stress amplitude of 24,000 psi (165 MPa). Calculate (a) the maximum and minimum stresses, (b) the stress ratio, and (c) the stress range.

To determine

The minimum and maximum stress.

Answer to Problem 30AAP

The maximum stress is $285\text{MPa}$ and the minimum stress is $45\text{MPa}$.

Explanation of Solution

Write the expression to calculate stress amplitude.

    ${\sigma }_a=\frac{\left({\sigma }_{\max }-{\sigma }_{\min }\right)}{2}$                                                               (I)

Here, the stress amplitude is ${\sigma }_a$, the maximum stress is ${\sigma }_{\max }$ and the minimum stress is ${\sigma }_{\min }$.

Write the expression to calculate mean stress.

    ${\sigma }_m=\frac{\left({\sigma }_{\max }+{\sigma }_{\min }\right)}{2}$                                                           (II)

Here, the mean stress is ${\sigma }_a$, the maximum stress is ${\sigma }_{\max }$ and the minimum stress is ${\sigma }_{\min }$.

Conclusion:

Substitute $120\text{MPa}$ for ${\sigma }_a$ in Equation (I).

    $\begin{array}{l}120\text{MPa}=\frac{\left({\sigma }_{\max }-{\sigma }_{\min }\right)}{2}\\ \left({\sigma }_{\max }-{\sigma }_{\min }\right)=240\text{MPa}\end{array}$                                         (III)

Substitute $165\text{MPa}$ for ${\sigma }_m$ in Equation (II).

    $\begin{array}{l}165\text{MPa}=\frac{\left({\sigma }_{\max }+{\sigma }_{\min }\right)}{2}\\ \left({\sigma }_{\max }+{\sigma }_{\min }\right)=330\text{MPa}\\ {\sigma }_{\max }=330\text{MPa}-{\sigma }_{\min }\end{array}$                                          (IV)

Substitute $330\text{MPa}-{\sigma }_{\min }$ for ${\sigma }_{\max }$ in Equation (III).

    $\begin{array}{l}\left(330\text{MPa}-{\sigma }_{\min }-{\sigma }_{\min }\right)=240\text{MPa}\\ 90\text{MPa}=\text{2}{\sigma }_{\min }\\ {\sigma }_{\min }=45\text{MPa}\end{array}$

Substitute $45\text{MPa}$ for ${\sigma }_{\min }$ in Equation (IV).

    $\begin{array}{c}{\sigma }_{\max }=330\text{MPa}-45\text{MPa}\\ =\text{285}\text{MPa}\end{array}$

Thus, the maximum stress is $285\text{MPa}$ and the minimum stress is $45\text{MPa}$.

To determine

The stress ratio.

Answer to Problem 30AAP

The stress ratio is $0.16$.

Explanation of Solution

Write the expression to calculate stress ratio.

    $R=\left|\frac{{\sigma }_{\min }}{{\sigma }_{\max }}\right|$                                                           (V)

Here, the stress ratio is $R$, the maximum stress is ${\sigma }_{\max }$ and the minimum stress is ${\sigma }_{\min }$.

Conclusion:

Substitute $285\text{MPa}$ for ${\sigma }_{\max }$ and $45\text{MPa}$ for ${\sigma }_{\min }$ in Equation (V).

    $\begin{array}{c}R=\left|\frac{45\text{MPa}}{285\text{MPa}}\right|\\ =0.16\end{array}$

Thus, the stress ratio is $0.16$.

To determine

The stress range.

Answer to Problem 30AAP

The stress range is $240\text{MPa}$.

Explanation of Solution

Write the expression to calculate stress ratio.

    ${\sigma }_r=\left({\sigma }_{\max }-{\sigma }_{\min }\right)$                                                            (VI)

Here, the stress range is ${\sigma }_r$, the maximum stress is ${\sigma }_{\max }$ and the minimum stress is ${\sigma }_{\min }$.

Conclusion:

Substitute $285\text{MPa}$ for ${\sigma }_{\max }$ and $45\text{MPa}$ for ${\sigma }_{\min }$ in Equation (VI).

    $\begin{array}{c}{\sigma }_r=\left(285\text{MPa}-45\text{MPa}\right)\\ =\left(240\text{MPa}\right)\end{array}$

Thus, the stress range is $240\text{MPa}$.