5.4: Maxima At and Above a Given Level

Now we look at the probability of any maximum amplitude \(a_{ia}\) reaching or exceeding a given level. We normalize the amplitude with the random process variance, i.e., \( \eta = a / \sqrt{M_0} \) and \( \bar{\eta} = A / \sqrt{M_0} \). The results are very useful for calculating extreme loads. First,

\[\begin{align} p( \eta = \bar{\eta}) \, &= \, \dfrac{\epsilon}{\sqrt{2 \pi}} e^{- \bar{\eta}^2 / 2 \epsilon^2} + \phi (\bar{\eta} q / \epsilon) \dfrac{\bar{\eta}q}{\sqrt{2 \pi}} e^{- \bar{\eta}^2 / 2} \end{align}\]

 where

\[\begin{align} q &= \, \sqrt{1 - \epsilon^2}, \\[4pt] \phi(\xi) \, &= \, \int\limits_{-\infty}^{\xi} e^{-u^2 / 2}  du \end{align}\]

related to the error function erf)

With large amplitudes being considered and small \(\epsilon\) (a narrow-banded process), we can make some approximations to find:

\begin{align} p(\eta = \bar{\eta}) \, &\approx \, \dfrac{2q}{1+q} \bar{\eta} e^{- \bar{\eta}^2 / 2} \longrightarrow \\[4pt] p(\eta > \bar{\eta}) \, &\approx \, \dfrac{2q}{1+q} e^{- \bar{\eta}^2 / 2}. \end{align}

The second relation here is the more useful, as it gives the probability that the (nondimensional) amplitude will exceed a given value. It follows directly from the former equation, since (roughly) the cumulative distribution is the derivative of the probability density.