## 2 Methodology for Calculation of Peak Factors and Extreme Values  
In this section, the translation process method based on the HPM is briefly introduced and employed to determine peak factors and extreme values of softening non-Gaussian wind pressures. Supposing a non-Gaussian process \(X(t)\), the normalization of \(X(t)\) is:  
\[Z(t)=\frac{X(t)-\mu}{\sigma}, \tag{1}\]  
where \(\mu\) is the mean value and \(\sigma\) is the standard deviation of \(X(t)\). \(Z(t)\) can be expressed as a translation of a normalized Gaussian process \(U(t)\) using:  
\[U(t)=g^{-1}(Z(t)), \tag{2}\]  
where \(g(\bullet)\) is a translation function. Then, the cumulative distribution function (CDF) of \(Z(t)\) is determined by:  
\[F_{\text{\tiny{2 ex}}}(z)=\exp\biggl{\{}-v_{0}T\exp\biggl{[}-\frac{g^{-1}(z)}{ 2}\biggr{]}\biggr{\}}, \tag{3}\]  
Figure 1: Configuration of L-shaped roof prototype. (**a**) Plan view, (**b**) cross section. and the probability density function \(p\) is determined by:  
\[p_{z_{\rm ex}}(z)=\exp(-\psi)\frac{d\psi}{dz},\psi=\
u_{0}T\exp\left[-\frac{g^{-1 }(z)}{2}\right]. \tag{4}\]  
Here, the subscript \(z_{\rm ex}\) represents the extreme value of \(Z(t)\) during the time \(T\), and \(\
u_{0}\) is the mean zero up-crossing rate, which can be determined by spectral moments as:  
\[\
u_{0}=\sqrt{\int_{0}^{\infty}f^{2}S_{\rm z}(f)df/\int_{0}^{\infty}S_{\rm z}( f)df}, \tag{5}\]  
where \(S_{\rm z}(f)\) is the auto-power spectral density of \(Z\) and \(f\) is the frequency in hertz. Accordingly, the expression of \(g(\bullet)\) is decided, and thus the CDF for \(Z(t)\) is determined. For hardening processes, whose kurtosises are less than 3, the peak factors are often smaller than those of the Gaussian processes. Peak factors calculated using Davenport's equation result in conservative extreme results thereof. However, kurtosis for most wind pressures on long-span roofs is larger than 3, indicating a softening process whose peak factor is larger than that of a Gaussian process. Therefore, the softening processes are of great concern in this study. The softening process \(Z\) can be expressed by the Hermite polynomial function of \(U\) as:  
\[Z=g(u)=He(u)=\kappa\left[u+\sum_{n=3}^{\infty}h_{n}He_{n-1}(u)\right]\approx \kappa[He_{1}(u)+h_{3}He_{2}(u)+h_{4}He_{3}(u)], \tag{6}\]  
where \(\kappa=\sqrt{1+2h_{3}^{2}+6h_{4}^{2}}\). \(h_{n}\) is the shape coefficient, and \(He_{n}(u)\) is the \(n\)-th order of the Hermite polynomial function which is expressed as:  
\[He_{n}(u)=(-1)^{n}\cdot\exp\left(-\frac{u^{2}}{2}\right)\frac{d^{n}}{du^{n}} \left[\exp\left(-\frac{u^{2}}{2}\right)\right]. \tag{7}\]  
Here, \(n=1\), 2, and 3 are substituted to Equation (7), respectively. \(He_{1}(u)\), \(He_{2}(u)\), and \(He_{3}(u)\) can be derived as:  
\[He_{1}(u)=u,He_{2}(u)=u^{2}-1,\ {\rm and}\ He_{3}(u)=u^{3}-3u. \tag{8}\]  
Then, Equation (8) is substituted to Equation (6). \(Z\) can be derived as:  
\[Z=g(u)=\kappa\left[u+h_{3}(u^{2}-1)+h_{4}\left(u^{3}-u\right)\right]. \tag{9}\]  
By these means, the expression of Equation (2) is termed the moment-based Hermite expression. Thus, \(g^{-1}(\bullet)\) is derived as:  
\[U(t)=g^{-1}(Z(t)), \tag{2}\]0  
where \(\zeta(z)=1.5b\left(a+\frac{z}{\kappa}\right)-a^{3}\), \(a=\frac{h_{3}}{3h_{4}}\), \(b=\frac{1}{3h_{4}}\) and \(c=\left(b-1-a^{2}\right)^{3}\). To make \(g(\bullet)\) always monotonically increasing, its slope ought to be positive, leading to:  
\[U(t)=g^{-1}(Z(t)), \tag{2}\]1  
The \(n\)th-order Hermite polynomial \(He_{n}(z)\) can be expanded at \(\kappa u\) using [PERSON]'s theorem as follows:  
\[U(t)=g^{-1}(Z(t)), \tag{2}\]2where \(He_{\rm n}(^{j})=nHe_{\rm n-1}\) is the \(j\)th-order derivative of \(He_{\rm n}\). Take the first four terms of the expansion in Equation (12) into consideration. Shape coefficients \(h_{3}\) and \(h_{4}\) are determined by skewness \(m_{3}\) and kurtosis \(m_{4}\) of the following process:  
\[\left\{