# Improved Simplified Engineering Fault Displacement Hazard Evaluation Method for On-Fault Sites
## 2 Development of the Improved, Simplified Hazard Evaluation Method
### Consideration of the Displacement Distribution\(-\)On-Fault Distance Ratio, \(l/l\)  
In the engineering hazard evaluations for on-fault sites, the displacement distribution along the rupture is neglected, and the average displacement is considered along the entire rupture length, e.g., in [4; 5; 31]. For the average displacement, empirical scaling relations can be used, as published for example in [35]:  
\[\log(D_{ave})=a+b\times m\pm\varepsilon \tag{4}\]  
where \(a=-6.32\), \(b=0.90\), and the standard deviation \(\varepsilon=0.28\) in log units. In [5], the weighted average of six empirical scaling relations is applied to evaluate displacements.  
The paper [26] provides three empirical equations for the dependence of the displacements on the variable \(l/L\) (Figure 1). These are bilinear, quadratic, elliptical and trigonometrical functions versus \(l/L\) scaled on the magnitude, \(m\). The elliptical and trigonometrical relations have the form:  
\[ln(D)=a\times m+b\times x^{*}(l/L)+c, \tag{5}\]  
For the elliptical model, \(a=1.7927\), \(b=3.3041\) and \(c=-11.2192\) are constants, and \(x^{*}(l/L)\equiv x^{*}(x_{s}-x_{i})/L)\) is as follows [26]:  
\[x^{*}(l/L)=x^{*}(\frac{x_{s}-x_{i}}{L})=\sqrt{1-0.5^{-2}\times\left[\frac{l}{ L}-0.5\rightConsidering all distribution forms, the displacements in the range of \([0;\sim L/3)\) are significantly smaller than the maximum, which is also valid for the range \((\sim 2L/3;L]\). The comparison of the empirical bilinear, quadratic, elliptical and trigonometrical functions for \(\ln(D)=F(m,l/L)\) shows that the elliptical and trigonometrical functions result in approximately identical distribution. In contrast, the bilinear seems to be slightly more conservative. The quadratic polynomial relation versus \(l/L\) does not have a maximum at \(l/L=0.5\) mid-point of the rupture, which should be evident from physical considerations. The paper [26] also proposes bilinear, quadratic or elliptical empirical relations for the normalised displacements \(D/D_{ave}\) that reveal the same features as the distribution of \(D\). Based on the comparison of numerical examples and the latest research results in [33], the elliptical distributions expressed by Equations (5) and (6) are preferred in the calculations below. The pragmatic reason for the selection is the easy-to-use analytical expression of \(x^{*}\) that also complies with geometrical limitations.  
As in [5], the use of a conservative weighted average scaling relation for displacement evaluation neglecting the dependence of the displacements on the site location, the distance from the end of the rupture to the site's position and the distance from the end of the rupture to the initiating point of the rupture results in essential conservatism, which is demonstrated in Figure 4.  
Here, the displacement distributions versus relative coordinate \(x_{i}/FL\) were calculated by Equations (5) and (6) for the magnitude of 6.5 and for the relative site locations \(x_{s}/FL\) equal to 0.5 (mid fault), 0.4, 0.3, 0.2 and 0.1 separately. The numbers 1-5 indicate the expected displacements that are defined assuming a uniform distribution of \(x_{i}\) along \([0;x_{s})\). The average displacement given by Equation (4) is also plotted in Figure 4. The black dotted line is the displacement value used in [5]. The black dashed line is the maximum displacement according to the bilinear distribution [26].