## 3 The Influence of the F\({}_{\text{y}}\) of YSD
### Single Variable Function Analysis  
As shown above, although we can achieve the designed layout of the bridge system by carrying out a comprehensive parametric analysis, it is time-consuming. Considering the result that the RD at each location decreases with the increase of the yielding strengths at any location, but the RCD is only influenced by the yielding strength of its own location, we can make the analysis easier by only using a one-variable function; that is, two of the variables (F\({}_{\rm{yt}}\) and F\({}_{\rm{yp}}\) or F\({}_{\rm{yt}}\) and F\({}_{\rm{ya}}\) or F\({}_{\rm{yp}}\) and F\({}_{\rm{ya}}\)) can remain unchanged to study the influence of the other variable (F\({}_{\rm{yt}}\) or F\({}_{\rm{yp}}\) or F\({}_{\rm{ya}}\)) on the structure's seismic response. Consequently, three series of cases are analyzed: (1) F\({}_{\rm{ya}}\) varies from 0 kN to 5000 kN, while F\({}_{\rm{yt}}\) and F\({}_{\rm{yp}}\) remain constant; (2) F\({}_{\rm{yp}}\) varies from 0 kN to 5000 kN, while F\({}_{\rm{ya}}\) and F\({}_{\rm{yt}}\) remain constant; and (3) F\({}_{\rm{yt}}\) varies from 1000 kN to 9000 kN, while F\({}_{\rm{ya}}\) and F\({}_{\rm{yp}}\) remain constant. In fact, it is required that the others should remain when analyzing the influence of yielding strength of the specified location. For illustration purposes, 2000 kN was set. It can be seen from Figures 20a, 21a and 22a that when the yielding strength is relatively small, with the increase of F\({}_{\rm{y}}\), the girder-pier/column RD at the auxiliary pier, transition pier and tower column decrease significantly. However, when the yielding strength is relatively large, the F\({}_{\rm{y}}\) only affects the displacement at its own location without a significant change at the other two locations. As shown in Figures 20b, 21b and 22b, with the increase of F\({}_{\rm{y}}\), there is a significant fluctuation of the RCD curve at its own location, but the RCD curves at the other two locations are approximately flat. In addition, the RCD curve at each location increases first and then decreases, indicating that a peak  
Figure 19: Contour of standard deviation of RD (F\({}_{\rm{ya}}\) = 1000 kN). Figure 18: Standard deviation of relative displacement (RD). must exist. Obviously, this result is consistent with that of the multivariate function analysis but much more concise. In addition, according to Figures 20b, 21b and 22b, the following Table 4 can be obtained to verify the correctness of the proportion of the above natural vibration response. As a matter of fact, the overall shear force transferred to the column end consists of two parts: one is from the inertial force developed by the girder and the other is from the self-vibration of the column itself. In order to determine the scope of parameter analysis, it is necessary to determine the proportion that the self-vibration response would take. Figures 16-18 can be obtained after the possible range of this proportion is determined, and one of the cases is taken as an example to illustrate that this self-vibration ratio is indeed around 0.3. It can be approximated that the increase of F\({}_{\mathrm{y}}\) generally constrains the RDs of pairs or towers, and only has a significant impact on its own location compared to the other two. Therefore, it can be considered that the selection of F\({}_{\mathrm{ya}}\), F\({}_{\mathrm{yp}}\) and F\({}_{\mathrm{yt}}\) is independent and not coupled. According to Figures 20b, 21b and 22b, the tower has the minimal RCD curve compared with other piers; to achieve one of the designed objectives as mentioned in multivariable analysis (large and uniform RCDs), F\({}_{\mathrm{yt}}\) should be set to its peak value of 2000 kN as shown in Figure 22b. Then, on the grounds of the intersection of the RCD curves of the auxiliary pier and of the tower in Figure 20b, F\({}_{\mathrm{ya}}\) should be 4000 kN, and F\({}_{\mathrm{yp}}\) should be approximately 4000 kN.