## 6 Conclusions  
In this study, load-displacement curves and mathematical models representing the LRGRF are obtained from the longitudinal resistance test of the guardrail fastener. These plots are then effectively introduced to the stock rail-guardrail-sleeper-bridge-pier integrated simulation model to examine the influences of the LRGRF and JRGR on track-bridge interaction. The main results obtained are summarized as follows:  
1. The Langmuir EXT1 function is employed as an effective mathematical model for the rising stage of the load-displacement curve that represents the LRGRF. The nominal  
Figure 19: Effect of the longitudinal stiffness of #6 pier. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Lower Limit of Piers**} & \multicolumn{3}{c}{**Upper Limit of Piers**} & \multicolumn{3}{c}{**Recommended**} \\ \multicolumn{1}{c}{**Spans of**} & \multicolumn{3}{c}{**Based on Equation (2) (kN/cm)**} & \multicolumn{3}{c}{**Based on Equation (3) (kN/cm)**} & \multicolumn{3}{c}{**Maximum**} \\
**CGB (N-m)** & \multicolumn{3}{c}{**Rail Temperature Variation**} & \multicolumn{3}{c}{**Rail Temperature Variation**} & \multicolumn{3}{c}{**Value (kN/cm)**} \\  & **30\({}^{\circ}\)C** & **40\({}^{\circ}\)C** & **50\({}^{\circ}\)C** & **30\({}^{\circ}\)C** & **40\({}^{\circ}\)C** & **50\({}^{\circ}\)C** & **Value (kN/cm)** \\ \hline
32 + 48 + 32 & - & - & - & 163.9 & 129.9 & 101.6 & - \\
40 + 64 + 40 & - & - & - & 652.0 & 617.3 & 612.2 & - \\
44 + 80 + 44 & 1177.9 & 1338.9 & 1605.8 & 1279.1 & 1245.1 & 1281.6 & - \\
50 + 92 + 50 & 400.0 & 400.0 & 400.0 & 2214.2 & 2147.3 & 2122.7 & 2120 \\
60 + 100 + 60 & 400.0 & 400.0 & 400.0 & 3207.2 & 3033.3 & 2968.0 & 2960 \\ \hline \hline \end{tabular}  
* “-” indicates that no pier stiffness exists that meets the requirements.  
\end{table} 
Table 3: Predicted results of the longitudinal stiffness for piers. longitudinal resistance and the elastic longitudinal displacement of the rail prior to slipping grow approximately linearly with the increase in the installation torque. As the installation torque varies from 40 N\(\cdot\)m to 60 N\(\cdot\)m, the nominal longitudinal resistance and longitudinal displacement of the rail prior to slipping change from 3.91 kN to 6.46 kN and from 0.15 mm to 0.29 mm, respectively. 2. The interaction between the stock rail and the guardrail could lessen the track-bridge interaction of the CWR on the bridge. Additionally, this results in a reduction of both the SRLF and the rail broke gap, with a stock rail braking force reduction of up to 28.0%. 3. The existence of a breathing zone of the guardrail can exacerbate the track-bridge interaction of the CWR near the abutment with moveable bearings. In a special case, when the spans of the CGB and the longitudinal stiffness of piers for simply supported bridges are small, the SRLF at the abutment position is even greater than the force at the CGB end position. This enables the abutment position to become a new control point for the design of CWR on bridges. 4. If the installation torque of the guardrail fastener is too high, the SRLF under expansion conditions greatly increases. Conversely, if the torque is too small, the stock rail braking force noticeably grows. Hence, in general, an installation torque of 40-60 N\(\cdot\)m is recommended. If the JRGR is too large, the SRLF under expansion conditions also increases significantly. If the JRGR is too small, the variation of the guardrail gap is large. Therefore, the JRGR is generally recommended to be in the interval of 370-430 kN. 5. If the temperature span of the CGB is lower than 124 m, the influence of guardrails must be considered in the design of the CWR on bridges. When the longitudinal stiffness of the pier for the CGB is less than the recommended maximum value proposed in the paper, the existing simplified model without guardrails can be effectively employed in the design of the CWR. Conceptualization, K.X. ; data curation, W.D. ; funding acquisition, K.X. and W.Z. ; investigation, K.X. and W.D. ; methodology, K.X.