# Study on Gage Widening Methods for Small-Radius Curves
## 3 Status of Gage Widening Standards  
Gage widening standards for small-radius curve sections in the Soviet Union, the United States, Japan, and Germany have been summarized below to analyze the applications of gage widening theories and standards. In 1936, based on a bogie with a wheelbase of 3.9 m and a rail gap of 11 mm, the Soviet Union adopted geometrically free inscribing to determine the gage widening value. As listed in Table 1, the minimum radius to be widened was 650 m, and the maximum widening amount was 16 mm, divided into four levels. Figure 4: Schematic of the wedge-shaped inscribing method. Figure 3: Gap calculation based on dynamically free inscribing. \begin{table}
\begin{tabular}{c c c c c} \hline Curve Radius (m) & \(>\)650 & 650–451 & 450–351 & \(<\)350 \\ \hline Widening Value (mm) & 0 & 6 & 11 & 16 \\ \hline \end{tabular}
\end{table} 
Table 1: Gage widening values used in the Soviet Union (before 1961). \begin{table}
\begin{tabular}{c c c c} \hline Curve Radius (m) & \(\geq\)350 & 349–300 & \(\leq\)299 \\ \hline Widening Value (mm) & 0 & 6 & 16 \\ \hline \end{tabular}
\end{table} 
Table 2: Rail gage widening value in the former Soviet Union (after 1961). \begin{table}
\begin{tabular}{c c c c c c} \hline Curve Radius (m) & \(\geq\)219 & 218–195 & 194–175 & 174–159 & 158–146 & 145–134 \\ \hline Widening Value (mm) & 0 & 3.12 & 6.24 & 9.36 & 12.5 & 15.6 \\ \hline \end{tabular}
\end{table} 
Table 4: Gage widening values used in American railways (mm). \begin{table}
\begin{tabular}{c c c c c} \hline Curve Radius (m) & \(>\)650 & 650–451 & 450–351 & \(<\)350 \\ \hline Widening Value (mm) & 0 & 6 & 11 & 16 \\ \hline \end{tabular}
\end{table} 
Table 1: Gage widening values used in the Soviet Union (before 1961). \begin{table}
\begin{tabular}{c c c c} \hline Curve Radius (m) & \(\geq\)350 & 349–300 & \(\leq\)299 \\ \hline Widening Value (mm) & 0 & 6 & 16 \\ \hline \end{tabular}
\end{table} 
Table 3: Rail gage widening value in Russia (Currently used). In 1958, the gage widening values for West German railways were calculated according to wedge-shaped inscribing, as shown in Equation (12) and listed in Table 6. The minimum radius to be widened was 200 m, the maximum widening amount was 20 mm, and the progressive amount was 5 mm, divided into five levels. \[\Delta S=1000\frac{l^{2}}{8R}-(S_{n}+S_{a}+S_{p}+S_{g}) \tag{12}\]  
Here, \(l\) represents the fixed distance of the bogie (m), \(R\) represents the curve radius (m), \(S_{n}\) represents the standard gap between the wheel and the rail (mm), \(S_{a}\) represents the transverse momentum of the intermediate shaft (mm), \(S_{p}\) represents the intermediate shaft flange wear (mm), and \(S_{g}\) represents the squeezing amount of the track at the position of the intermediate shaft (mm). During the operation process, it was found that wedge-shaped inscribing caused relatively large driving resistance. In 1963, the gage was adjusted, and the curve radius was widened. As listed in Table 7, the widening amount and the number of levels remained unchanged while the minimum curve radius was increased to 300 m.  
According to the current standards of the German railway, when the curve radius is greater than 175 m, the minimum gage shall not be less than 1430 mm. When the curve \begin{table}
\begin{tabular}{c c c c c c} \hline Curve Radius (m) & \textgreater{}300 & 300–200 & 200–150 & 150–120 & 120–100 \\ \hline Widening Value (mm) & 0 & 5 & 10 & 15 & 20 \\ \hline \end{tabular}
\end{table} 
Table 7: Gage widening values in West German railways (mm). Figure 5: Schematic of gage widening calculations utilized in Japan. \begin{table}
\begin{tabular}{c c c c c c c} \hline Curve Radius (m) & \textgreater{}600 & 600–440 & 440–320 & 320–240 & 240–200 & 200–170 & \textless{}170 \\ \hline Widening Value (mm) & 0 & 5 & 10 & 15 & 20 & 25 \\ \hline \end{tabular}
\end{table} 
Table 5: Gage widening values used in Japanese railways (mm). radius is in the range of 150 to 175 m, the minimum gage shall not be less than 1435 mm.