# Assessment of Simplified Rating Factor Methodology for Shallow Cover Metal Culverts
## 5 Definition of \"Span\" for Long-Span Structures  
According to [PERSON] (2016) and [PERSON] et al. (2011), the circumferential thrust develops in proportion to the radius of curvature in the wall of a non-circular pipe arch culvert (Figure 2):  
\[P=\frac{T}{R} \tag{4}\]  
thus, \(T=P\times R\) (5)  
where \(P\) is the soil pressure, \(R\) is the radius of curvature at the point under consideration, and \(T\) is the circumferential thrust. Moreover, in an arch culvert (Figure 3), the vertical reaction \(V\) at the springline is defined as:  
\[V=P\times\frac{S}{2} \tag{6}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{\(h\)/\(L\)} & \multicolumn{2}{c}{**Three-Hinged Arch**} & \multicolumn{2}{c}{**Two-Hinged Arch**} & \multicolumn{2}{c}{**Fixed Arch**} \\ \cline{2-7}  & \(qL^{3}\)/EI & \(HL^{2}\)/EI & \(qL^{3}\)/EI & \(HL^{2}\)/EI & \(qL^{3}\)/EI & \(HL^{2}\)/EI \\ \hline \multicolumn{7}{c}{Parabolic arches subjected to vertical load uniformly distributed on a horizontal projection} \\
0.10 & 22.5 & 28.1 & 29.1 & 36.3 & 60.9 & 76.2 \\
0.15 & & & 39.5 & 32.9 & 85.1 & 70.9 \\
0.20 & 39.6 & 24.8 & 46.1 & 28.8 & 103.1 & 64.5 \\
0.25 & & & 49.2 & 24.6 & 114.6 & 57.3 \\
0.30 & 49.5 & 20.6 & 49.5 & 20.6 & 120.1 & 50.0 \\
0.35 & & & 47.8 & 17.1 & 120.6 & 43.1 \\
0.40 & 45.0 & 14.1 & 45.0 & 14.1 & 117.5 & 36.7 \\
0.50 & 38.2 & 9.6 & 38.2 & 9.6 & 105.3 & 26.3 \\ \hline \end{tabular}
\end{table}
Table 2: Critical load parameter for uniform elastic arches in pure compression (2016).  
Figure 2: Circumferential thrust in non-circular culvert (2016).  
Thus, the tangential thrust \(C\) is:  
\[C=\frac{V}{\cos\phi}=P\times\frac{S}{2}/\cos\phi=P\times R \tag{7}\]  
where \(\phi\) is the angle between the arch axis and the vertical line at the footing, \(S\) is the arch span, and \(R\) is the radius of arch curvature. Therefore, for both shapes (pipe arch culvert in Figure 2 and arch culvert in Figure 3), the thrust is directly proportional to the top radius rather than the culvert span.  
Accordingly, twice the value of the top radius must be used to define the thrust in lieu of using the span dimension. This is defined by 12.7.2.1 of the AASHTO Standard Specification [19], AASHTO LRFD 12.8.3.2 [14], and _NCSPA Design Data Sheet No. 19_ section II-A-3 for all long-span structures (horizontal ellipse, low and high profile arches, inverted pear shapes and pear arches) as well as other horizontal ellipses with a relatively large radius of curvature greater than 3.96 m (13.0 ft) in the crown or side plates. However, the actual field-measured span must be used in the formulae for other calculations in long-span culverts. Using twice the top arc radius in these equations for long-span structures would artificially amplify the responses and thereby result in rating factors that are lower than the actual representation of capacity. This is associated with the reduction factor \(F_{1}\) in AASHTO LRFD 12.7.2.2-5 [14], the critical load defined by [PERSON] [5], the parameter \(C\) in the RF equation based on minimum cover requirements \(\frac{H^{2}}{Ch^{2}}\), and the moment calculations in the MDOT spreadsheet, etc.