# Assessment of Simplified Rating Factor Methodology for Shallow Cover Metal Culverts
## 11 Culvert Under Rigid Pavement  
The rigid pavement considered in [PERSON]'s study consisted of plain cement concrete (PCC), under which there was a cement-treated base course, a lime-treated subgrade, and a natural subgrade above the culvert. The elastic modulus of concrete pavement can be determined using the correlation given by ACI 318-19 [27]:  
\[E_{c}=4700\sqrt{{f_{c}}^{\prime}}\ (\text{Metric})\ \text{or}\ E_{c}=57000 \sqrt{{f_{c}}^{\prime}}\ (\text{US}) \tag{13}\]  
where \(E_{c}=\) elastic modulus of the concrete (Mpa or psi) and \({f_{c}}^{\prime}=\) compressive strength of the concrete (Mpa or psi). The typical concrete pavement design strength is 27.6 MPa (4000 psi), so the calculated elastic modulus of the concrete is 24,820 MPa (3.6 \(\times\) 10\({}^{6}\) psi). The averageelastic moduli of 5170 MPa (750,000 psi) and 310 MPa (45,000 psi) are adopted for cement-treated base course and lime-treated subgrade, respectively, based on the AASHTO _Guide for Design of Pavement Structures_[28]. The average elastic modulus of the natural subgrade is obtained from the elastic soil property of the top layer of various backfills by [PERSON] [29] (Table 3). The vertical pressure distribution angles from the concrete pavement to the cement-treated base course, from the cement-treated base course to the lime-treated subgrade, and the lime-treated subgrade to the natural subgrade are determined by Equation (12). 2 \(\times\) tan\(\alpha_{0}\) is taken as 1.15 per AASHTO LRFD Table 3.6.1.2.6a-1 [14] or 1.75 per AASHTO Standard 6.4.1 [19]. Thus, the \(\alpha_{0}\) is taken as 30\({}^{\circ}\) per AASHTO LRFD [14] or 41.2\({}^{\circ}\) per the AASHTO Standard Specification [19], respectively. The distribution angle calculated for each rigid pavement layer is provided in Table 4.  
Hence, the length of the rectangular distribution area on the metal culvert along the wheel axle due to a single wheel load \"\(a\)\" can be determined as follows:  
\[a=t_{l}+2(h_{1}\text{tan }a_{1}+h_{2}\text{tan }a_{2}+h_{3}\text{tan }a_{3}+h_{4}\text{tan }a_{4}) \tag{14}\]  
where \(a\) = length of the distributed area along the axle of wheels by one single wheel load; \(t_{l}\) = length of the tire footprint along the axle of wheels; \(h_{1}\), \(h_{2}\), \(h_{3}\), and \(h_{4}\) = thicknesses of concrete pavement, cement-treated base, lime-treated subgrade, and natural subgrade, respectively; and \(a_{1}\), \(a_{2}\), \(a_{3}\), and \(a_{4}\) = distribution angles between concrete pavement, cement-treated base, lime-treated subgrade, and natural subgrade, respectively. The thicknesses of the cement-treated base layer \(h_{2}\) and the lime-treated subgrade layer \(h_{3}\) are fixed at 10 cm (4 inches) and 15 cm (6 inches), respectively, which refers to the typical cross sections of the culvert models under rigid pavements field-tested by the Kansas Department of Transportation [11]. The thicknesses of the concrete layer \(h_{1}\) and the natural subgrade layer \(h_{4}\) are variable. A schematic of the vertical pressure distribution on the culvert under typical concrete pavement is shown in Figure 7. The length of the combined distribution area by the two-wheel loads along the axle of the wheels can be determined as follows:  
\[b=a+s \tag{15}\] \begin{table}
\begin{tabular}{c c c c} \hline
**Pavement Layer** & \(E_{bx}\) **MPa (psi)** & \(E_{\text{sq}}\) **MPa (psi)** & **Distribution Angle tan\(\alpha_{1}\)** \\ \hline Concrete pavement & 24,800 (3,600,000) & 5170 (750,000) & 1.02 \\ \hline Cement-treated base & 5170 (750,000) & 310 (45,000) & 2.41 \\ \hline Lime-treated subgrade & 310 (45,000) & 12.4 (1800) & 3.39 \\ \hline Natural subgrade & 12.4 (1800) & — & 0.58 \\ \hline \end{tabular} Note: \(\alpha_{0}\) is taken as 30\({}^{\circ}\), and the elastic modulus of the natural subgrade is selected as 12.4 MPa (1800 psi) from Ml.95 in Table 3.  
\end{table} 
Table 4: Distribution angle between pavement layers.