# Assessment of Simplified Rating Factor Methodology for Shallow Cover Metal Culverts
## 12 Culvert Under Flexible Pavement  
Flexible pavement consists of a hot mixed asphalt (HMA) layer, under which there is typically a lime-treated subgrade and a natural subgrade above the culvert. The elastic modulus of the asphalt layer is approximated as 1380 MPa (200,000 psi). The average elastic modulus of 310 MPa (45,000 psi) is adopted for lime-treated subgrade as recommended by the _Guide for Design of Pavement Structures_[28]. The average elastic modulus of the natural subgrade is obtained from the elastic soil property in the top layer of various backfills by [PERSON] [29] (Table 3).  
Figure 8: Distributed pressure over the culvert under a rigid pavement.  
Figure 7: Vertical pressure distribution under rigid pavement [11].  
The vertical pressure distribution angles from the asphalt concrete to the lime-treated subgrade and the lime-treated subgrade to the natural subgrade are determined by Equation (12), where 2 \(\times\) tan\(\alpha_{0}\) is taken as 1.15 per AASHTO LRFD Table 3.6.1.2.6a-1 or 1.75 per AASHTO Standard 6.4.1. Thus, the \(\alpha_{0}\) is taken as 30\({}^{\circ}\) per AASHTO LRFD or 41.2\({}^{\circ}\) per the AASHTO Standard Specification, respectively. The distribution angle calculated for each flexible pavement layer is shown in Table 5.  
Hence, \"\(\alpha\)\" the length of the rectangular distribution area on the metal culvert along the wheel axle due to a single wheel load can be determined as follows:  
\[a=t_{l}+2(h_{1}\text{tan}a_{1}+h_{2}\text{tan}a_{2}+h_{3}\text{tan}a_{3}) \tag{17}\]  
where \(\alpha\) = 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; \(h_{1}\), \(h_{2}\), and \(h_{3}\) = thicknesses of asphalt pavement, lime-treated subgrade, and natural subgrade, respectively; and \(a_{1}\), \(a_{2}\), and \(a_{3}\) = distribution angles between asphalt pavement, lime-treated subgrade, and natural subgrade, respectively. The thickness of the lime-treated subgrade layer \(h_{2}\) is fixed at 15 cm (6 inches), which refers to the typical cross sections of the culvert models under flexible pavements field-tested by the Kansas Department of Transportation [11]. The thicknesses of the asphalt layer \(h_{1}\) and the natural subgrade layer \(h_{3}\) are variable. A schematic of the vertical pressure distribution on the culvert under pavement is shown in Figure 9.  
Figure 9: Vertical pressure distribution through flexible pavement layers.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Pavement Layer** & **\(E_{\text{bx}}\) MPa (psi)** & **\(E_{\text{sg}}\) MPa (psi)** & **Distribution Angle tan\(\alpha_{1}\)** \\ \hline Asphal concrete & 1380 (200,000) & 310 (45,000) & 0.98 \\ \hline Lime-treated subgrade & 310 (45,000) & 12.4 (1800) & 3.39 \\ \hline Natural subgrade & 12.4 (1800) & — & 0.58 \\ \hline \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 MI.95 in Table 3.  
\end{table}
Table 5: Distribution angle between pavement layers.  
For flexible pavement, the applied loads are assumed to be distributed independently over the distribution area from an individual wheel load onto the culvert. The uniform vertical pressure from a single wheel load is:  
\[P_{c1}=\frac{P}{a(a-t_{l}+t_{w})} \tag{18}\]  
where \(P_{c1}\) = the distributed vertical pressure from an individual wheel load over the culvert. Within the overlapped area (Figure 10), the vertical pressure over the culvert can be determined by superposing the pressures due to two individual wheels:  
\[P_{c2}=2P_{c1} \tag{19}\]  
where \(P_{c2}\) = distributed vertical pressure within the overlapped area over the culvert.  
The pavement benefit is only accounted for in the calculation of moment due to live load in the load rating because the spreading effect of pavement affects the moment much more than the thrust due to live load in the metal structure.