# Assessment of Simplified Rating Factor Methodology for Shallow Cover Metal Culverts
## Appendix A Example and Comparison
### Refined ODOT Approach
Calculate the Critical Intensity of Distributed LoadCritical intensity of distributed load [5]: \(q_{\sigma}=\gamma_{4}\frac{EI}{5^{3}}\)  
\[\text{E}=\text{E}_{\text{m}}=\text{29,000 ksi}\]  
\[\text{q}_{\text{cr}}=\text{0.0285 ksi}\]  
Calculate the Maximum Distributed Load over the Crown  
Distribution of wheel load through earth fill (AASHTO LRFD 3.6.1.2.6),  
Live load patch width at cover depth H over the crown: \(\text{W}_{\text{L}}=\text{W}_{\text{T}}+\text{\varphi}_{\text{E}}(\text{H})+ 0.06\text{S}=\text{3.65 ft}<\text{6 ft}\) (no overlapping in a single axle)  
Live load patch length at cover depth H over the crown: \(\text{L}_{\text{L}}=\text{L}_{\text{T}}+\text{\varphi}_{\text{E}}(\text{H})= \text{2.17 ft}<\text{4 ft}\) (no overlapping between adjacent axles)  
Live load patch area at cover depth H over the crown: \(\text{A}_{\text{LL}  
Secant modulus of backfill for H = 0 ft (based on ML95 from Table 7 in Duncan and Drawsky, [7]): E\({}_{\rm S}\) = 45.578 k/ft\({}^{2}\)  
Flexibility number for H = 0 ft: \(N_{f}=\frac{E_{S}S^{3}}{ET}\)  
\[\rm E=E_{m}=29,000\,\,\rm{ksi}\,\times\,12^{2}=4176000\,\,\rm{k/ft}^{2}\]  
\[\rm N_{f}=1283\]  
Determine the Moment Coefficients for The Condition of Zero Cover Depth  
For N\({}_{\rm f}\leq 5000\), K\({}_{\rm M1}\) = 0.0046-0.0010 \(\times\) log\({}_{10}\)N\({}_{\rm f}\)  
For N\({}_{\rm f}\geq 5000\), K\({}_{\rm M1}\) = 0.0009  
Then, K\({}_{\rm M1}\) = 0.0014918  
For 0.2 \(\leq\) R/S \(\leq\) 0.35, R\({}_{\rm B}\) = 0.67 + 0.87(R/S-0.2),  
For 0.35 \(\leq\) R/S \(\leq\) 0.50, R\({}_{\rm B}\) = 0.80 + 1.33(R/S-0.35),  
For 0.5 \(\leq\) R/S \(\leq\) 0.60, R\({}_{\rm B}\) = 2R/S,  
Then, R\({}_{\rm B}\) = 0.7220  
Calculate the Maximum Moment at Both the Quarter Point and the Crown for the  
Condition of Zero Cover Depth and No Live Load  
Maximum moment with zero cover depth: M\({}_{\rm 1}\) = K\({}_{\rm M1}\)R\({}_{\rm B}\)yS\({}^{3}\)  
\[\rm\gamma=\sigma=0.112\,\,\rm{k/ft}^{3}\]  
\[\rm M_{\rm 1}=0.559\,\,\rm{k^{*}ft/ft}\]  
Calculate the Flexibility Number for Cover Depth over the Quarter Point  
Cover depth over the crown: H = 1.08 ft  
Depth of cover over the quarter point: H + R/2 = 1.08 ft + 4.33 ft/2 = 3.25 ft  
Secant modulus for backfill at the depth of cover over quarter point (based on ML95 from Table 7 in Duncan and Drawsky, [7]): E\({}_{\rm S}\) = 49.369 k/ft\({}^{2}\)  
Flexibility number for the depth of cover over the quarter point: \(N_{f}=\frac{E_{S}S^{3}}{ET}=1390\)  
Determine the Moment Coefficients for Cover Depth over the Quarter Point  
For N\({}_{\rm f}\leq 5000\), K\({}_{\rm M2}\) = 0.018-0.004 \(\times\) log\({}_{10}\)N\({}_{\rm f}\),  
For N\({}_{\rm f}\geq 5000\), K\({}_{\rm M2}\) = 0.0032  
Then, K\({}_{\rm M2}\) = 0.005428  
For N\({}_{\rm f}\leq 100\),000, K\({}_{\rm M3}\) = 0.120-0.018 \(\times\) log\({}_{10}\)N\({}_{\rm f}\),  
For N\({}_{\rm f}\geq 100\),000, K\({}_{\rm M3}\) = 0.030,  
Then, K\({}_{\rm M3}\) = 0.0634  
\[R_{L}=\frac{0.265-0.053\,\log_{10}N_{f}}{\left[\left(H+R/2\right)/S\right]^{0. 75}}=0.3356\]  
\[\rm M_{\rm 2}=K_{\rm M2}R_{\rm B}\)yS\({}^{2}\)H = 0.132 k\({}^{*}\)ft/ft  
Calculate the Maximum Moment at Quarter Point due to Earth Fill  
The maximum moment at the quarter point due to earth fill: M\({}_{\rm B}\) = M\({}_{\rm 1}\)-M\({}_{\rm 2}\)  
\[\rm\gamma=\delta=0.112\,\,\rm{k/ft}^{3}\]  
\[\rm H=1.08\,\,\rm{ft}\]  
\[\rm M_{\rm B}=0.427\,\,\rm{k^{*}ft/ft}\]  
Compare the factored M\({}_{\rm B}\) = \(\rm\gamma_{EVmax}\)\(\rm\gamma_{IR}\)M\({}_{\rm 1}\)-(\(\rm\gamma_{EVmin}/\)\(\rm\gamma_{IR}\))M\({}_{\rm 2}\) in simplified approach with CANDE result: \begin{tabular}{c c} \hline \hline
**Simplified** & **CANDE** \\ \hline
1.03 k*ft/ft & 0.40 k*ft/ft \\ \hline \hline \end{tabular}   
The CANDE value is obtained at the quarter point of the culvert. Calculate the Equivalent Line Load with Consideration of Pavement for the Calculation of Moment due to Live Load