# Assessment of Simplified Rating Factor Methodology for Shallow Cover Metal Culverts
## Appendix A Example and Comparison
### Refined ODOT Approach
\begin{tabular}{c} \hline Elastic modulus of the asphalt pavement [11]: E\({}_{1}\) = 200,000 psi \\ Elastic modulus of the lime-treated subgrade [11]: E\({}_{2}\) = 45,000 psi \\ Elastic modulus of the natural subgrade (the elastic modulus in top layer of ML95 in [29]): E\({}_{3}\) = 1800 psi \\ Thicknesses of asphalt pavement: h\({}_{1}\) = H\({}_{\text{pave}}\) = 2.5 in \\ Thicknesses of lime-treated subgrade [11]: h\({}_{2}\) = 6 in \\ Thicknesses of natural subgrade: h\({}_{3}\) = H + R/2-h\({}_{1}\)-h\({}_{2}\) = 30.48 in \\ Reference vertical pressure distribution angle in the backfill (AASHTO LRFD Table 3.6.1.2.6a-1): tan\(\alpha_{0}\) = tan300 = LLDF/2 = 1.15/2 = 0.575 \\ Vertical pressure distribution angle in the asphalt pavement ([PERSON] et al., 2013): tan\(\alpha_{1}\) = tan\(\alpha_{0}\Big{[}1+0.204\Big{(}\frac{E_{1}}{E_{2}}-1\Big{)}\Big{]}\) = 0.98 \\ Vertical pressure distribution angle in the lime-treated subgrade: tan\(\alpha_{2}\) = tan\(\alpha_{0}\Big{[}1+0.204\Big{(}\frac{E_{2}}{E_{3}}-1\Big{)}\Big{]}\) = 3.39 \\ Vertical pressure distribution angle in the natural subgrade: \\ \(\text{tan}\alpha_{3}\) = tan\(\alpha_{0}\) = 0.58 \\ Length of the distributed area at the quarter point by one single wheel load: \\ \(\text{q}_{i}\) = W\({}_{\text{T}}\) + 2(h\({}_{1}\)tan\(\alpha_{1}\) + h\({}_{2}\)tan\(\alpha_{2}\) + h\({}_{3}\)tan\(\alpha_{3}\)) = 8.13 ft \\ Wheel spacing on axle: 6 ft \\ Equivalent line load with consideration of pavement: \\ If \(\text{q}_{i}\) \(>\) 6 ft (overlapping in a single axle), K\({}_{4}\) = q\({}_{i}\) + 6 ft \\ If \(\text{q}_{i}\) \(\leq\) 6 ft (no overlapping in a single axle), K\({}_{4}\) = 2q\({}_{i}\) \\ Then, K\({}_{4}\) = q\({}_{i}\) + 6 ft = 14.13 ft (overlapping in a single axle) \\ The load supported on a single axle or on tandem axles if the spacing between the axles is less than one-third the span of the culvert: AL = 50 kips (because the spacing between the axles 4 ft is less than one-third the span of the culvert 16.67/3 = 5.56 ft) \\ The equivalent line load: \(LL=\frac{AL}{K_{4}}=3.54\) k/ft \\ \hline \end{tabular}   
Calculate the Additional Moment at Quarter Point due to Live Load  
The pavement effect is only accounted for in the calculation of moment due to live load because it affects moment much more than thrust. Additional moment at quarter point due to live load: \(\Delta\text{M}_{\text{L}}\) = K\({}_{\text{M}}\)R\({}_{\text{L}}\)S(LL) = 1.26 k*ft/ft \\ Calculate the Operating Load Rating Factor Based on Plastic Moment Capacity \begin{tabular}{c} \hline \hline Rating factor based on plastic moment capacity: \(RF_{\text{O}-M}=\frac{\varphi_{\text{r}}\varphi_{\text{r}}\varphi_{\text{f}}M _{\text{p}}-\gamma_{\text{E}V\text{max}}\eta_{\text{R}}M_{\text{B}}}{\gamma_{ \text{LL}}M_{\text{(L+1M)}k\text{m}=\mu j}}\) \\ Resistance factor for flexure (AASHTO LRFR 6.5.4.2): \(\varphi_{\text{f}}\) = 1.0 \\ Live load dynamic load allowance at quarter point: IM = 33(1.0 \(-\) 0.125D\({}_{\text{E}}\)) \\ \(\text{D}_{\text{E}}\) = H + R/2 = 3.25 ft \\ IM = 19.6\% \\ (1 + IM) = 1.196 \\ \hline \hline \end{tabular}   
Additional moment at quarter point due to live load plus dynamic load allowance and adjusted multiple presence factor: \(\text{M}_{\text{(L+IM)}\text{\&maadj}}=(1+\text{IM)}\text{m}_{\text{adj}} \text{\Delta M}_{\text{L}}=1.62\ \text{k}^{\text{*}}\text{ft}/\text{ft}\)  
Compare the factored \(\text{M}_{\text{(L+IM)}\text{\&maadj}}=\gamma_{\text{LL}}\text{M}_{\text{(L+IM) \&maadj}}\) in simplified approach with its CANDE result: \begin{tabular}{c c} \hline \hline
**Simplified** & **CANDE** \\ \hline
2.11 k\({}^{\text{*}}\)ft/ft & 0.40 k\({}^{\text{*}}\)ft/ft \\ \hline \hline \end{tabular}   
The CANDE value is obtained at the quarter point of the culvert with the second axle stopped over the quarter point. \[\text{RF}_{\text{O-M}}=1.57\]