# Assessment of Simplified Rating Factor Methodology for Shallow Cover Metal Culverts
## Appendix A Example and Comparison
### Rating Factors Based on [PERSON]'s Equation  
According to Figure 6, to incorporate the simpler moment equation by [PERSON] for load rating, the live load effects should be calculated based on the cover depth over the crown: H = 13 in. Calculate the Pipe Wall Thrust due to Live Load plus Dynamic Load Allowance  
Maximum distributed load at the crown due to live load plus dynamic load allowance and adjusted multiple presence factor: \(\rho_{\text{(L+IM)\&madj}}=\text{P}_{\text{(L+IM)\&madj}}/\text{A}_{\text{ LL}}=12.5\text{k}\times 1.285\times 1.08/7.92\text{ft}^{2}=2.19\) ksf  
Pipe wall thrust due to live load plus dynamic load allowance and adjusted multiple presence factor (AASHTO LRFD 12.7.2.2): \(\text{T}_{\text{(L+IM)\&madj}}=\text{P}_{\text{(L+M)\&madj}}(\text{C}_{ \text{L}})\text{F}_{1}/2\)  
\[\text{C}_{\text{L}}=\text{L}_{\text{L}}=2.17\text{ ft}<\text{S}=16.67\text{ ft}\]  
\[F_{1}=Max\left(\frac{0.75(S)}{L_{L}},\frac{15}{12(S)},1\right)=\left(\frac{0.7 5(S)}{L_{L}}\right)=5.77\]  
\[\text{T}_{\text{(L+IM)\&madj}}=2.19\text{ ksf}\times 2.17\text{ ft}\times 5.77/2=13.70\text{ k/ft}\]  
Compare the factored \(\text{T}_{\text{(L+IM)\&madj}}=\text{Y}_{\text{LL}}\text{T}_{\text{(L+IM)\&madj}}\) in simplified approach with CANDE result: \begin{tabular}{c c} \hline \hline
**Simplified** & **CANDE** \\ \hline
17.81 k/ft & 11.67 k/ft \\ \hline \hline \end{tabular}   
The CANDE value is obtained at the footing with the second axle stopped over the crown. Calculate The Maximum Moment 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. Thicknesses of natural subgrade: h\({}_{3}\) = H-h\({}_{1}\)-h\({}_{2}\) = 4.50 in  
Length of the distributed area at the crown by one single wheel load: a = W\({}_{\text{T}}\) + 2(h\({}_{1}\)tan\(\alpha_{1}\) + h\({}_{2}\)tan\(\alpha_{2}\) + h\({}_{3}\)tan\(\alpha_{3}\)) = 5.64 ft  
Width of the distributed area at the crown by one single wheel load:  
a-(W\({}_{\text{T}}\)-L\({}_{\text{T}}\)) = 5.15 ft  
Distributed area at the crown by one single wheel load:  
a[a-(W\({}_{\text{T}}\)-L\({}_{\text{T}}\))] = 29.03 ft2  
Wheel spacing on axle: 6 ft  
Pressure at the crown due to the second axle approaching the pipe from the left side of the centerline plus dynamic load allowance and adjusted multiple presence factor with consideration of pavement:  
If a > 6 ft (overlapping within the distributed area), \(\rho_{(L+IM)\delta m_{adj}\delta p_{grave}}=2\times\frac{P_{(L+IM)\delta m_{adj}} }{\mu[a-(W_{\text{T}}-L_{T})]}\)  
If a \(\leq\) 6 ft (no overlapping within the distributed area), \(\rho_{(L+IM)\delta m_{adj}\delta p_{grave}}=\frac{P_{(L+IM)\delta m_{adj}} }{\mu[a-(W_{\text{T}}-L_{T})]}\)  
Live load dynamic load allowance at crown:  
IM = 33(1.0 - 0.125D\({}_{\text{E}}\))  
D\({}_{\text{E}}\) = H = 1.08 ft  
IM = 28.5%  
(1 + IM) = 1.285  
Then, \(\rho_{(L+IM)\delta m_{adj}\delta p_{grave}}=\frac{P_{(L+IM)\delta m_{adj}} }{\mu[a-(W_{\text{T}}-L_{T})]}=\frac{12.5\times 1.285\times 1.08}{29.03}=0.60\) ksf (No overlapping)  
Maximum moment on the right side of the centerline due to live load plus dynamic load allowance and adjusted multiple presence factor: M\({}_{(\text{L+IM)\delta madj}}\) = 0.022 Pr2 = 0.022 \(\times\) \(\rho_{(\text{L+IM)\delta madj}\delta p_{grave}}\)\(\times\) R\({}_{\text{T}}\) = 1.24 k*ft/ft  
Compare the factored M\({}_{(\text{L+IM)\delta madj}}\) = \(\gamma_{\text{LL}}\)M\({}_{(\text{L+IM)\delta madj}}\) in simplified approach with CANDE result: \begin{tabular}{c c} \hline \hline
**Simplified** & **CANDE** \\ \hline
1.61 k/ft & 1.97 k/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.