# Assessment of Simplified Rating Factor Methodology for Shallow Cover Metal Culverts
## 16 Rating Factor Equations for Culverts with Shallow Cover  
The rating factor based on the minimum cover requirement used by ODOT and the rating factor based on moment in the MDOT method is overly simplified and too conservative. The derivation of the rating factor based on the minimum cover requirement  
Figure 14: Live load is over the middle point between the crown and the quarter point (outside values denote outer fiber in tension, and inside values denote inner fiber in tension). in the ODOT approach is presented in Appendix E of the NCSPA _Design Data Sheet No. 19_. The following demonstrates the derivation and discusses its flaws. For metal culverts buried under the minimum cover required by AASHTO, \(h\), to withstand the load imposed by \(AL\) (axle load for design vehicle), the structure requires a plastic moment capacity of:  
\[M_{p}=\frac{ALdF_{P}}{c}\bigg{(}\frac{S}{h}\bigg{)}^{2} \tag{28}\]  
Thus, for a given structure designed with the plastic moment capacity \(M_{p}\) defined by Equation (28), as the thickness of the cover over the crown \(H\) becomes greater, a greater load \(RF(AL)\) can be applied to the culvert to maintain the same required plastic moment capacity:  
\[\frac{ALdF_{P}}{c}\bigg{(}\frac{S}{h}\bigg{)}^{2}=\frac{RF(AL)dF_{P}}{c} \bigg{(}\frac{S}{H}\bigg{)}^{2} \tag{29}\]  
After the parameters cancel out,  
\[RF=\frac{H^{2}}{h^{2}} \tag{30}\]  
However, the axle load \(AL\) applied to minimum cover \(h\) for design on the left side of the equation is not necessarily the same as the \(AL\) on \(H\) for load rating on the right side. Plus, the factor of safety \(F_{p}\) that depends on the cover-to-span ratio \(H/S\) for minimum cover \(h\) on the left side of the equation is not equal to \(F_{p}\) for greater cover \(H\) on the right side. Thus, these two parameters, \(AL\) and \(F_{p}\), will not cancel out. Moreover, the culvert is not necessarily designed to have the moment capacity defined by Equation (28), which depends on the section properties of the metal culvert. In this sense, the rating factor Equation (30) that is based on the NCSPA Design Data Sheet No. 19 definition of cover heights came from an overly simplified derivation. For the rating factor based on moment in the MDOT method, the RF only defines how the maximum moment approaches its plastic moment capacity but ignores the limit state for the combined effect of axial force and moment:  
\[RF_{M}=\frac{A_{1}M_{C}-A_{2}M_{D}}{A_{3}M_{L}} \tag{31}\]  
In other words, Equation (31) does not reflect any influence from thrust that might decrease the rating factor. Two alternative rating factor definitions are therefore proposed to compensate for the deficiencies of the ODOT and MDOT rating factor methodologies. The first comes from the factor of safety based on the development of a plastic hinge [7]:  
\[RF_{Duncan}=F_{P}=0.5\frac{P_{P}}{P}\Bigg{[}\sqrt{\bigg{(}\frac{M}{M_{P}} \bigg{)}^{2}\bigg{(}\frac{P_{P}}{P}\bigg{)}^{2}+4}-\bigg{(}\frac{M}{M_{P}} \bigg{)}\bigg{(}\frac{P_{P}}{P}\bigg{)}\Bigg{]} \tag{32}\]  
where \(P\) = factored thrust due to both dead load and live load; \(P_{P}\) = factored wall yield strength; \(M\) = factored moment due to both dead load and live load; and \(M_{P}\) = factored plastic moment capacity. Even though the \(F_{P}\) does not represent a typical rating factor expression, (capacity-dead load demand)/(live load demand), its value less than 1 implies insufficient resistance against the development of plastic hinge, while a factor greater than 1 indicates safety in this regard, which serves the same use as the rating factor. Furthermore, the capacity and load demand are factored to conform to the LRFR analysis. In addition, the reciprocal of \(F_{P}\) could be considered as load demand and developed into a typical rating factor equation; however, it would overly complicate the calculation in the simplified approach. Therefore, using \(F_{P}\) as the rating factor is a compromise between simplicity and accuracy.