# Assessment of Simplified Rating Factor Methodology for Shallow Cover Metal Culverts
## 15 Worst-Case Position for Load Rating  
[PERSON] [17] concluded that the effects of backfill and live load with load positioned over the quarter point produce the greatest moment at the quarter point compared to the moment at any other location in the culvert. However, the live load position over the quarter point will not necessarily result in the lowest rating factor. The live load position over the crown or live load over the point between the crown and the quarter point where the moment due to backfill is approximately zero could also be the possible worst-case location (produces the lowest rating factor). Thus, theoretically, the rating factors should be calculated at all three positions of live load (over the crown, over the quarter point, and over the middle point between the crown and the quarter point) to obtain the critical value. This is because, for shallow buried culverts, the backfill will result in moment with tension in the inner fiber (culvert interior) at the quarter point and tension in the outer fiber at the crown of the structure, as illustrated in Figure 11, and the live load applied at different positions over the culvert will result in different techniques for computing the rating factors. The rating factors based on the three possible worst-case positions of live load should, therefore, be calculated as follows. **(1) When live load is over the crown** as in Figure 12, the moment due to live load causes tension in the inner fiber at the crown that is opposite to the fiber strain due to backfill. In this way, the live load must compensate for the moment due to backfill before the crown moves inward as is caused by the moment due to live load. In other words, the moment due to backfill at the crown tends to increase the resistance of the structure when the live load is over the crown. Hence, the rating factor based on moment should be calculated by:  
\[RF_{M}=\frac{A_{1}M_{C}+A_{2}M_{D}}{A_{3}M_{L}} \tag{23}\]  
where \(M_{C}=\) plastic moment capacity of the culvert section; \(M_{D}=\) maximum moment at the crown due to dead load; \(M_{L}=\) additional moment at the crown due to live load; and \(A_{1},A_{2},\)\(A_{3}=\) factors for moment capacity, dead load and live load, respectively. Figure 11: Moment distribution due to backfill at final cover without live load (outside values denote outer fiber in tension, inside values denote inner fiber in tension). Per AASHTO LRFD section 3.4 and CANDE-2022 _Solution Methods and Formulations_ section 6.2 [30], the load factor \(\gamma\) and load modifier \(\eta_{EB}\) for earth fill loading should be selected to produce the worst-case structural response scenario. Thus,  
\[A_{2}M_{D}=\frac{\gamma_{\min}}{\eta_{EB}}M_{1}-\gamma_{\max}\eta_{EB}M_{2}= \frac{0.9}{1.05}M_{1}-1.95\times 1.05M_{2} \tag{24}\]  
where the definitions of \(M_{1}\) and \(M_{2}\) refer to Equations (8) and (9). The parameters in the computation for \(M_{L}\) and its corresponding impact factor should be calculated based on the minimum cover depth \(H\) over the crown, where \(H\) denotes the minimum cover over the crown. \(\gamma_{\max}\) = maximum load factor dependent on load case and culvert type; \(\gamma_{min}\) = minimum load factor dependent on load case and culvert type; \(\eta_{EB}\) = net load modifier for earth fill load case = ((ductility) (redundancy) (importance)). **(2) When a live load is over the quarter point,** as illustrated in Figure 13, the moment due to the live load causes tension in the inner fiber at the quarter point that is the same as the fiber strain due to backfill [7]. In this way, the moment due to backfill at the quarter point tends to decrease the resistance of the structure when the live load is over the quarter point. Hence, the rating factor based on moment should be calculated by:  
\[RF_{M}=\frac{A_{1}M_{C}-A_{2}M_{D}}{A_{3}M_{L}} \tag{25}\]  
Figure 12: Live load is over the crown (outside values denote outer fiber in tension, inside values denote inner fiber in tension).