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10 yield limit state that are calibrated with respect to safety factors that prevailed for the former allowable stress-based design (ASD). Table 7 is a summary of resistance factors for the yield limit state as presented in the current AASHTO specifications. The ASD employed safety factors of 1.8 (i.e., 1/0.55) or 2.1 (i.e., 1/0.48) relative to yield of strip-type reinforcements or grid- type reinforcements, respectively. The higher safety factor for grid reinforcing members corresponds to a lower resistance fac- tor and is intended to ensure that no individual wire is stressed to more than 0.55Fy. This compensates for interior longitudinal elements that carry higher load compared to exterior elements due to load transfer through the transverse members of the bar mat. The safety factor of 2.1, and corresponding resistance fac- tor of 0.65, is appropriate for bar mats with four or more longi- tudinal elements but should be higher for elements with only three longitudinal elements. However, this point is not Figure 3. Statistical model of limit state equation. addressed in the current AASHTO specifications. D'Appolonia (2007) assessed strength reduction factors for the yield limit state via reliability-based calibration, but did not Resistance Factors for Design consider metal loss from corrosion as a variable. This project of Earth Reinforcements extends these studies to consider variability of metal loss and Reliability-based calibration of the strength reduction fac- the impact that this has on computed levels of reliability using tor for LRFD modeling is focused on the design of MSE wall existing design methodologies and methods for computing systems, since the AASHTO LRFD specifications for MSE the load transferred to the reinforcements. Calibration of the walls include metal loss as an explicit part of the design. resistance factors uses load factors from the AASHTO LRFD Ground anchor systems described in the AASHTO specifica- specifications and calibration methodology recommended tions incorporate a Class I corrosion protection system, by Allen et al. (2005). The resistance factor is calibrated with therefore metal loss is not incorporated into the design calcu- respect to a target reliability index, T, (i.e., probability of lations. Current AASHTO specifications include resistance occurrence), which accounts for the redundancy of the system factors for the structural resistance of ground anchors that and load redistribution inherent to the yield limit state. consider variations inherent to steel manufacturing and fab- rication. The value of varies depending on steel type as 0.9 Probability of Occurrence (Exceeding Yield) for mild steel (ASTM A-615) and 0.8 for high-strength steel for Existing Construction tendons (ASTM A-722). The AASHTO specifications do not specifically address design calculations in support of rock-bolt Generally, MSE wall systems are prefabricated, resulting installations. To address this need, service life estimates and in distinct reinforcement and reinforcement spacing. Thus, example calibrations of resistance factors for rock bolts are reinforcement yield resistance is available in discrete incre- also included in this report. ments determined by the distinct size of the reinforcement The current AASHTO (2009) LRFD Bridge Design Specifica- and reinforcement spacing selected for the project. Reinforce- tions for design of MSE walls include resistance factors for the ment sizes and spacings are selected based on particular design locations, often near the base of the wall; and unless the wall is very tall, these dimensions are held constant throughout. Table 6. Relationship Therefore, yield resistance is not optimized with respect to the between and pf. yield limit state, and for many reinforcement locations, there is a large disparity between reinforcement loads and resistance. Reliability Index Probability of () occurrence D'Appolonia (2007) studied this case using data that included (pf) measurements of reinforcement load that could be compared 2.0 2.275 x 10-2 2.5 6.210 x 10-3 with the available yield resistance. Essentially, the results 3.0 1.350 x 10-3 reported by D'Appolonia describe the probability of occur- 3.5 2.326 x 10-4 4.0 3.167 x 10-5 rence for as-built conditions, rather than for a conceptual 4.5 3.398 x 10-6 design for which yield resistance is optimized with respect to 5.0 2.867 x 10-7 the limit state.

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11 Table 7. Resistance factors for yield resistance for MSE walls with metallic reinforcement and connectors from Table 11.5.6-1, AASHTO (2009). Reinforcement Type Loading Condition Resistance Factor Strip reinforcements1 Static loading 0.75 Combined static/earthquake loading 1.00 Grid reinforcements1,2 Static loading 0.65 Combined static/earthquake loading 0.85 1 Apply to gross cross section less sacrificial area. For sections with holes, reduce gross area in accordance with AASHTO (2009) Article 6.8.3 and apply to net section less sacrificial area. 2 Apply to grid reinforcements connected to rigid facing element, for example, a concrete panel or block. For grid reinforcements connected to a flexible facing mat or that are continuous with the facing mat, use the resistance factor for strip reinforcements. Results from Monte Carlo simulations of the limit state yield resistance. Furthermore, for tall walls there may be a function and comparison with closed form solutions as number of locations where yield resistance is selected to meet reported by D'Appolonia indicate that the probability of a given load. Thus, locally, the probability of occurrence may occurrence for as-built conditions is very low, corresponding be much higher than that predicted by D'Appolonia. to > 3.5 and pf < 0.0001. These results are insensitive to Alternatively, this report describes reliability-based cali- metal loss and do not depend on the choice of resistance fac- bration for resistance factors considering that the yield limit tor. This leads to the conclusion that reinforcement yield is state function is explicitly applied at every reinforcement very unlikely given the as-built conditions of MSE walls, and location. Thus, the potential for overdesign is not directly the yield limit state does not appear to have a significant included in the analysis; however, a target reliability index, impact on performance. T of 2.3 corresponding to pf = 0.01, is adopted considering The D'Appolonia model assumes that the difference between the large redundancy inherent to the system (Allen et al., yield resistance and reinforcement load is randomly distrib- 2005). Considering as-built conditions, the resistance factors uted. In reality this is not the case. For example, the difference computed by this technique are conservative, although they may be much smaller for reinforcements located near the base are in the range of those incorporated into AASHTO (2009) of the wall or other locations that may govern the required as shown in Table 7.