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• #### D: Comparison of ASD- and LRFD-Based Designsof Soil Nail Walls 127-136

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14 COVQ = coefﬁcient of variation of the bias of Qmax; In addition, Rn mean = λR Ro, where λR is the normal mean of zi = standard normal variable, which is the inverse the bias of Rn and Ro is the non-random scaling value deﬁned Φ−1(uia) of the normal function Φ; and previously. uia = a random number between 0 and 1 (represent- ing a random probability of occurrence). Step 3: Select Target Reliability Index In addition, Qmax mean = λQ Qo, where λQ is the normal mean Target reliability indices are selected based on the type of of the bias of Qmax, and Qo is a non-random scaling value. structure, importance of structure (i.e., related to conse- If the variable Qmax is lognormal, random values of this quences of failure), and the structural redundancy. Structural variable can be generated as: redundancy refers to the ability of a structure to transfer loads to other members if one of its supporting members fails. Tar- Qmax i = exp ( μ ln Q + σ ln Q z i ) (3-21) get reliability indices typically range between 2 and 3 for typi- cal geotechnical design (Barker et al., 1991). Allen et al. (2005) where recommend selecting βT close to 2, the lower end of the typical μln Q = lognormal mean of Qmax and range, when the structural component is not critical or it is σln Q = lognormal standard deviation of Qmax. redundant, and close to 3, the upper end of the range, when the The above parameters can be obtained from the normal structural component is critical or it is non-redundant. parameters deﬁned previously as: Zhang et al. (2001) suggested that it is acceptable to assign to individual structural elements participating in a group a prob- σ ln Q 2 μ ln Q = ln (Qmax mean ) − ability of failure that is higher than that of the group. Allen (3-22) 2 et al. (2005) suggested that an individual element of a substruc- ture can be considered redundant if the reliability index of the and entire system is signiﬁcantly lower (i.e., 0.5 lower) than that of individual components. This situation may occur in geotech- σ ln Q = ln (COVQ + 1) (3-23) nical systems that rely on numerous structural elements (e.g., various layers of geosynthetic or steel reinforcement in a retain- If the resistance is modeled as a lognormal variable, the ﬁrst ing structure or various driven piles in a pile group). Systems term of Equation 3-19 can be randomly generated as: with various structural elements tend to have greater structural redundancy and thereby result in a higher overall reliability γQ exp ( μ ln R + σ ln R z i ) Rni = (3-24) index than systems with few resisting elements. For example, a φR pile group is signiﬁcantly more redundant than a single drilled pile. This concept will be applied to SNWs, as discussed in the where following paragraph. μln R = lognormal mean of Rn; Resistance factors for shallow foundations have been cali- σln R = lognormal standard deviation of Rn; brated using βT = 3.0 (corresponding to Pf = 0.14%, a relatively zi = standard normal variable, which is the inverse Φ−1(uib) low value), as these systems are not highly redundant (Baker of the normal function Φ; and et al, 1991). Resistance factors for deep foundations have been uib = a random number between 0 and 1 (representing a calibrated for βT = 2.33 (corresponding to Pf = 1%), as driven random probability of occurrence, and being inde- piles and drilled shafts are typically installed as part of pile/shaft pendently generated from uia). groups (Paikowsky et al., 2004) and thereby carry some struc- tural redundancy. D’Appolonia (1999) used βT = 2.50 to cali- The above parameters can be obtained from the normal parameters for Rn as: brate resistance factors for pullout in geogrids, which is a system that tends to be redundant as multiple reinforcement layers are σ ln R 2 installed with a typical vertical spacing of 1 to 1.5 ft. Allen et al. μ ln R = ln ( Rn mean ) − (3-25) (2001) adopted βT = 2.33 for the calibration of pullout resis- 2 tance factors in MSE walls. and Step 4: Establish Load Factors σ ln R = ln (COVR + 1) (3-26) An estimate of the load factor needs to be performed to evaluate whether the load factors [typically those used in where Rn mean = mean of Rn and AASHTO (2007)] are applicable or whether different load COVR = coefﬁcient of variation of the bias of Rn. factors need to be proposed.

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54 Table 3-11. Statistics of bias for nominal bond strength. Resistance Parameters Number Coefficient Log Log of Points Mean of Standard of Mean of Standard Material Distribution in Bias Deviation Variation Bias Deviation Type Database λR σR μln σln N COVR Sand and 82 Lognormal 1.050 0.25 0.24 0.02 0.24 Sand/Gravel Clay/ 45 Lognormal 1.033 0.05 0.05 0.03 0.05 Fine-Grained Rock 26 Lognormal 0.920 0.18 0.19 –0.10 0.19 All 153 Lognormal 1.050 0.22 0.21 0.03 0.21 variate on Figure 3-32. The distribution selected to ﬁt the data ulations were conducted to improve initial values presented was also a lognormal distribution that was adjusted to match previously in this chapter. the upper tail of the load bias distribution. The bias calculated for each of these cases is presented 3.7.2 Description of Calibration Process in Table 3-13. Statistical parameters are summarized in Table 3-14. These parameters are also used in the calibration. The calibration was performed using the following steps: 3.7 Calibration of Pullout Step 1: Establish a limit state function; Resistance Factors Step 2: Develop PDFs and statistical parameters for loads and resistances; 3.7.1 Introduction Step 3: Select a target reliability index for SNW design; Step 4: Establish load factors; This section presents the results of the calibration of pull- Step 5: Best-ﬁt cumulative density functions to data points; out resistance factors. The calibration was conducted apply- Step 6: Conduct Monte Carlo simulation; ing the calibration framework developed by Allen et al. (2005), Step 7: Compare computed and target reliability indices; and which was presented earlier in this chapter. Monte Carlo sim- 0 Data Approximation 0.2 Normalized Depth, h/H 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 T Normalized Nail Force, K γ H S S a hv Figure 3-30. Summary of tensile forces measured in instrumented SNWs.

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Table 3-12. Characteristics of monitored soil nail walls. Feature Case Swift- Swift- IH-30, IH-30, Feature Peasmarsh, Guernsey, San Cumberland Oregon Delta Delta Polyclinic Rockwall, Rockwall, I-78, Allentown U.K. U.K. Bernardino Gap, 1988 Station 1 Station 2 Section A Section B Height (m) TBC 5.3 5.6 16.8 11 20 5.2 4.3 7.6 7.9 12.2 Face slope (deg) TBC 0 0 0 20 30 0 0 6 0 3 m bench 55 kN/m 75 kN/m Back slope (deg) TBC 27 0 0 0 0 5 33 33 surcharge surcharge Type of facing TBC shotcrete shotcrete shotcrete geogrid geogrid shotcrete shotcrete shotcrete shotcrete concrete panels Nail length (m) TBC 6.4 5.2 10.7 6–7 10 6.1 6.1 6.7 13.4 6.1–9.2 Nail inclination TBC 15 15 15 20 20 5 5 12 15 10 (deg) Nail diameter (mm) TBC NA NA NA NA NA 152 152 203 114 89 Steel diameter (mm) TBC 29 29 36 25 25 19 19 25 29 25–32 Spacing, H x V (m) TBC 1.4 x 1 1.4 x 1 1.8 x 1.8 1.5 x 1.5 1.5 x 1.25 0.75 x 0.75 0.75 x .75 1.5 x 1.5 1.5 x 1.2 1.5 x 1.5

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57 2 11 Instrumented SNWs (13 Points) 1.5 1 Standard Normal Variable, z 0.5 0 -0.5 -1 -1.5 -2 0.5 0.75 1 1.25 1.5 Bias λQ of Tm Figure 3-32. Bias for maximum load in soil nails. Q Table 3-13. Summary of normalized measured and predicted maximum nail load. Normalized Normalized Bias of No. Case Measured Predicted Load Load, Tm Load, Tp 1 Cumberland Gap, 1988 0.54 1.05 0.51 2 Polyclinic 0.56 0.94 0.59 3 I-78, Allentown 0.68 1.07 0.63 4 Guernsey, U.K. 0.51 0.71 0.72 5 Swift-Delta Station 2 1.11 1.43 0.78 6 Oregon – 3-A 0.81 0.98 0.82 7 Swift-Delta Station 1 0.81 0.97 0.84 8 Peasmarsh, U.K. 0.58 0.65 0.89 9 Oregon – 2-B 1.05 1.10 0.95 10 IH-30, Rockwall, Section B 1.06 0.99 1.01 11 Oregon – 1-A 0.96 0.80 1.11 12 San Bernardino (R) 1.08 0.83 1.20 13 San Bernardino (L) 1.13 0.83 1.36 Table 3-14. Statistics of bias for maximum nail loads. Load Parameters Number Coefficient Log Log of Points Mean of Standard of Mean of Standard Distribution in Bias Deviation Variation Bias Deviation Type Database λQ σQ μ ln σ ln N COVQ 13 Lognormal 0.912 0.290 0.32 -0.140 0.31

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