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126 Table 68. Calibrated resistance factors for different datasets of resistance bias obtained using Goodman's (1989) method. No. of Bias Resistance factor ( T = 3) Dataset cases Mean COV MCS Recommended All data 119 1.35 0.535 0.336 0.30 Measured friction angle, f 98 1.41 0.541 0.346 0.35 Measured spacing, s 83 1.43 0.461 0.437 0.40 Measured friction angle, f, and s 67 1.51 0.459 0.464 0.45 data for all cases as well as for the cases with measured f, The -squared GOF tests have been carried out for all the respectively. The minimum bias observed for both of these data subsets, classified according to rock RMR values, to check datasets is 0.19, and the second lowest is 0.29 (for both, the rock the suitability of the assumption that the datasets can be mod- discontinuity spacing s is based on AASHTO [2007]). A rough eled by lognormal distributions. The -squared values obtained estimate of the equivalent factor of safety for a given calibrated for the normal distribution (N) and the lognormal distribution resistance factor is given by Equation 122, while the equivalent (LN), respectively are the following: (1) 481.64 for N and 16.22 minimum allowable bias for which the design will be safe for for LN for the total cases for rocks (n = 119); (2) 15.87 for N and the given resistance factor is given by the reciprocal of the 15.61 for LN for RMR 85 (n = 23); (3) 18.97 for N and 31.82 equivalent factor of safety (Equation 123). Thus, the minimum for LN for 65 RMR < 85 (n = 57); (4) 11.58 for N and 9.12 for allowable biases for the recommended resistance factors are the LN for 44 RMR < 65 (n = 17); and (5) 13.34 for N and 10.43 following: (1) 0.21 for = 0.30, (2) 0.25 for = 0.35, (3) 0.28 for LN for 3 RMR < 44 (n = 22). The -squared values at the for = 0.40, and (4) 0.32 for = 0.45, respectively. Except for 1% and 5% significance levels are 21.66 and 16.92, respectively; the single case of the minimum bias of 0.19 (which can be taken hence, the GOF tests show that a majority of the data subsets as a marginal case), the results imply safe design when = 0.30 follow lognormal distributions and that no outliers need to be is taken, i.e., all the data result in safe design on the application identified. of the recommended resistance factors. 4.12.2 Calibration of Resistance Factors 4.12 Carter and Kulhawy's (1988) Based on the datasets, for a majority of which the GOF tests Semi-Empirical Bearing show that lognormal distributions can be assumed to model Capacity Method for the bias distribution, the resistance factors have been calibrated Footings in/on Rock using MCS using one million samples. These factors are pre- 4.12.1 Identification of Outliers sented in Table 69. If no RMR information is available, the rec- The information and analyses presented in Section 3.8.2 suggest that the bearing resistance bias obtained using the 4 Carter and Kulhawy (1988) method depends on the type of = 3.42 COV = 0.554 foundation, i.e., a rock socket drilled into rock or a shallow Standard normal quantile foundation in/on the rock. It is also observed that a systematic 2 variation exists in the bearing resistance bias with the rock quality. When examining both factors, the data suggested (Sec- tion 3.8.2.2) that the bias variation attributed to the foundation 0 type is actually controlled by the bias relation to the rock qual- Bearing Capacity of all cases in rocks using Carter ity within the independent databases for each of the founda- and Kulhawy (1988) -2 tion types. As such, GOF tests have been carried out on the Total data (n = 119) Normal distribution datasets categorized according to the rock RMR and the resis- Lognormal distribution tance factors developed for each of these subgroups. -4 Comparisons of the standard normal quantiles of the data- 0 5 10 15 20 25 30 35 40 45 50 55 sets for (1) the total cases in/on rocks, (2) the cases in/on rocks Bias with RMR 85, and (3) the cases in/on rocks with 65 RMR Figure 117. Comparison of the unfiltered bias < 85 are presented in Figures 117, 118 and 119, respectively. for bearing capacity calculated using the Carter Except in the case of Figure 119, it can be observed that the and Kulhawy (1988) method for total cases in/on lognormal distribution fits the data better than the normal rocks in the database and the theoretical normal distribution. and lognormal distributions.