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127 3 4 = 2.93 = 3.78 COV = 0.651 COV = 0.463 2 Standard normal quantile Standard normal quantile 2 1 0 0 Bearing Capacity of all cases Bearing Capacity of all cases in rocks using Carter in rocks using Carter -1 and Kulhawy (1988) and Kulhawy (1988) RMR 85 -2 65 RMR < 85 Total data (n = 23) Total data (n = 57) -2 Normal distribution Normal distribution Lognormal distribution Lognormal distribution -3 -4 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 9 Bias Bias Figure 118. Comparison of the unfiltered bias Figure 119. Comparison of the unfiltered bias for bearing capacity calculated using the Carter for bearing capacity calculated using the Carter and Kulhawy (1988) method for all cases in and Kulhawy (1988) method for all cases in rocks with RMR ­> 85 and the theoretical normal rocks with 65 < ­ RMR < 85 and the theoretical and lognormal distributions. normal and lognormal distributions. ommended is 0.35. When the rock has RMR 85 the recom- quality (expressed via RMR), and a calibration was required mended is 0.50. For rocks with RMR lower than 85, = 1.00. following the rock quality designation. The relatively higher resistance factors are a byproduct of the large bias of the method and, hence, do not represent efficient design as expressed by the 4.13 Summary of Recommended low efficiency factor of the method's application compared to Resistance Factors for Shallow Goodman's (1989) method. Foundations in/on Rock Table 70 summarizes (based on the information presented in Tables 68 and 69) the recommended resistance factors to be 4.14 Sliding Friction Resistance used in evaluation of the bearing capacity of shallow founda- 4.14.1 Parametric Study Evaluating the tions on rock. The resistance factors for both examined meth- Resistance Factor as a Function of ods are presented along with the efficiency factors providing a the Ratio of Dead to Live Load measure for the relative efficiency of the methods. Goodman's (1989) method performed exceptionally well The probabilistic characteristics of the parameter contribut- consistently, regardless of rock quality. Improvement in the ing directly to the sliding friction resistance, the friction coeffi- method's performance with an increase in knowledge trans- cient ratio ( fc), have been presented in Section 3.9 and lates into an increase in the resistance factor and the associated summarized in Table 48. The uncertainties in the friction coef- method efficiency. ficient ratio ( fc) follow one-to-one transformation to the slid- The performance of the Carter and Kulhawy (1988) method ing resistance, i.e., the mean of sliding resistance = vertical has a "built-in" safety that increases as the rock quality load × (mean of fc × tan f ) and the standard deviation (s.d.) decreases. As such, the method's bias changes with the rock of sliding resistance = vertical load × (s.d. of fc × tan f ). Hence, Table 69. Calibrated resistance factors for different datasets of resistance bias obtained using Carter and Kulhawy's (1988) method. No. of Bias Resistance factor ( T = 3) Dataset cases Mean COV MCS Recommended All cases 119 8.00 1.240 0.372 0.35 RMR 85 23 2.93 0.651 0.535 0.50 65 RMR < 85 57 3.78 0.463 1.149 1.00 44 RMR < 65 17 8.83 0.651 1.612 1.00 3 RMR < 44 22 23.62 0.574 5.295 1.00

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128 Table 70. Recommended resistance factors for foundations in/on rock based on T = 3.0 (pf = 0.135%). Efficiency Method of Equation Application factor analysis / (%) All 0.35 4.4 Carter and RMR 85 0.50 17.1 Kulhawy qult qu m s 65 RMR < 85 26.5 (1988) 44 RMR < 65 1.00 11.3 3 RMR < 44 4.2 For fractured rocks: All 0.30 22.2 qult qu N 1 Goodman Measured f 0.35 24.8 For non-fractured rocks: (1989) (N 1) N 1 s Measured s 0.40 28.0 qult qu N 1 N 1 B Measured s and f 0.45 29.8 Table 71. Resistance factors obtained from MCS simulations for footings, either cast in place or prefabricated, in soils with various friction angles, along with the effect of ratios of lateral dead load to lateral live load. (a) Cast-in-place footings Resistance factor from MCS ( MCS) f obtained At-rest earth pressure Active earth pressure from LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL =2 =4 =5 =7 =2 =4 =5 =7 SPT 0.469 0.455 0.452 0.447 0.507 0.498 0.496 0.492 CPT 0.516 0.499 0.494 0.488 0.558 0.545 0.542 0.537 Lab test 0.558 0.535 0.530 0.523 0.603 0.585 0.581 0.576 (b) Prefabricated footings Resistance factor from MCS ( MCS) f obtained At-rest earth pressure Active earth pressure from LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL LFD/LFL =2 =4 =5 =7 =2 =4 =5 =7 SPT 0.195 0.193 0.193 0.191 0.211 0.212 0.211 0.211 CPT 0.217 0.213 0.212 0.210 0.234 0.233 0.232 0.232 Lab test 0.239 0.234 0.232 0.230 0.258 0.256 0.255 0.253 Table 72. Recommended resistance factors for sliding resistance () for soil friction angles based on different tests and lateral pressure due to at-rest or active earth pressure for cast-in-place and prefabricated footings. Resistance factor for sliding friction ( ) ( T = 3) f obtained At-rest earth pressure Active earth pressure from Cast in- Cast in- Prefabricated 2 Prefabricated 2 place 1 place 1 SPT 0.40 0.45 CPT 0.45 0.20 0.50 0.20 Lab test3 0.50 0.55 1 2 3 tan s = 0.91 tan f ; tan s = 0.53 tan f , Any laboratory shear strength measurement of f

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129 the form of the limit state function for sliding resistance is Based on the loadings for the design example bridges consid- essentially the same as that for the bearing resistance (see Equa- ered in the current research study, it is found that the ratios of tion 118), which can be expressed as LFD to LFL range from 4 to 7. As a result, the resistance factors for sliding resistance have been calibrated for LFD to LFL ratios Z = R - LFD - LFL (125) varying from 2 to 7 and the corresponding results are presented in Table 71 for cast-in-place and prefabricated footings. where Z is the load combination for sliding, R is sliding resis- tance of a footing, LFD is lateral load due to dead load, and LFL 4.14.2 Resistance Factors is lateral load due to live load. A summary of the uncertainties in the lateral loads and the load factors as recommended in The calculated resistance factors presented in Table 71 suggest AASHTO (2007) are presented in Section 4.2.3.3. that the ratio of LFD to LFL does not have a pronounced effect Analogous to the calibration of resistance factors for the on the magnitude of the resistance factors. As a result, selected bearing resistance, the influence of the ratio of lateral dead load resistance factors are recommended for use for sliding resistance to the lateral live load has been studied and presented here. of footings on granular materials as presented in Table 72.