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85 F3 [kN] [kip] 0.16 PeE1.2 - e3=-0.0225m PeE1.4 - e3=0.0225m 0.03 PeE1.3 - e3=-0.0225m PeE1.5 - e3=0.0225m 0.12 MoE1.4 - e3=0.01m MoE1.1 - e3=-0.01m 0.02 PeE - steplike MoE - radial 0.08 0.01 0.04 u3 [in] 0.12 0.08 0.04 0 0.04 0.08 0.12 F1 [kip] 0 0 u3 [mm] 4 3 2 1 0 0 0.2 0.4 0.6 F1 [kN] 0.4 0.02 0.8 0.04 1.2 0.06 1.6 u1 [mm] [in] Figure 73. Loaddisplacement curves for inclined-eccentric loading with different loading directions utilizing data from Perau (1995) and Montrasio (1994). the analysis also reveals that the gain of capacity is relatively provides detailed examples for the calculations performed for small, and, for vertical load levels greater than or equal to 0.3, each analysis. Sections G.5 and G.6 relate to the utilization the effect of loading direction is negligible. of Goodman's (1989) method, and Section G.7 relates to the utilization of Carter and Kulhawy's (1988) method in the traditional way (i.e., using Equation 82a). This section sum- 3.8 Uncertainty in the Bearing marizes the results of the analyses for the examined methods: Capacity of Footings in/on Rock the semi-empirical mass parameters procedure developed by Carter and Kulhawy (1988) and the analytical method pro- 3.8.1 Overview posed by Goodman (1989). The ratio of the measured/interpreted bearing capacity to The consistency of the rocks in the database, the types the calculated shallow foundation bearing capacity (the bias ) of foundation, and the level of knowledge of the rock were was used to assess the uncertainty of the selected design categorized, when applicable, while examining their influ- methods for the 119 case histories of database GTR-UML ence on the bias. In addition, histograms and PDFs of the RockFound07. Section 1.7 details the methods of analysis bias obtained by the different methods are presented and selected for the bearing capacity calculations. Appendix G discussed.

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86 F1/F10= 0.08 F1/F10= 0.13 F1/F10= 0.145 0.12 F1/F10= 0.26 F1/F10= 0.33 F1/F10= 0.41 0.1 F1/F10= 0.55 F1/F10= 0.08 F1/F10= 0.13 0.08 F1/F10= 0.15 F1/F10= 0.26 F1/F10= 0.33 F2/ F10 F1/F10= 0.5 0.06 0.04 0.02 0 -0.12 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 M3/(F10b2) Figure 74. Influence of loading direction on capacity in the case of inclined-eccentric loading (Lesny, 2001). 3.8.2 Carter and Kulhawy's (1988) 100000 Semi-Empirical Bearing qL2 = 16.14 (qult)0.619 Capacity Method (n = 119; R2 = 0.921) qL2 = 36.51 (qult)0.600 (Revised) 3.8.2.1 Presentation of Findings Interpreted Foundation Capacity qL2 (ksf) 10000 (n = 119; R2 = 0.917) qL2 = qult Carter and Kulhawy's (1988) method is described in Sec- tion 1.7.6 and its application is demonstrated in Section G.7 1000 in Appendix G. Table E-2 of Appendix E presents the calculated bearing capacity values and the associated bias for each of the 119 case histories of database UML-GTR RockFound07 (Table E-2 includes all 122 original cases and the excluded 100 3 cases as noted). The relationships between the bearing capacities (qult) calculated using the two Carter and Kulhawy (1988) semi-empirical procedures (Equation 82a and the 10 58 Footing cases revised relations given by Equation 82b) and the interpreted 61 Rock Socket cases 119 All cases with bearing capacity (qL2) are presented in Figure 75. Equation 109a revised equation provides the best fit line generated using regression analysis 1 of all data using Equation 82a and results in a coefficient of 0.01 0.1 1 10 100 1000 10000 100000 determination (R2) of 0.921. Equation 109b represents the Carter and Kulhawy (1988) Bearing Capacity qult (ksf) best fit line generated using regression analysis of all data using Figure 75. Relationship between calculated bearing Equation 82b for calculating the bearing capacity and results capacity (qult) using two versions of Carter and in a coefficient of determination (R2) of 0.917. Kulhawy (1988) and interpreted bearing capacity (qL2 ).

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87 Table 37. Summary of the statistics for the ratio of measured (qL2) to calculated bearing capacity (qult) for all foundations on rock using the Carter and Kulhawy (1988) method. Cases n No. of sites m COV All (measured qu) 119 78 8.00 9.92 1.240 Measured discontinuity spacing (s) 83 48 8.03 10.27 1.279 Fractured with measured discontinuity spacing (s) 20 9 4.05 2.42 0.596 All non-fractured 99 60 8.80 1066 1.211 Non-fractured with measured discontinuity spacing (s) 63 39 9.29 11.44 1.232 Non-fractured with s based on AASHTO (2007) 36 21 7.94 9.22 1.161 n = number of case histories, m = mean of biases, = standard deviation, COV = coefficient of variation Calculated capacity based on Equation 82a qL 2 = 16.14 ( qult ) 0.619 (109a) (measured bearing capacity, qL2, to calculated bearing capacity, qult) using Carter and Kulhawy's (1988) semi-empirical method qL 2 = 36.51 ( qult ) 0.600 (109 b) are summarized in Table 37. In Table 37, the statistics are categorized according to the joint spacing and the source of It can be observed in Figure 75 (and in Equations 109a and the data (i.e., measured discontinuity spacing versus spacing 109b) that the revised expression provided by Equation 82b assumed based on the specifications). In Table 38, the data gives systematically higher resistance biases than those biases are subcategorized according to type of foundation (footings obtained using Equation 82a. The bias mean and COV obtained versus rock sockets) and the source of the joint spacing data. using Equation 82b for all data (n = 119) are found to be 30.29 Table 39 is a summary of the statistics for the ratio of the and 1.322, respectively, versus 8.00 and 1.240, respectively, measured bearing capacity (qL2) to calculated bearing capacity obtained using Equation 82a. Both relations provide close to (qult) categorized according to foundation type and rock quality parallel lines when compared to the measured capacities. Equa- ranges for each type and all types combined. tions 109a and 109b suggest that Equation 82b roughly predicts The distribution of the ratio of the interpreted bearing capac- half the capacity of Equation 82a as its multiplier to match the ity to the calculated bearing capacity (the bias ) for the 119 case measured capacity is about double. As the relations pro- histories (detailed in Table E-2 of Appendix E) is presented in vided by Equation 82a are already consistently conservative, Figure 76. The distribution of the bias has a mean (m) of 8.00 Equation 82a is preferred over Equation 82b, and the results and a COV of 1.240 and resembles a lognormal random vari- processed and analyzed are those obtained using Equation 82a. able. The distribution of the bias for foundations on fractured Statistical analyses were performed to investigate the effect rock only (20 cases) is presented in Figure 77 and has an m of of the joint or discontinuity spacing (s) either measured or 4.05 and a COV of 0.596. The distribution of the bias for the determined based on AASHTO (2008) tables (see Section 1.8.3) foundations on fractured rock resembles a lognormal random and the effect of the friction angle (f) of the rock on the variable and has less scatter, reflected by the smaller COV when calculated bearing capacity. Statistics for the ratio of the bias compared with the distribution of for all 119 case histories. Table 38. Summary of the statistics for the ratio of measured (qL2) to calculated bearing capacity (qult) of rock sockets and footings on rock using the Carter and Kulhawy (1988) method. Cases n No. of sites m COV All rock sockets 61 49 4.29 3.08 0.716 All rock sockets on fractured rock 11 6 5.26 1.54 0.294 All rock sockets on non-fractured rock 50 43 4.08 3.29 0.807 Rock sockets on non-fractured rock with measured discontinuity spacing (s) 34 14 3.95 3.75 0.949 Rock sockets on non-fractured rock with s based on AASHTO (2007) 16 13 4.36 2.09 0.480 All footings 58 29 11.90 12.794 1.075 All footings on fractured rock 9 3 2.58 2.54 0.985 All footings on non-fractured rock 49 26 13.62 13.19 0.969 Footings on non-fractured rock with measured discontinuity spacing (s) 29 11 15.55 14.08 0.905 Footings on non-fractured rock with s based on AASHTO (2007) 20 11 10.81 11.56 1.069 n = number of case histories, m = mean of biases, = standard deviation, COV = coefficient of variation Calculated capacity based on Equation 82a

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88 Table 39. Summary of the statistics for the ratio of measured (qL2) to calculated bearing capacity (qult) using the Carter and Kulhawy (1988) method categorized by the rock quality and foundation type. Foundation Cases n No. of sites m COV Type RMR > 85 23 23 2.93 1.908 0.651 65 < RMR < 85 57 36 3.78 1.749 0.463 All 44 < RMR < 65 17 10 8.83 5.744 0.651 3 < RMR < 44 22 9 23.62 13.550 0.574 RMR > 85 16 16 3.42 1.893 0.554 Rock 65 < RMR < 85 35 24 3.93 1.769 0.451 Sockets 44 < RMR < 65 9 8 6.82 6.285 0.921 3 < RMR < 44 1 1 8.39 -- -- RMR > 85 7 7 1.81 1.509 0.835 65 < RMR < 85 22 13 3.54 1.732 0.489 Footings 44 < RMR < 65 8 5 11.09 4.391 0.396 3 < RMR < 44 21 8 24.34 13.440 0.552 n = number of case histories, m = mean of biases, = standard deviation, COV = coefficient of variation Calculated capacity based on Equation 82a 3.8.2.2 Observations higher than those obtained using Equation 82a, with very similar COVs. As both equations are by and large conser- The presented findings of Carter and Kulhawy's (1988) vative, only the traditional equation (Equation 82a) was methods for the prediction of bearing capacity suggest the used for further analysis and method evaluation. following: 2. The method (Equation 82a) substantially underpredicts (on the safe side) for the range of capacities typically lower 1. The bias of the estimated bearing resistances obtained using than 700 ksf. The bias increases as the bearing capacity the revised equation (Equation 82b) are systematically decreases. This provides a logical trend in which founda- 0.15 5 0.25 119 Rock sockets and Footing cases 20 Foundation cases on Fractured Rocks Carter and Kulhawy (1988) Carter and Kulhawy (1988) 16 mean = 8.00 mean = 4.05 COV = 1.240 4 0.2 COV = 0.596 Number of observations Number of observations 12 0.1 3 0.15 Frequency Frequency lognormal lognormal distribution 8 distribution 2 normal 0.1 normal distribution 0.05 distribution 4 1 0.05 0 0 0 0 0 4 8 12 16 20 24 28 32 36 40 44 48 52 0 1 2 3 4 5 6 7 8 9 10 Bias, = qu,meas / qu,calc Bias, = qu,meas / qu,calc Figure 76. Distribution of the ratio of the interpreted Figure 77. Distribution of the ratio of the interpreted bearing capacity (qL2) to the bearing capacity (qult ) bearing capacity (qL2) to the bearing capacity (qult ) calculated using Carter and Kulhawy's (1988) method calculated using Carter and Kulhawy's (1988) method (Equation 82a) for the rock sockets and footings in (Equation 82a) for foundations on fractured rock in database UML-GTR RockFound07. database UML-GTR RockFound07.

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89 tions on lower bearing capacity materials are provided with 3.8.3 Goodman's (1989) Analytical a higher margin of safety while for foundations on harder Bearing Capacity rock with higher bearing capacities, the bias is smaller than 3.8.3.1 Presentation of Findings one (1.0) (i.e., measured capacities are lower than calculated capacities). The bearing capacity values on the higher Goodman's (1989) method is described in Section 1.7.5 capacity sides are controlled by the strength of the foun- and its application is demonstrated in Sections G.5 and G.6 dation material (i.e., concrete), and, therefore, the results of Appendix G. Table E-3 of Appendix E presents the calculated in that range are not necessarily translated into unsafe bearing capacity values for each of the 119 case histories. The practice. relationship between the bearing capacity calculated using 3. Comparison of the statistics obtained for shallow foun- Goodman's (1989) analytical procedure (qult) and the inter- dations (n = 58, m = 11.90, COV = 1.075 and number preted bearing capacity (qL2) is presented in Figure 78. Equa- of sites = 29) with the statistics obtained for rock sockets tion 110 represents the best fit line that was generated using (n = 61, m = 4.29, COV = 0.716 and number of sites = 49) regression analysis and resulted in a coefficient of determination may suggest that the method better predicts the capacity (R2) of 0.897. of rock sockets than the capacity of footings. This obser- qL 2 = 2.63 ( qult ) 0.824 vation might also suggest that the use of load-displacement (110) relations for the tip of a loaded rock socket is not analogous Statistical analyses were performed to investigate the effect to the use of load-displacement relations for a shallow of the measured and AASHTO-based joint (2007) or dis- foundation constructed below surface; hence, the data continuity spacing (s) and friction angle (f) of the rock on related to the tip of a rock socket should not be employed the bearing capacity calculations. Table 40 summarizes the for shallow foundation analyses. This observation must be statistics for the ratio of the measured bearing capacity (qL2) re-examined in light of the varied bias of the method with to calculated bearing capacity (qult) using Goodman's (1989) the rock strength, as is evident in Figure 75 and detailed in analytical method for the entire database. Table 41 provides the Table 39. The varying bias of the method, as observed in statistics for subcategorization based on foundation type and Figure 75 and described in Number 2 above, results in a available information. Table 42 is a summary of the statistics relatively high scatter (COV = 1.240 for all cases). When the for the ratio of the measured bearing capacity (qL2) to the evaluation is categorized based on rock quality, the scatter calculated bearing capacity (qult) categorized according to (COV) systematically decreases to be between about 0.5 foundation type and rock quality ranges for each type. to 0.6, as detailed in Table 39. However, the changes in the mean of the bias with rock quality for the footings are 100000 much more pronounced than the changes for the rock qL2 = 2.16 (qult)0.868 sockets because most of the footings were tested on rock (n = 119; R2 = 0.897) that was of lower quality than the rock existing at the tip of qL2 = qult Interpreted Foundation Capacity qL2 (ksf) 10000 the rock sockets. For example, of the 22 cases of the lowest rock quality (3 RMR < 44), 21 cases involved a shallow foundation and 1 case involved a rock socket. In contrast, of the 23 cases of the highest quality rock (RMR 85), 1000 only 7 cases involve footings and 16 cases involve rock sockets. The conclusion, therefore, is that the variation in the method application is more associated with the rock 100 type/strength and its influence on the method's predic- tion than the foundation type. This conclusion is further confirmed by examination of the Goodman (1989) method, 10 in which the bias is not affected by rock quality and, hence, 58 Footing cases similar statistics are obtained for the rock socket and the 61 Rock Socket cases footing cases. 1 4. No significant differences exist between the cases for 1 10 100 1000 10000 100000 which discontinuity spacing (s) was measured in the Goodman (1989) Bearing Capacity qult (ksf) field and the cases for which the spacing was deter- Figure 78. Relationship between Goodman's (1989) mined based on generic tables utilizing rock description calculated bearing capacity (qult) and the interpreted (Tables 37 and 38). bearing capacity (qL2)

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90 Table 40. Summary of the statistics for the ratio of measured (qL2) to calculated bearing capacity (qult) of rock sockets and footings on rock subcategorized by data quality using the Goodman (1989) method. Cases n No. of sites m COV All 119 78 1.35 0.72 0.535 Measured discontinuity spacing (s) and friction angle ( f) 67 43 1.51 0.69 0.459 Measured discontinuity spacing (s) 83 48 1.43 0.66 0.461 Measured friction angle ( f) 98 71 1.41 0.76 0.541 Fractured 20 9 1.24 0.34 0.276 Fractured with measured friction angle ( f) 12 7 1.33 0.25 0.189 Non-fractured 99 60 1.37 0.77 0.565 Non-fractured with measured s and measured f 55 37 1.55 0.75 0.485 Non-fractured with measured discontinuity spacing (s) 63 39 1.49 0.72 0.485 Non-fractured with measured friction angle ( f) 86 64 1.42 0.81 0.569 Spacing s and f, both based on AASHTO (2007) 5 3 0.89 0.33 0.368 Discontinuity spacing (s) based on AASHTO (2007) 36 21 1.16 0.83 0.712 Friction angle ( f) based on AASHTO (2007) 21 7 1.06 0.37 0.346 n = number of case histories, m = mean of biases, = standard deviation, COV = coefficient of variation The distribution of the ratio of the interpreted measured 3.8.3.2 Observations bearing capacity to the calculated bearing capacity () for The presented findings of Goodman's (1989) method for the 119 case histories in Table E-3 of database UML-GTR the prediction of bearing capacity suggest the following: RockFound07 is presented in Figure 79. The distribution of has a mean (m) of 1.35 and a COV of 0.535 and resembles a lognormal random variable. The distribution of for only 1. The method is systematically accurate, as demonstrated the foundations on fractured rock is presented in Figure 80 by the proximity of the best fit line to the perfect match and has an m of 1.24 and a COV of 0.276. line (measured qL2 = predicted qu) presented in Figure 78 Table 41. Summary of the statistics for the ratio of measured (qL2) to calculated bearing capacity (qult) of rock sockets and footings on rock subcategorized by foundation type and data quality using the Goodman (1989) method. Cases n No. of sites m COV All rock sockets 61 49 1.52 0.82 0.541 Rock sockets with measured friction angle ( f) 46 48 1.64 0.90 0.547 All rock sockets on fractured rock 11 6 1.29 0.26 0.202 Rock sockets on fractured rock with measured friction angle ( f) 7 5 1.23 0.18 0.144 All rock sockets on non-fractured rock 50 43 1.58 0.90 0.569 Rock sockets on non-fractured rock with measured s and measured f 26 26 1.58 0.79 0.497 Rock sockets on non-fractured rock with measured discontinuity spacing (s) 34 14 1.49 0.71 0.477 Rock sockets on non-fractured rock with measured friction angle ( f) 39 43 1. 72 0.96 0.557 Rock sockets on non-fractured rock with discontinuity spacing (s) based on 13 12 1.99 1.22 0.614 AASHTO (2007) and measured friction angle ( f) Rock sockets on non-fractured rock with measured discontinuity spacing (s) and 8 3 1.19 0.21 0.176 friction angle ( f) based on AASHTO (2007) Rock sockets on non-fractured rock with discontinuity spacing (s) based on 3 2 0.75 0.36 0.483 AASHTO (2007) and friction angle ( f) based on AASHTO (2007) All footings 58 29 1.23 0.66 0.539 Footings with measured friction angle ( f) 52 23 1.27 0.69 0.542 All footings on fractured rock 9 3 1.18 0.43 0.366 Footings on fractured rock with measured friction angle ( f) 5 2 1.47 0.29 0.200 All footings on non-fractured rock 49 26 1.24 0.70 0.565 Footings on non-fractured rock with measured s and measured f 29 11 1.51 0.73 0.481 Footings on non-fractured rock with measured discontinuity spacing (s) 29 11 1.51 073 0.481 Footings on non-fractured rock with measured friction angle ( f) 47 21 1.25 0.72 0.573 Footings on non-fractured rock with discontinuity spacing (s) based on AASHTO 18 10 0.82 0.45 0.543 (2007) and measured friction angle ( f) Footings on non-fractured rock with discontinuity spacing (s) based on AASHTO 2 1 1.10 0.13 0.115 (2007) and friction angle ( f) based on AASHTO (2007) n = number of case histories, m = mean of biases, = standard deviation, COV = coefficient of variation

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91 Table 42. Summary of the statistics for the ratio of measured (qL2) to calculated bearing capacity (qult) using the Goodman (1989) method categorized by rock quality and foundation type. Foundation Cases n No. of sites m COV type RMR > 85 23 23 1.55 0.679 0.438 65 < RMR < 85 57 36 1.33 0.791 0.595 All 44 < RMR < 65 17 10 1.27 0.746 0.586 3 < RMR < 44 22 9 1.24 0.529 0.426 RMR > 85 16 16 1.59 0.809 0.509 Rock 65 < RMR < 85 35 24 1.40 0.722 0.515 Sockets 44 < RMR < 65 9 8 1.47 0.916 0.624 3 < RMR < 44 1 1 1.27 -- -- RMR > 85 7 7 1.46 0.204 0.140 65 < RMR < 85 22 13 1.22 0.896 0.738 Footings 44 < RMR < 65 8 5 1.06 0.461 0.437 3 < RMR < 44 21 8 1.24 0.542 0.437 n = number of case histories, m = mean of biases, = standard deviation, COV = coefficient of variation and the bias of about 1.2 to 1.5 for all types of major variation of bias with rock strength resulted in a similar subcategorization. COV only when each range of rock strength was examined 2. The consistently reliable performance of the method for separately. This observation enforces the notion of incor- all ranges of rock strength (and hence RMR) provides a porating rock quality categorization (e.g., RMR) within the COV of 0.535 for all cases. The variation of the bias mean bearing capacity predictive methodology when necessary. and COV with rock quality is essentially absent, as can be 3. Similar statistics were obtained for shallow foundations observed in Table 42. This is in contrast to the perform- (n = 58, m = 1.23, COV = 0.539) and rock sockets (n = 61, ance of Carter and Kulhawy's (1988) method, in which the m = 1. 52, COV = 0.541). These observations suggest that 12 0.6 119 Rock sockets and Footing cases 20 Foundation cases on Fractured Rocks 0.35 40 Goodman (1989) Goodman (1989) mean = 1.35 mean = 1.24 10 0.5 COV = 0.535 0.3 COV = 0.276 Number of observations Number of observations 30 0.25 8 0.4 lognormal Frequency Frequency 0.2 distribution lognormal 6 0.3 20 distribution 0.15 normal distribution normal 4 0.2 distribution 0.1 10 2 0.1 0.05 0 0 0 0 0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 Bias, = qu,meas / qu,calc Bias, = qu,meas / qu,calc Figure 79. Distribution of the ratio of the interpreted Figure 80. Distribution of the ratio of the interpreted bearing capacity (qL2) to the bearing capacity (qult) bearing capacity (qL2) to the bearing capacity (qult) calculated using Goodman's (1989) method for the calculated using Goodman's (1989) method for rock sockets and footings in database UML-GTR foundations on fractured rock in database UML-GTR RockFound07. RockFound07.