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OCR for page 107
107 2 FOSM QL = 1.15 QD = 1.05 1.6 COVQL = 0.2 COVQD = 0.1 QD/QL = 2.0 = 3.00, 3.25, 3.50 D = 1.25 L = 1.75 Resistance Factor () 1.2 0 3 .0 5 = 3. 2 = 0.1 0 = 3. 5 0.8 V = CO 0.3 V= CO 0.4 0.6 COV = COV = 0.9 0 0 1 2 3 Bias () Figure 92. Calculated resistance factors as a function of the bias and COV of the resistance for the chosen vertical loading distributions and ratios under the range of the examined target reliabilities. and collision loads). The load combination for Strength I (Z) 4.4 Examination of the Factor N was therefore applied in its primary form, as shown in the fol- as a Source of Uncertainty in lowing limit state: Bearing Capacity Analysis Z = R - D - LL (118) 4.4.1 Overview Section 3.5 examined the uncertainty in the bearing capac- where R = the strength or resistance of the footing, D = dead ity of footings in/on granular soils subjected to vertical-centric load, and LL = vehicular live loads. The probabilistic char- loading. This load type pertains to 173 case histories of data- acteristics of the random variables D and LL are provided base UML-GTR ShalFound07. A summary of the bias is pre- in Table 49 for vertical loads and in Section for lat- sented as a flow chart in Figure 60 and histograms and relations eral loads. For the strength or resistance (R), the probabilis- between measured and calculated capacities in Figures 61 to 65. tic characteristics are developed in Chapters 3 and 4, based The analysis of the data indicated the following: on the databases for the various methods and conditions of analysis. Paikowsky et al. (2004) examined the influence of the ratio 1. Overall, the mean bias (measured over predicted capacity) of dead load to live load, demonstrating very little sensitivity of was greater than 1 (m = 1.59 for n = 173) pointing out a the resistance factors to that ratio, with overall decrease of the systematic capacity underprediction. resistance factors with the increase in the ratio of dead load to 2. The mean bias (m) of the footings on natural soil condi- live load. Large ratios of dead load to live load represent condi- tions was 1.0, and the mean bias (m) of the footings on tions of bridge construction typically associated with very long controlled soil conditions was 1.64. bridge spans. The relatively small influence of the ratio of dead 3. Previous findings suggested resistance factors based on the load to live load on the resistance factor led Paikowsky et al. separation between natural and controlled soils, using the (2004) to use a typical ratio of 2.0, knowing that the obtained above findings (Paikowsky et al., 2008; Amatya et al., 2009; factors are by and large applicable for long span bridges, being Paikowsky et al., 2009b). on the conservative side. This ratio was adopted, therefore, for the present study calibrations as well. Discussion of the ratio of A clear variation exists between the cases of the foundations dead load to live load for lateral loads is presented later in this on natural soils and the cases of the foundations on controlled chapter. soils by a factor of 1.6. The source of this large variation in the

OCR for page 107
108 bias was further investigated, especially other parameters that bias in N obtained for soils with friction angles between 42 could affect this variation and could be the source for the large and 46. The data points representing the bias in N presented bias in the prediction. Section 1.5.2 discusses the fact that no in Figure 93 suggest a clear trend in which the bias N increased closed-form analytical solution exists for the bearing capacity as the soil's internal friction increased beyond about f 43. problem formulation once the soil weight effect beneath the The best fit line of the bias N versus internal friction f , as foundation is considered. The factor N has been, therefore, expressed in Figure 93, can be used to develop an expression evaluated by many researchers with varying results, as demon- for a modified bearing capacity factor N that would better strated in Figure 11. The investigation of the factor N using match the experimental data: the robust database assembled for this study is presented in the following section in view of the aforementioned bias findings. N Exp = exp ( 0.205 f - 8.655 ) N Vesic for 42.5 f 46 (120) 4.4.2 The Uncertainty in the Bearing Capacity Factor N The large scatter of the data results in a coefficient of deter- mination (R2) of 0.351 for Equation 120. For foundations tested on the surface of granular soils, the bearing capacity (Equation 19) becomes a function of the term N only, as the cohesion and embedment terms are zeroed. 4.4.3 Re-examination of the Uncertainty The bearing capacity factor N can then be back-calculated and in Bearing Capacity of Footings the obtained factor (termed NExp) can be evaluated against that in/on Granular Soils Accounting proposed by Vesic (1973) (termed NVesic) and used in this for the Bias in the Factor N study (see Equation 29 and Table 26). The bias of the term N The effect of the bias in N established in Section 4.4.2 is can be defined as the following: examined in this section by comparing the bias of the calcu- lated bearing capacity under different loading conditions to the NExp qu ( 0.5 Bs ) bias established for N. Figures 94 to 98 describe the bias of the N = = (119) NVesic 2 ( N q + 1) tan f calculated bearing capacity for soil friction angles between 42.5 and 46.0 (for which Equation 120 is valid) for different One hundred and twenty five relevant cases were investigated loading conditions. For the case of vertical-centric loading in which the foundation was tested on the ground surface, and (Figure 94), the bias of the bearing capacity calculation over- the groundwater was below the zone of the foundation influ- laps that of N , suggesting that the bias observed for the inves- ence. Figure 93 presents the scatter and exponential fit of the tigated cases can be mostly attributed to the bias in N. This 3 3 Data Bearing Capacity bias (n = 131) load test data; n = 125 Bearing Capacity (BC) bias, = exp(0.205 f 8.655) (R2 = 0.351) 2.5 2.5 N bias, = [qu / (0.5 B s )] / N Vesic 2 2 Bias 1.5 1.5 1 1 N 0.5 0.5 0 0 42 43 44 45 46 43 44 45 46 Friction Angle, f (deg) Friction Angle, f (deg) Figure 93. The ratio (N) of the back-calculated Figure 94. The ratio between measured and calculated bearing capacity factor N (based on experimental bearing capacity (bias ) compared to the bias in the data) and the bearing capacity factor proposed by bearing capacity factor N (N) versus the soil friction ' (1973) versus soil friction angle. Vesic angle for footings under vertical-centric loadings.

OCR for page 107
109 3 5 Data Bearing Capacity bias (n = 26) Bearing Capacity (BC) bias, 4.5 2.5 N bias, N 4 Data Bearing Capacity bias (n = 8) 3.5 Bearing Capacity (BC) bias, 2 bias, 3 Bias, Bias, 1.5 2.5 2 1 1.5 1 0.5 0.5 0 0 43 44 45 46 44 44.5 45 45.5 46 Friction Angle, f (deg) Friction Angle, f (deg) Figure 95. The ratio between measured and calculated Figure 97. The ratio between measured and calculated bearing capacity (bias ) compared to the bias in the bearing capacity (bias ) compared to the bias in the bearing capacity factor N (N) versus the soil friction bearing capacity factor N (N) versus the soil friction angle for footings under vertical-eccentric loadings. angle for footings under inclined-eccentric, positive moment loadings. conclusion is subjected, however, to the fact that most of the cases are related to surface loading, hence, used for establish- involved in inclined-eccentric loading (Figures 97 and 98) have ing the bias in N. For the cases related to vertical-eccentric and a small number of data cases and the bearing capacity is highly inclined-centric loading (Figures 95 and 96), the data suggests sensitive to the loading conditions. Overall, the data presented that the trends are similar, and, hence, the bias in N may be in Figures 94 to 98 suggest that the bias in the bearing capacity a significant contributor to the bias in the bearing capacity factor N is a major contributor to the uncertainties in the bear- calculations. The biases do not overlap because the cases ing capacity estimation regardless of the load combinations involved in eccentric and inclined loading are highly sensitive acting on the footing. to many other factors that affect the bearing capacity. The cases 8 3 Data Bearing Capacity bias (n = 29) 7 Bearing Capacity (BC) bias, Data Bearing Capacity bias (n = 7) 2.5 6 Bearing Capacity (BC) bias, N bias, bias, 5 2 Bias, 4 Bias, 1.5 3 1 2 1 0.5 0 0 44 44.5 45 45.5 46 43 44 45 46 Friction Angle, f (deg) Friction Angle, f (deg) Figure 98. The ratio between measured and calculated Figure 96. The ratio between measured and calculated bearing capacity (bias ) compared to the bias in the bearing capacity (bias ) compared to the bias in the bearing capacity factor N (N) versus the soil friction bearing capacity factor N (N) versus the soil friction angle for footings under inclined-eccentric, negative angle for footings under inclined-centric loadings. moment loadings.