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76 when analyzing data for natural soil deposits result in layer This means that the load inclination was no longer constant variation (as expressed by the COV) and reduction in the mean during the test but varied from zero up to the maximum bias. Further investigation as to the source of the obtained load inclination at failure, ult. The step-like load paths were bias is presented in Section 4.4. applied in tests under inclined-centric and inclined-eccentric loadings only. 3.6 Uncertainty in the Bearing 3.6.2 Determination of the Measured Capacity of Footings in/on Strength Limit State for Foundations Granular Soils Subjected Under Inclined Loading to Vertical-Eccentric, Inclined-Centric, and The procedure to determine the failure loads from the Inclined-Eccentric Loading model tests depends on the load paths applied in the tests. The analysis shows that in the case of a test with a radial load 3.6.1 Scope and Loading Procedures path it is sufficient to consider only the vertical load versus of the Case Histories vertical displacement curve. This curve already includes the The analysis of failure under vertical-eccentric, inclined- unfavorable effect that a horizontal load or a bending moment centric, and inclined-eccentric loading is based on test results has on the bearing capacity of a shallow foundation, leading from DEGEBO, Perau (1995), Montrasio (1994), and Gottardi to smaller vertical failure loads compared to the case of centric (1992). The test conditions of the various data sources are vertical loading. summarized in Table 31. The following analysis is based on Figure 65 provides an example using test results with inclined the loading convention shown in Figure 64(a). loading performed by Montrasio (1994) under different load The application of loadings in the tests varied. In the tests inclination angles. Both vertical load/vertical displacement with radial load paths, both the vertical and the horizontal and horizontal load/horizontal displacement curves are shown loads were increased up to failure, maintaining a constant for each test with inclined load. The load displacement relation- ratio of F3/F1 during the test, i.e., the load inclination () was ship in Figure 65 indicates that the vertical failure load, F1,ult, constant (see Figure 64(b)). The same applies to the tests decreases with the increase of the load inclination. with eccentric loading; the eccentricity, e = M2/F1, was main- Applying the minimum slope criterion to the centric tained constant during the test, because the vertical load was vertical load test results ( = 0, MoA2.1) provides the fail- applied eccentrically at one location. On the other hand, in the ure load F10,ult = 0.956 kip (4.25kN). The failure loads for tests with step-like load paths, the vertical load was increased the tests with inclined loading decrease to F1,ult = 0.738kip up to a certain level and then kept constant while the hori- (3.28kN) for a load inclination angle of = 3 (MoD2.1) zontal load was increased up to failure (see Figure 64(c)). and F1,ult = 0.677 kip (3.01kN) for = 8 (MoD2.2) and further Table 31. Test data used for failure analysis. Footing size Source Soil conditions Footing base Loading1 Load application1 ft2 (m) Fine to medium Eccentric radial load path sand, loose to 1.6 6.6 (0.5 2.0) Inclined radial load path medium dense, 3.3 3.3 (1.0 1.0) medium rough DEGEBO dense; 3.3 9.8 (1.0 3.0) (prefabricated) Inclined- radial load path gravel, medium 2.0 6.9 (0.6 2.09) eccentric dense, dense Eccentric radial load path Medium to coarse rough (base Perau 0.3 0.3 (0.09 0.09) Inclined step-like load path sand, dense to very glued with (1995) 0.2 0.2 (0.05 0.15) Inclined- F1-M2: radial load path dense sand) eccentric F1-F3: step-like load path Eccentric radial load path Medium to coarse 0.3 0.3 (0.08 0.08) rough (base Montrasio Inclined radial load path sand (Ticino Sand), 0.5 0.3 (0.16 0.08) glued with (1994) Inclined- F1-F3: step-like load path dense 0.8 0.3 (0.24 0.08) sand) eccentric F3-M2: radial load path Eccentric radial load path Medium to coarse rough (base Inclined radial or step-like load path Gottardi sand (Adige Sand), 1.6 0.3 (0.5 0.1) glued with F1-M2: radial load path (1992) Inclined- dense sand) F1-F3: radial or step-like eccentric load path 1 See Figure 64 for details

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77 x2 F1 D M2 M1 F3 F2 M3 b2 x3 b3 g , f x1 (a) Loading convention F1 F1 F1 F1,const. = constant arctan e = constant increasing F3 M2 F3 (b) Radial load path (c) Step-like load path Figure 64. Loading convention and load paths used during tests. Vertical load, F1, F10 (kN) Horizontal load, F3 (kN) 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2 0 0 0 0 MoD2.1 = 3 Horizontal displacement, u3 (mm) Vertical displacement, u1 (mm) MoD2.2 = 8 2 MoD2.3 = 14 0.1 2 0.1 u1 (in) u3 (in) 4 0.2 MoD2.1 = 3 4 6 MoD2.2 = 8 MoD2.3 = 14 0.2 MoA2.1 = 0(F10) 0.3 8 6 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 F1 (kips) F3 (kips) (a) (b) Figure 65. Loaddisplacement curves for model tests conducted by Montrasio (1994) with varying load inclination: (a) vertical load versus vertical displacement and (b) horizontal load versus horizontal displacement.

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78 decreases to F1,ult = 0.425kip (1.89kN) when the load inclination could always be used and therefore seemed to have a distinct increases to = 14 (MoD2.3). Consequently, the correspond- advantage. ing horizontal component of the failure load, F3,ult, increases with the increase in the load inclination. Overall, the horizontal 3.6.3 Summary of Mean Bias Statistics loads are significantly smaller than the vertical failure loads for Vertical-Eccentric Loading due to limited soil-foundation frictional resistance. This pro- cedure results in vertical failure loads (F1,ult) that can be directly Table 32 presents a summary of the statistics of the bias for related to the theoretical failure loads determined by the calcu- the footings under vertical-eccentric loading. Section G.2 in lation model for the relevant load inclination, hence making it Appendix G presents the details of the bias calculation for a possible to obtain the bias of the model for the bearing capacity single relevant case history (ID #471) of database UML-GTR of foundations under inclined loads. ShalFound07. The total number of cases under vertical- In the case of a step-like load path, a different procedure has eccentric loading from all sources was 43, including all outliers to be applied. In these tests, the vertical load was kept constant to be addressed in Chapter 4. Seventeen cases from DEGEBO, up to failure, hence the vertical load/vertical displacement 14 cases from Montrasio (1994) and Gottardi (1992) and curves are not meaningful. The failure is analyzed on the basis 12 cases from Perau (1995) could be analyzed. Figure 66 pres- of the horizontal load/horizontal displacement curves result- ents a histogram and a PDF of the bias as well as the relation- ing in horizontal failure loads, F3,ult. The vertical failure loads, ship between measured and calculated bearing capacities for F1,ult, are the ones corresponding to the horizontal failure all vertical, eccentrically loaded foundation cases summarized loads, F3,ult, and coincide with the constant vertical load in in Table 32. DEGEBO results show the highest mean and COV each test. As the load inclination is increased during the test, of the bias when using any of the failure criteria. Table 33 the maximum load inclination reached is the load inclination summarizes the statistics of the bias associated with bearing at failure, tan ult = F3,ult/F1,ult. The theoretical (vertical) failure capacity calculations when using the full geometrical size of load is then calculated for the load inclination at failure, ult, the foundation width (B). Table 33 was added in order to gain and compared to the measured vertical failure load, F1,ult, to perspective on the bias in cases where the influence of the determine the bias. Additionally, the theoretical horizontal effective width is neglected. failure loads are calculated using the respective load inclina- Comparing Tables 32 and 33, it can be seen that the mean tion at failure and the theoretical vertical failure loads. It can bias of the ultimate strength estimation decreases and the COV be shown that the resulting biases of the horizontal failure of the bias increases when full footing geometry (B) is used loads coincide with the biases of the vertical failure loads and instead of the effective footing dimensions (B). This is an confirm this procedure. expected outcome considering the larger B would result in In both procedures, the minimum slope criterion and the a higher bearing capacity (and hence decreased bias) while two-slope criterion were examined for the failure load inter- the methodology is incorrect, contributing to the increased pretation. In most cases, the results were found to be com- uncertainty (being represented by the COV). The decreased parable. However, in some cases, the two-slope criterion was bias and increased COV would necessitate a significant increase not applicable (FOTIDs #251 and #266, DEGEBO tests on in the resistance to ensure a specified safety, i.e., utilizing eccentric loading, FOTIDs #301 and #317, and DEGEBO lower resistance factors. For example, considering all cases, tests on inclined loading) while the minimum slope criterion the resistance factor obtained is 0.60 when B is used and 0.30 Table 32. Summary of the statistics for biases of the test results for vertical-eccentric loading when using effective foundation width (B). No. of Minimum slope criterion Two-slope criterion Tests cases Mean Std. dev. COV Mean Std. dev. COV DEGEBO radial 17 2.22 0.754 0.340 2.04 0.668 0.328 load path (15)1 Montrasio (1994)/Gottardi (1992) 14 1.71 0.399 0.234 1.52 0.478 0.313 radial load path Perau (1995) radial 12 1.43 0.337 0.263 1.19 0.470 0.396 load path 43 All cases 1.83 0.644 0.351 1.61 0.645 0.400 (41)1 1 Number of cases for two-slope criterion

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79 1000 12 Vertical-eccentric loading Vertical-eccentric loading using Minimum Slope criterion (Vesic, 1963) Data (n = 43) n = 43 0.25 mean = 1.83 Data best fit line Interpreted bearing capacity, qu,meas 10 COV = 0.351 100 No bias line 0.2 Number of observations 8 lognormal distribution Frequency (ksf) 0.15 10 6 normal distribution 0.1 4 1 2 0.05 0.1 0 0 0.1 1 10 100 0.4 1.2 2 2.8 3.6 Calculated bearing capacity, qu,calc Bias, = qu,meas / qu,calc (Vesic, 1975 and modified AASHTO) (a) (ksf) (b) Figure 66. (a) Histogram and probability density function of the bias and (b) relationship between measured and calculated bearing capacity for all vertical, eccentrically loaded shallow foundations. when B is used. Thus, Tables 32 and 33 indicate that the bear- presents a histogram and PDF of the bias as well as the relation- ing capacity obtained using the full footing width (B) is unsafe ship between measured and calculated bearing capacity for all when compared to the bearing capacity obtained when using inclined, centrically loaded shallow foundations. the effective width (B). There are no differences in the biases obtained from the two-slope and the minimum slope failure criteria for the cases of step-like load paths. Gottardi's tests with radial load paths 3.6.4 Summary of Mean Bias Statistics sometimes seem to result in smaller biases than the other tests, for Inclined-Centric Loading but overall, no significant differences exist in the biases of the The mean and standard deviation of the calculated biases in step-like and radial load path tests. The biases determined for the case of inclined loading are summarized in Table 34 for the the DEGEBO tests are also in the same order of magnitude two failure criteria. Section G.3 of Appendix G presents the as the ones from the small-scale model tests although they details of the bias calculations for a single relevant case history were carried out on foundations significantly larger in size. (ID #547) of database UML-GTR ShalFound07. Figure 67 DEGEBO tests were carried out on foundations of 1.6 ft 3.3 ft Table 33. Summary of the statistics for biases of the test results for vertical-eccentric loading when using the full foundation width (B). No. of Minimum slope criterion Two-slope criterion Tests cases Mean Std. dev. COV Mean Std. dev. COV DEGEBO radial 17 1.30 0.464 0.358 1.20 0.425 0.355 load path (15)1 Montrasio (1994)/Gottardi (1992) 14 0.97 0.369 0.380 0.86 0.339 0.396 radial load path Perau (1995) radial 12 0.79 0.302 0.383 0.64 0.296 0.465 load path 43 All cases 1.05 0.441 0.420 0.92 0.423 0.461 (41)1 1 Number of cases for two-slope criterion

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80 Table 34. Summary of the statistics for biases of the test results for inclined-centric loading when using foundation width (B). No. of Minimum slope criterion Two-slope criterion Tests cases Mean Std. dev. COV Mean Std. dev. COV DEGEBO/ Montrasio 26 1.56 0.346 0.222 1.35 0.452 0.334 (1994)/Gottardi (1992) (24)1 radial load path Perau (1995)/Gottardi (1992) step-like load 13 1.17 0.537 0.459 1.17 0.537 0.459 path 39 All cases 1.43 0.422 0.295 1.29 0.455 0.353 (37)1 1 Number of cases for two-slope criterion (0.5 m 1.0 m) to 3.3 ft 9.8 ft (1 m 3 m) versus the small scale shallow foundation cases. As in the inclined-centric loading models having foundation sizes of 2 in. 6 in. (5 cm 15 cm) cases, there is no significant difference in the tests results to 4 in. 20 in. (10 cm 50 cm). between the radial and the step-like load paths. The bearing capacity calculations of these case histories were noticeably affected by using the effective foundation width (B) versus 3.6.5 Summary of Mean Bias Statistics the geometrical actual foundation width (B). Table 36 sum- for Inclined-Eccentric Loading marizes the statistics associated with the bearing capacity Table 35 presents a summary of the statistics of the bias for calculations using the full geometrical foundation width (B) footings subjected to inclined-eccentric loadings, with both in order to gain perspective on the bias in cases where the influ- radial and step-like load paths and including the effective ence of the effective width is neglected. The biases presented foundation width, B. Figure 68 presents a histogram and PDF in Table 36 indicate that for the examined case histories the of the bias as well as the relationship between measured and calculated bearing capacity using the effective width resulted in calculated bearing capacity for all inclined, eccentrically loaded a bias about two times larger (i.e., a bearing capacity two times 100 Inclined-centric loading Inclined-centric loading Data (n = 39) using Minimum Slope criterion (Vesic, 1963) 12 n = 39 0.3 Data best fit line mean = 1.43 Interpreted bearing capacity, qu,meas No bias line COV = 0.295 10 Number of observations 8 0.2 Frequency (ksf) lognormal normal distribution distribution 1 4 0.1 0.1 0 0 0.1 1 10 100 0.2 0.6 1 1.4 1.8 2.2 2.6 Calculated bearing capacity, qu,calc Bias, = qu,meas / qu,calc (Vesic, 1975 and modified AASHTO) (a) (ksf) (b) Figure 67. (a) Histogram and probability density function of the bias and (b) relationship between measured and calculated bearing capacity for all inclined, centrically loaded shallow foundations.

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Table 35. Summary of the statistics for biases of the test results for inclined-eccentric loading when using effective foundation width (B). No. of Minimum slope criterion Two-slope criterion Tests cases Mean Std. dev. COV Mean Std. dev. COV DEGEBO/Gottardi (1992) 8 2.06 0.813 0.394 1.78 0.552 0. 310 radial load path Montrasio (1994)/ 6 2.13 0.496 0.234 2.12 0.495 0.233 Gottardi (1992) Perau (1995) positive 8 2.16 1.092 0.506 2.15 1.073 0.500 Step-like eccentricity load path Perau (1995) negative 7 3.43 1.792 0.523 3.39 1.739 0.513 eccentricity All step-like load 21 2.57 1.352 0.526 2.56 1.319 0.516 cases All cases 29 2.43 1.234 0.508 2.34 1.201 0.513 100 9 Inclined-eccentric loading 0.3 using Minimum Slope criterion (Vesic, 1963) n = 29 8 Interpreted bearing capacity, qu,meas mean = 2.43 COV = 0.508 0.25 7 10 Number of observations 6 0.2 Frequency (ksf) 5 0.15 4 lognormal 1 3 0.1 distribution Inclined-eccentric loading 2 normal Data (n = 29) distribution Data best fit line 0.05 1 No bias line 0.1 0 0 0.1 1 10 100 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 6 6.6 7.2 Calculated bearing capacity, qu,calc Bias, = qu,meas / qu,calc (Vesic, 1975 and modified AASHTO) (a) (ksf) (b) Figure 68. (a) Histogram and probability density function of the bias and (b) relationship between measured and calculated bearing capacity for all inclined, eccentrically loaded shallow foundations. Table 36. Summary of the statistics for biases of the test results for inclined-eccentric loading when using foundation width (B). No. of Minimum slope criterion Two-slope criterion Tests cases Mean Std. dev. COV Mean Std. dev. COV DEGEBO/Gottardi (1992) 8 1.07 0.448 0.417 0.94 0.365 0. 387 radial load path Montrasio (1994)/ 6 1.18 0.126 0.106 1.18 0.125 0.106 Gottardi (1992) Perau (1995) positive 8 0.70 0.136 0.194 0.70 0.135 0.194 Step-like eccentricity load path Perau (1995) negative 7 1.09 0.208 0.191 1.08 0.208 0.193 eccentricity All step-like load 21 0.97 0.267 0.276 0.96 0.267 0.277 cases All cases 29 1.00 0.322 0.323 0.96 0.290 0.303