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OCR for page 82
82 smaller) than that obtained using the full geometrical width an eccentric vertical load with "negative" eccentricity. The of the foundation. The ramifications of these findings are resultant moment, which is negative in case of loading eccen- relevant to design practices in which the loading details are not tricity along footing width b3 (a) and positive in case of loading known at the time of the design. This issue was touched upon eccentricity along footing length b2 (b)(refer to Figure 69 for in Section 3.1.7 and will be further discussed in Chapter 4. sign conventions), then acts in the opposite direction to the The change in variability between the two cases as well as the horizontal load. The induced rotations counteract the dis- mean bias are greatly affected by a few outliers and will be placements forced by the horizontal load, leading to a higher further discussed in Chapter 4. The effects of the moment resistance of the footing compared with the inclined-centric direction (or load eccentricity) with respect to the horizontal load case and, thus, to higher failure loads. In contrast, the load, noted in Tables 35 and 36 as positive and negative footing in the lower part of Figure 69 is loaded by an eccentric moments for tests conducted by Perau (1995), are discussed vertical load with "positive" eccentricity. This leads to a pos- in the following sections. itive moment in the case of loading eccentricity along footing width b3 (a), and a negative moment in the case of loading eccentricity along footing length b2 (b), which acts in the same 3.7 Loading Direction Effect direction as the horizontal load. The induced rotations enforce for Inclined-Eccentric Loading the horizontal displacements; hence, the footing resistance is The loading direction in the case of inclined-eccentric load- smaller than in the case of inclined-centric loading, leading to ing affects the failure loads. Figure 69 presents the definitions smaller failure loads. established for the loading direction along the footing width In a different approach, when the moment is in the "opposite" (a) and along the footing length (b) (see also Butterfield et al., direction, it induces higher contact stresses between the foun- 1996) depending on the eccentricity direction in relation to the dation and the soil in the "front" of the foundation where the direction of the applied lateral load. The footing in the upper lateral load is applied. As the foundation-soil friction is pro- part of Figure 69 (a) and (b) is loaded by a horizontal load and gressive, the higher contact stress results in a higher friction resistance and, hence, the overall layer capacity. In contrast, when the moment acts in the "same" direction, the contact e3 M2 stress at the "front" of the footing decreases, thereby reducing the friction and resulting in a decrease in the total foundation F1 F1 resistance (bearing capacity). The effect of the loading direction F3 F3 expressed in Tables 35 and 36 is demonstrated in a graphical b3 b3 format in Figures 70 and 71. Figures 70 and 71 present a his- Moment acting in direction opposite to the lateral loading negative eccentricity togram and PDF of the bias as well as the relationship between e3 M2 measured and calculated bearing capacity for inclined-eccentric loading under positive and negative moments, respectively. A F1 F1 comparison of Figures 70 and 71 shows an increase of the bias F3 F3 for the negative moment cases. b3 b3 The effect of loading direction is further demonstrated by the Moment acting in the same direction as the lateral loading positive eccentricity results of two tests carried out by Gottardi (1992) and shown e3 M2 in Figure 72. The failure loads in the case of loading in the same direction (positive loading eccentricity) are significantly F1 F1 smaller than the failure loads in the case of opposite loading F3 F3 direction (negative loading direction). The influence on the b3 b3 bias is substantial--0.37 versus 0.64 for the two-slope criterion Moment acting in direction opposite to the lateral loading negative eccentricity and 0.37 versus 0.66 for the minimum slope criterion. Hence, e3 M2 it appears that this difference cannot be neglected and needs to be considered. F1 F1 Figure 73 shows the load-displacement curves for two F3 F3 double tests (positive and negative loading eccentricity) con- b3 b3 ducted by Perau (1995) and one double test by Montrasio Moment acting in the same direction as the lateral loading positive eccentricity (1994), applying different loading directions at the same level Figure 69. Loading directions for the case of of vertical loading. The results of Perau's and Montrasio's tests inclined-eccentric loadings: (a) along footing show a similar trend. Montrasio's test leads to a bias of 1.86 width and (b) along footing length. versus 1.97 (positive versus negative loading eccentricity),

OCR for page 82
83 10 0.4 Inclined-eccentric loading using Minimum Slope criterion (Vesic, 1963) 3 Positive eccentricity n=8 Interpreted bearing capacity, qu,meas mean = 2.16 COV = 1.092 0.3 Number of observations 2 Frequency (ksf) 1 0.2 lognormal distribution 1 normal Inclined-eccentric loading distribution Positive eccentricity 0.1 Data (n = 8) Data best fit line No bias line 0.1 0 0 0.1 1 10 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 Calculated bearing capacity, qu,calc Bias, = qu,meas / qu,calc (Vesic, 1975 and modified AASHTO) (a) (ksf) (b) Figure 70. (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 under positive moment. 10 0.5 Inclined-eccentric loading using Minimum Slope criterion (Vesic, 1963) Negative eccentricity 3 n=7 Interpreted bearing capacity, qu,meas mean = 3.43 0.4 COV = 0.523 Number of observations 0.3 Frequency (ksf) 2 1 0.2 lognormal distribution Inclined-eccentric loading 1 normal Negative eccentricity distribution Data (n = 7) 0.1 Data best fit line No bias line 0.1 0 0 0.1 1 10 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 71. (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 under negative moment.

OCR for page 82
84 F3 [kN] [kip] 1.6 0.3 1.2 0.2 0.8 0.1 0.4 u3 [in] 0.2 0.15 0.1 0.05 0 1 2 3 F1 [kip] 0 0 u3 [mm] 6 4 2 0 4 8 12 F1 [kN] 2 0.1 GoE6.1, e=-0.0167 GoE6.2, e=+0.0167 4 0.2 6 0.3 8 u1 [mm] [in] Figure 72. Loaddisplacement curves for inclined-eccentric loading with different loading directions utilizing data from Gottardi (1992). indicating a minor effect of the loading direction. However, this adopted in order to differentiate the failure load of vertical- effect is more significant in Perau's tests, where the evaluation centric loading from the vertical component F1 of the inclined of the failure loads leads to a mean bias of 1.79 (COV 0.206) failure loads (refer to Figure 65 and Section 3.6.2). In this for a horizontal load and moment acting in the same direc- context, small load inclinations coincide with relatively high tion (positive loading eccentricity) and 2.76 (COV 0.152) for vertical load levels. Figure 74 shows an evaluation of the bear- a moment in an opposite loading direction (negative loading ing capacity in the F2/F10 - M3/(F10 b2) plane performed by eccentricity). Lesny (2001) using Perau's (1995) test results. In reference to In general, it can be stated that the effect of the loading Figure 64, F2 is the horizontal component of the inclined load direction is less pronounced if the vertical load (F1) is relatively and b2 is the footing length in the same direction. Different high (i.e., the load inclination is relatively small) because this loading directions and different load levels have been ana- effect is predominantly determined by the load inclination lyzed in Figure 74, resulting in distorted trend lines due to the and not by the load eccentricity. The level of the vertical load existence of a higher capacity if horizontal load and moment (F1) can properly be expressed by relating it to the failure load act in the opposite direction (i.e., both load components are for centric vertical loading (F10). The notation F10 has been positive and the loading eccentricity is negative). However,