Geotechnical Engineering Circular No. 9 Design, Analysis, and Testing of Laterally Loaded Deep Foundations that Support Transportation Facilities
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- Figure 10-2: Slope divided into n slices (after Liang and Zeng 2002). Figure 10-3: Forces acting on slice (Liang and Zeng 2002).
| 10.3.3 Liang and Zeng (2002) Method In the Liang and Zeng (2002) method, the slope is divided into n slices, as shown by Figure 10-2, to facilitate the computation of internal forces and the calculation of the factor of safety, which at 1.0 is the same as the geotechnical resistance factor. The resultant interslice force is assumed to be parallel to the inclination of the base of the previous slice with respect to the direction of slope movement, as shown by the method of slices in Figure 10-3. Development of a permanent arching effect is essential to this method, which maybe inhibited by liquefiable sand that can flow around the drilled shafts, very soft soil that can squeeze through the shaft spacing, or shafts that are too widely spaced to develop arching. As the program is based upon a dimensional finite element method (FEM) model, limitations include 3- dimensional effects and the uncertain validity of extrapolation to model shaft diameters larger than about feet or cohesion values in excess of about 6 psi. 134 Figure 10-2: Slope divided into n slices (after Liang and Zeng 2002). Figure 10-3: Forces acting on slice (Liang and Zeng 2002). Dividing the slipping mass into n slices (method of slices) and applying force equilibrium results to each slice i, results in the following relationship ππ ππ = sin πΌπΌ ππ β 1 πΉπΉππ [πΆπΆβ² πΌπΌ ππ ππ + (cos πΌπΌ ππ β π’π’ ππ ππ ππ ) tan β ππ ] + Equation 10-1) 135 Where i = 1, 2, β¦, n W i = Weight of slice i. Ξ± i = Inclination with respect to the horizontal of the base of each slice. FS = Factor of safety. C i βI i = Cohesion intercept at the slip surface. l i = Length of the base of the slice. u i = Pore pressure at the base of the slice. Ο = Friction angle of the soil at the slip surface. K i = Coefficient. R = Reduction factor. P i I i = Interslice force. K i is obtained as follows πΎπΎ ππ = cos πΌπΌ ππ ) β sin tan Equation 10-2) The reduction factor R is a factor that considers the soil arching effect. When the shaft spacing (s) is comparable to the shaft diameter (d), i.e., the extreme case where s/d = 1, R = 1, the arch effect is largest, and the entire driving force of the slipping mass is transmitted to the shafts. Conversely, when s is much larger than d (i.e., s/d >> 1), the arch effect is negligible. At this point, R approaches R p , defined as the percent of residual soil pressure acting on the soil mass between shafts. In the extreme case that s/d β β, R = R p . A general expression can be used to obtain R: πΉπΉ = 1 π π ππ οΏ½ + οΏ½1 β 1 π π ππ οΏ½ οΏ½ Equation 10-3) Values of R p were obtained in a D finite element method (FEM) parametric study (Liang and Zeng 2002), included in Table 10-1 (a) through 10-1 (c) for shaft diameters of 1, 2, and 3 feet, respectively. Equations 10-1, 10-2, and 10-3 are applied to each slice, resulting in a recursive formula for determining P i with initial value P o . First, an initial value of FS is assumed. For example, FS = 1/0.75 = 1.33 or FS= 1/0.65 = 1.53 for LRFD resistance factors Ο = 0.75 or 0.65 per AASHTO (2014) 11.6.2.3. The FS value is used in the iterative formula. If Equation 10-1 results in tension for P i (i.e., P i < 0) at any computational step, P i should beset to zero in the next step to calculate P i +1 because stability should not rely on the typically small tension resistance of soils. Figure 10-4 shows that two cases of possible pressure distributions that need to be evaluated, because of the possible relative movement between the shaft and soil above the shear surface. In Case I, the earth pressure down-slope of the shaft is equal to the residual value, i.e., the upslope interslice force (Pi, which is reduced for the presence of the shaft per Equation 10-1, multiplied by the percent of residual soil pressure (Rp) occurring between the shafts per Table a) through c Case II arises when the earth pressure down-slope of the shaft is equal to the at-rest value. Equations 10-4 and 10-5 are used to calculate the net shaft force for Cases I and II, respectively. |