Numerical examples and verification

Based on the present formulation and the formulation by Huang and Tsai (2000), Cheng et al. (2005) and Cheng and Yip (2007) have developed the program SLOPE3D and several examples are used for the study of the proposed formulations. In this chapter, function f(x, y) is taken to be 1.0 for the Morgenstern–Price analysis and the method is hence actually Spencer analysis.

Cheng and Yip (2007) have tried f(x, y) for limited cases and the results from the use of f(x) = 1.0 and f(x, y) = sin(x, y) are virtually the same which is similar to the corresponding 2D analysis. It is expected that, for a highly irregular failure surface, the results may be sensitive to the choice of f(x, y). The first example is a laterally symmetric slope (Figure 5.8) considered by Baligh and Azzouz (1975) with an assumed spherical sliding surface, and the results of analysis are shown in Table 5.2. For Example 1, SLOPE3D gives a sliding angle exactly equal to 0 and the results are also very close to those by other researchers except the one by Hungr (1987). Cheng and Yip (2007) view that the result by Hungr (1987), 1.422, is actually not correct as he obtained this result based on eq. (5.27):

(5.27) F=

PðWitan φ_i+ CiAicos g_zÞ=ma

PWisin ay

Figure 5.8 Slope geometry for Example 1.

c = 0.1 kPa, φ = 0 γ = 1 kN/m³

Moment Arm, R = 1m

0.5R R

x y z Gradient 2H : 1V

(x0, y0, z0)

Table 5.1 Summary of some 3D limit equilibrium methods

Method Related 2D method Assumptions Equilibrium Hovland (1977) Ordinary method No inter-column Overall moment

of slice force equilibrium

Chen and Spencer method Constant Overall moment

Chameau inclination equilibrium

(1982) Overall force

equilibrium Hungr Bishop simplified Vertical Overall moment

(1987) method equilibrium equilibrium

Vertical force equilibrium Lam and General limit Inter-column Overall moment

Fredlund equilibrium force function equilibrium

(1993) Overall force

equilibrium Huang and Bishop simplified Consider Overall moment

Tsai (2000) method direction equilibrium

of slide Vertical force equilibrium Cheng and Bishop and Janbu Consider Overall force

Yip (2007) simplified, direction equilibrium, and Morgenstern–Price of slide overall moment

equilibrium for Morgenstern–Price

Equation (5.27) is not correct because moment equilibrium is considered about the centre of rotation of the spherical failure surface. Moment is a vector and should be defined about an axis instead of a point so that the moment contribution from each section cannot be added directly as in eq.

(5.27). To correct eq. (5.27), the moment equilibrium should be considered

about an axis passing through the centre of rotation and the smaller radius at each section should be adopted and is given by the following:

(5.28) where and R is the radius of the spherical failure mass. In eq. (5.28), the radius at each section, r_i, is smaller than the global radius of rotation R and cannot be cancelled out because r_iis changing at different sections. Cheng and Yip (2007) have tried eq. (5.27) and have obtained the value 1.42 (same as Hungr), whereas an answer of 1.39 is obtained by eq.

(5.28) for Example 1.

ri= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R²− y² p F=

PðWitan φ_i+ CiAicos g_zÞri=m_a PWirisin ay ,

Table 5.2 Comparison of Fsfor Example 1

Method Baligh and Hungr Lam and Huang SLOPE3D SLOPE3D Azzouz et al. Fredlund and Tsai (Huang’s (Cheng’s (1975) (1989) (1993) (2000) approach) approach)

Bishop 1.402 1.422 1.386 1.399 1.390 1.390

simplified (1.39) (1200c) (5300c) (8720c) (8720c)

Janbu — — — — 1.612 1.612

simplified (8720c) (8720c)

Note: Number of columns in the analyses. For Hungr’s result, the factor of safety after correction is 1.39.

Example 2 (Huang and Tsai, 2000) is a vertical cut slope (Figure 5.9) with an assumed spherical sliding surface. The failure mass is symmetrical about an axis inclined at 45° to the x axis, and this result is predicted with both Huang and Tsai’s formulation and the present formulation. The results by the present formulations agree well with the results by Huang and Tsai, which have demonstrated that the new formulation gives results close to those from Huang and Tsai’s formulation.

Example 3 is a vertical cut slope (Figure 5.10) in which a wedge-like failure is considered in the analysis. The FOS for this rigid block failure is determined explicitly from the simple rigid block failure as 0.726. Similar results are deter-mined by the present 3D Janbu and Morgenstern–Price analyses (Table 5.3).

This has demonstrated that if the correct failure mode is adopted, the present formulation can give reasonable results for 3D analysis. For 3D Bishop analy-sis, the FOS based on the moment point (x₀, y₀, z₀) is not correct as the pres-ent failure mode is a sliding failure while the Bishop method does not fulfil horizontal force equilibrium. The value of 0.62 from the Bishop method should not be adopted because the Bishop method does not satisfy horizontal force equilibrium while the wedge actually fails by sliding. If any moment point is chosen for the Bishop analysis, the FOS will be different and this is a

well-known problem for the Bishop method. For the determination of the 3D FOS, the failure mechanism should be considered in the selection of a suitable method of analysis. Example 4 is an asymmetric rigid block failure with a size 2 m×4 m, shown in Figure 5.11. The FOS and the sliding with respect to the x axis can be determined explicitly as 0.2795 and 63.4°, respectively, which are also determined exactly from SLOPE3D with the 3D Janbu and Morgenstern–

Price analyses. The factors of safety for Examples 3 and 4 are correctly pre-dicted by the present formulation that is a support to the present formulation.

For the FOS of a simple slope where c′= 0, F_sis given as tanφ′/tanθ, where θis the slope angle. The critical failure surface is a planar surface parallel to the ground surface. From a spherical/elliptical optimization search, the mini-mum factors of safety for a simple slope from the present formulations are

R

R R

x z y

Sliding Direction on Plan

(x0, y0, z0)

c' = 5 kPa φ' = 30°

γ = 20 kN/m³

Calculated Dip Angle = 54.736°

R = 5m

Figure 5.10 Slope geometry for Example 3.

Figure 5.9 Slope geometry for Example 2.

c' = 24.5 kPa φ' = 20°

γ = 17.64 kN/m³ Moment Arm, R = H

R

x z y A-A axis

Slope Height, H = 5m (x0, y0, z0)

equal to tanφ/tanθ for the Bishop, Janbu and Morgenstern–Price analyses, and the results comply well with the requirement of basic soil mechanics.

The results in Examples 1 to 4 show that the present theory gives sliding directions similar to those computed by Huang and Tsai’s (2000) method.

Cheng and Yip (2007) have also tried many other examples, and the dif-ferences between the FOS and sliding direction from Huang and Tsai’s (2000) formulation and the present formulation are small in general, if there is no transverse load.

5.2.4 Comparison between Huang and Tsai’s method and the present