Stability with FE Stresses

Transcription

1 1 Introduction Stability with FE Stresses In the finite element stresses stability analysis method, the stresses in the ground can be computed using SIGMA/W, and then SLOPE/W uses the SIGMA/W stresses to compute safety factors. This is a completely different approach than the limit equilibrium method. The purpose of this example is to highlight the use of the finite element stresses stability analysis method using the SIGMA/W and SLOPE/W. Features of this example include: Use of finite element stresses from SIGMA/W Use of pore water pressure computed from SIGMA/W Computation of stability factor from SLOPE/W Comparison with limit equilibrium method Multiple analyses 2 Configuration and setup A simple homogenous embankment under gravity loading is model, as shown in Figure 1. To establish the stresses in the ground using SIGMA/W, the Insitu analysis type is used and an initial water table is drawn to determine pore-water pressures within the ground. By defining an initial water table, both the total and effective stresses can be determined. The bottom boundary condition is pinned and rollers are used along the vertical extents of the finite element mesh. An unstructured mesh with quads and triangular elements is used in the analysis. Figure 1 SIGMA/W mesh used to determine stresses in the ground SLOPE/W Example File: Stability with FE stresses.docx (pdf) (gsz) Page 1 of 6

2 Table 1 Material property parameters used in the SIGMA/W analysis Parameter 3 SIGMA/W stresses Value Young s modulus, E (kpa) 100,000 Poisson s ratio, 0.4 Unit weight,, (kn/m 3 ) 20 Cohesion, c (kn/m 2 ) 8 Friction angle,, (degrees) 30 The total stress contours computed by the SIGMA/W analysis are shown in Figure 2. A Mohr circle showing the stress state is drawn for node 110 (Figure 3). Figure 2 Total stress contours within the embankment Figure 3 A Mohr circle of a Gauss region SLOPE/W Example File: Stability with FE stresses.docx (pdf) (gsz) Page 2 of 6

3 Figures 4,5 and 6 show the Y-total stress, pore-water pressure and Y-effective stress of the profile for the nodes at x = 10 m respectively. Note that in this example the pore-water pressure is assumed to vary hydrostatically below and above the water table with a maximum negative pressure head of 100m. Figure 4 Total stress profile for the nodes at x=10 m Figure 5 Pore-water pressure profile for the nodes at x=10 m For a level ground with a material unit weight of 20 kn/m 3, and an embankment height of 40 m, the maximum Y total stress at the bottom of the embankment is (20 x 40 = 800 kpa). For this example at x=10 m, the maximum Y total stress is computed to be 780 kpam as shown in Figure 4. The difference is due to the fact that the embankment is not level. The vertical distance of the water table to the bottom of the embankment at x=10 m is about 31.6 m, therefore the maximum pore water pressure should be about (31.6 x = 310 kpa), as shown in Figure 5. The maximum Y effective stress for the same profile should be ( =470 kpa), as shown in Figure 6. SLOPE/W Example File: Stability with FE stresses.docx (pdf) (gsz) Page 3 of 6

4 Figure 6 Effective stress profile for the nodes at x=10 m 4 SLOPE/W stability factor FE stress method Now that the actual stresses in the ground have been computed, the results of the SIGMA/W analysis can be used within SLOPE/W to determine a stability factor for the embankment. To do this you must have both SIGMA/W and SLOPE/W installed in your computer. You can add a new SLOPE/W analysis to the problem and specify in shear strength properties in SLOPE/W. In this example, the pore water pressure computed by SIGMA/W is also used. A single search center and a line of radii has been defined. During the solve process within SLOPE/W, the stability factor is determined by dividing the total available shear resistance(s r ) by the total mobilized shear (S m ) along the entire length of the slip surface as shown in the following equation: FS S S r m The critical slip surface and stability factor is shown in Figure 7 SLOPE/W Example File: Stability with FE stresses.docx (pdf) (gsz) Page 4 of 6

5 Figure 7 Stability factor and critical slip surface using the finite element stress method One of the primary assumptions with a limit equilibrium analysis is that the global factor of safety is the same as the factor of safety for every slice. In a limit equilibrium analysis, the factor of safety does not change with position across the slip surface. In a finite element stability analysis, the same condition does not need to be satisfied and the stability factor varies across the slip surface, as shown in Figure 8. Figure 8 Stability factor as it varies across the slip surface SLOPE/W Example File: Stability with FE stresses.docx (pdf) (gsz) Page 5 of 6

6 6 SLOPE/W factor of safety LE method For illustration purposes, the same example is also solved with the limit equilibrium approach. Figure 9 shows the factor of safety and the critical slip surface. In this example, the pore-water pressure computed from SIGMA/W is used, but the Morgenstern-Price method is selected. Note that the factor of safety is slightly lower than the finite element stress approach. Figure 9 Stability factor and critical slip surface using limit equilibrium method For more information on the differences between limit equilibrium and the finite element method, refer to the chapter on Factor of Safety Methods. SLOPE/W Example File: Stability with FE stresses.docx (pdf) (gsz) Page 6 of 6