Strength Reduction Stability

Transcription

1 1 Introduction Reduction Stability Stability by strength reduction is a procedure where the factor of safety is obtained by weakening the soil in steps in an elastic-plastic finite element analysis until the slope fails. The factor of safety is deemed to be the factor by which the soil strength needs to be reduced to reach failure (Dawson et al., 1999; Griffiths and Lane, 1999). Numerically, the failure occurs when it is no longer possible to obtain a converged solution. The finite element equations for a stress-strain formulation are in essence equations of equilibrium. Not being able to obtain a converged solution therefore infers the system is beyond the point of limiting equilibrium. An alternative way to define failure is the point at which the deformations become excessive. This is a much more subjective criterion than the non-non-convergence criterion put one which comes into play with the Reduction method. SIGMA/W can be used to do a -Reduction stability analysis by using the Stress Redistribution analysis option. The Stress Redistribution type of analysis in SIGMA/W, is a special algorithm for redistributing stresses due to perhaps some overstressing in some zones. A Linear-Elastic analysis may, for example, give some zones where the computed stresses are greater than the available shear strength. The Stress Redistribution option can be used to alter the stresses so that there is no overstressing. In the Reduction method, the soil strength is artificially reduced, and so there is a need to redistribute the stresses. This can be done by the Stress Redistribution algorithm, and so this option can be indirectly used to do a Reduction stability analysis. This example illustrates how a Reduction stability analysis can be done with SIGMA/W. In addition, the results are discussed in the context of an alternate and preferred procedure whereby the SIGMA/W results are used in conjunction with a SLOPE/W analysis to compute a safety factor. 2 Analysis procedure The analysis procedure is presented in Figure 1. The first step is to do an insitu analysis to establish the state of stress in a 2h:1v slope 1 m high, as shown in Figure 2. This step uses Linear-Elastic soil properties. The factor of safety is then computed by using the finite element results in a SLOPE/W analysis. The soil strength is then reduced in steps. For each strength case, the SLOPE/W factor of safety is computed. Each strength reduction analysis uses the previous or Parent analysis as its initial conditions. SIGMA/W Example File: reduction stability.doc (pdf )(gsz) Page 1 of 9

2 Figure 1 Analysis procedure m Elevation - m Figure 2 Slope configuration 3 reduction factor The strength reduction factor (SRF) is defined as, tanφ c SRF = = tanφ f c f Distance - m where Ø f and c f are the effective stress strength parameters at failure, or the reduced strength. The strength reduction approach generally uses the same SRF for all material and for all strength parameters, so that the stability factor reduces to one number in the end. This means c and Ø are reduced by the same factor. SIGMA/W Example File: reduction stability.doc (pdf )(gsz) Page 2 of 9

3 4 Conventional factor of safety The SIGMA/W computed stresses can be used by SLOPE/W to calculate a factor of safety. The resulting factor of safety based on the insitu stresses is shown in Figure m Figure 3 Factor of safety based on insitu stresses Figure 4 presents the distribution of the resistance and driving or mobilized shear along the slip surface. Note that the shear resistance or shear strength is locally greater than the mobilized shear along the entire slips surface. The overall factor of safety is the area under the resistance curve divided by the area under the mobilized shear curve. From the computed factor of safety the area under the resistance curve is times greater than the area under the mobilized shear curve Figure 4 strength and mobilized shear along the slip surface SIGMA/W Example File: reduction stability.doc (pdf )(gsz) Page 3 of 9

4 5 Reduced strengths The reduced strengths used here are as follows: Case Ø c SRF Base The following figures show the resistance and mobilized shear distributions for each of the five cases. Notice how the resisting shear and mobilized shear essentially become identical as the factor of safety migrates towards unity Figure 5 Case 1 resisting and mobilized shear distributions; F of S = 1.32 SIGMA/W Example File: reduction stability.doc (pdf )(gsz) Page 4 of 9

5 Figure 6 Case 2 resisting and mobilized shear distributions; F of S = Figure 7 Case 3 resisting and mobilized shear distributions; F of S = 1.98 SIGMA/W Example File: reduction stability.doc (pdf )(gsz) Page 5 of 9

6 Figure 8 Case 4 resisting and mobilized shear distributions; F of S = Figure 9 Case 5 resisting and mobilized shear distributions; F of S =.988 SIGMA/W Example File: reduction stability.doc (pdf )(gsz) Page 6 of 9

7 6 Factors of safety versus Reduction Factors The following table compares the SLOPE/W computed safety factors with the Reduction Factors (SRF). Ideally, when the SRF is equal to the original factor of safety, the computer safety factor should be 1.. The actual computed factor of safety is.988 which is remarkably close to the ideal 1. value. Case Ø c SRF F of S Base Use of the SRF method in isolation It can be rather arbitrary to determine the factor of safety of a slope when the Reduction method is used in isolation. In this case converged solutions can be obtained for SRF values greater than the original factor of safety. Also there is no distinct sharp break in the crest settlement curve as shown in Figure 1. The rate of settlement increases as the SRF increases but there is no distinct break to help with deciding on the point of failure. This illustrates the difficulty of using the SRF method in isolation. SIGMA/W Example File: reduction stability.doc (pdf )(gsz) Page 7 of 9

8 Crest settlement -.2 Y-Displacement (m) Time (sec) Figure 1 Crest settlements as the SRF increases 8 Concluding remarks This example demonstrates that SIGMA/W can be used to do a Reduction method of stability analysis. While it is possible to do a Reduction stability analysis with SIGMA/W, we recommend that you combine this with the SIGMA/W-SLOPE/W strength summation approach inherent in GeoStudio. Combining the results from the two methods greatly helps with understanding the stability analysis. Due to the numerical difficulties that can arise with the Reduction method, we recommend and prefer using the SIGMA/W computed stress in SLOPE/W to compute margins of safety. Even Linear- Elastic stresses, although not perfect, give acceptable factors of safety as point out by Krahn (3). Other independent studies such the one by Stinson, Chan and Fredlund (4) arrived at the same conclusion. Using Linear-Elastic analyses to establish the stress state is very appealing from a practical point of view, since there are no convergence issues. Also, in an integrated environment like GeoStudio it is easier and more reliable to use SIGMA/W together with SLOPE/W than to take the Reduction approach in isolation. SIGMA/W Example File: reduction stability.doc (pdf )(gsz) Page 8 of 9

9 9 References Dawson, E.M., Roth, W.H. and Drescher, A. (1999). Slope Stability Analysis by Reduction, Geotechnique, 49(6), Griffiths, D.V. and Lane, P.A. (1999). Slope Stability Analysis by Finite Elements, Geotechnique, 49(3), Krahn, John 3. The 1 R.M. Hardy Lecture: The Limits of Limit Equilibrium Analyses. Canadian Geotechnical Journal, Vol. 4, pp Krahn, John (7). Limit Equilibrium, Summation and Reduction Methods for Assessing Slope Stability. Proceeding, 1st Canada-U.S. Rock Mechanics Symposium, Vancouver, B.C. May Stianson, J. R., Chan, D. and Fredlund, D.G. (4). Comparing Slope Stability Analysis Based on Linear-Elastic or Elasto-Plastic Stresses using Dynamic Programming Techniques. Proceeding, 4 Canadian Geotechnical Conference, Quebec City, Quebec, Canada SIGMA/W Example File: reduction stability.doc (pdf )(gsz) Page 9 of 9