Critical acceleration versus static factor of safety in stability analysis of earth dams and embankments

Transcription

1 TECHNICAL NOTES Critical acceleration versus static factor of safety in stability analysis of earth dams and embankments S. K. SARMA, BTech, PhD, DIC* and M. V. BHAVE BSc, DIG* It has been shown (Sarma, 1973) that the computation of the critical acceleration for a given earth dam section is much simpler than the computation of the static factor of safety. Even though the term acceleration suggests some kind of dynamics, Sarma s method of solution is a static one. The critical acceleration factor K, is obtained as first part of the solution towards the static factor of safety and Sarma suggested the use of Kc as a measure of the static factor of safety. Certain advantages of Sarma s Kc method of solution over other factor of safety analysis methods are listed in the following text. The method of solution is rigorous but simple and easy to understand. It can be applied to slip surfaces of any shape. No iterative computations are necessary to obtain Kc, although these cannot be avoided if a factor of safety is needed. Whereas in other methods, the solutions may not converge to give a value of the factor of safety, the Kc method does not suffer from this problem. Within the bounds of the assumptions involved, the solution of K, is unique. The uniqueness of the solution of the factor of safety by other methods cannot be proved. In the process of obtaining the factor of safety by the Kc method, the values obtained in each of the iterations are useful whereas, in other methods, the intermediate results are useless. Since no iterations are necessary to obtain Kc, the time saved in computation could be enormous, irrespective of whether a desk calculator or computer is used. Over and above these computational advantages, there are also features which are physically more accurate and therefore can be considered as advantageous. In the Kc method, the curve for the interslice body forces is obtained by considering the stress conditions inside the mass; these are therefore not arbitrary. Although these contain certain assumptions, the assumptions can be modified when more information is obtained. It is known that the factor of safety on a slip surface is not uniform except when it is equal to unity (Wright et. ul., 1973). The value of Kc is obtained when the factor of safety is equal to one; this is therefore more accurate in the physical sense than a value of the factor of safety greater than one. It can therefore be said that the K, method of solution is both a physically accurate and computationally advantageous method. In the design of earth dams and embankments, a designer faces the following two problems. First, he has to find the most critical surface and, second, to find the value of the factor of safety for that surface. To solve these problems, he selects hundreds of surfaces and computes the factor of safety for each one of them. The su.rface that produces the minimum factor of safety is considered to be the critical surface. If Kc can be used through this search pattern to obtain the critical surface, then the amount of computation will be drastically reduced and valuable time saved. Accordingly, a homogeneous * Civil Engineering Department, Imperial College, London.

2 662 TECHNICAL NOTES R,= -0 Scale of feet Fig. 2. Typical cross-section of the dam dam of cross-section and material properties shown in Fig. 1 is chosen. Several hundreds of slip surfaces (circles) were tested for both the factor of safety (determined by Sarma s method) and Kc. After a thorough search, it was found that the surfaces which gave the minimum factor of safety and the minimum Kc were very nearly identical. The two surfaces and their results are shown in Fig. 1. It can be seen that the differences appear in the third decimal place; these are smaller than differences that are obtained by different methods of solution or from varying assumptions in the same method of solution. This conclusion appears to be true even for heterogeneous sections. Therefore Kc can be used directly to obtain the most critical surface and then determine the factor of safety for that surface. As the purpose of the factor of safety and Kc is to give an idea of the reserved strength, K, can be used more advantageously by the designer provided an idea can be given as to the relationship of K, to the factor of safety. With this in mind, an attempt is made to work out a relationship for earth dams and embankments. For this purpose, a homogeneous earth dam section is chosen for analysis (Fig. 2). In this section, the height, the crest width and the depth of the bedrock are kept constant. Properties of the soil both above and below the ground level are the same. The following combinations of the soil parameters and side slopes are chosen : c = 0, 200,400, 600, 800, 1000 lbf/ftz tan + = 0,0*2,0~4,0*6,0*8, 1-O slope = 23: 1,3: 1,33: 1 R, = 0,0-4

3 TECHNICAL NOTES 663 Fig. 3 0 Slope i 3:l R, i 0 vslope=3:iru=@4 Fig. 4 Several circular and non-circular slip surfaces are chosen in each case. The simplified Bishop s method (1955) is used for most of the circular slip surfaces, mainly because this method is most commonly used by designers and also because this gives quite accurate results (Spencer, 1967). Therefore, a comparison of results by this method with the KC method is most appropriate. For the rest of the surfaces, Sarma s method is used. The large amount of data from about 150 critical surfaces giving both Kc and the factor of safety are plotted in Fig. 3. It is obvious that a factor of safety equal to one corresponds to zero Kc. Keeping this point fixed, a linear relationship can be found between the static factor of safety F and the critical acceleration factor Kc. This gives where F= l.o+bk, (1) b = 3.33

4 664 TECHNICAL NOTES IO tiomogeneour Heinhr earth dam = looft rm.+i= I 0 06 ran J n. f o I c Fig. 5 Fig. 6

5 TECHNICAL NOTES 665 Within the range of values of practical interest up to Kc =0*4, b has a scatter of Around a factor of safety of 2.0, this represents a scatter of k 0.27 which is not much greater than the scatter obtained from different methods of stability analysis. This scatter is much less when the factor of safety approaches one. The scatter in the value of b appears to have come from the geometry of the dam, though the soil properties have some effect; this is apparent from Fig. 4. The steeper the slope of the dam, the smaller is the value of 6; even then the points are quite close together. The figure also shows that the same value of the factor of safety implies different amounts of reserved strength as regards to its capability to withstand horizontal loads which may be imparted by, say, an earthquake. Figures 5 and 6 show the relationship of the critical acceleration factor K,, factor of safety F, soil parameters c and 4 for a slope of 3 : 1 and two values of R,. Although equation (1) is derived for uniform, homogeneous earth dams, it seems to be valid for multi-zoned dams as well. Several results for dams which are multi-zoned but of various cross-sections are shown in the graph by full circles. From the results already mentioned, it may be concluded that a value of Kc may be safely used as a design parameter for earth dam designs in place of the factor of safety. ACKNOWLEDGEMENTS The work mentioned in this Note was carried out in the Civil Engineering Department of Imperial College, London. This forms part of the main line of research into the stability of slopes and foundations supported by the Science Research Council. The computations were carried out in the CDC 6400 computer at Imperial College. Acknowledgements are due to the Government of India and the Government of Maharashtra for sponsoring Mr Bhave to study at Imperial College and to the British Council for financial assistance to him. REFERENCES Bishop, A. W. (1955). The use of the slip circle in the stability analysis of slopes. Ge ootechnique 5, No. 1, Sarma, S. K. (1973). Stability analysis of embankments and slopes. G&ootechnique 23, No. 3, Spencer, E. (1967). A method of analysis of the stability of embankments assuming parallel inter-slice forces. Ge otechnique 17, No. 1, 1 l-26. Wright, S. G., Kulhawy, F. H. & Duncan, J. M. (1973). Accuracy of equilibrium slope stability analysis. Jnl Soil Mech. Fdn Engng Am. Sot. Civ. Engrs 99, SMlO, The paraffin method-triaxial testing without a rubber membrane K. IVERSEN* and J. MOUM* In normal triaxial testing of soil samples the triaxial cell is filled with water to transmit the desired all-round pressure to the soil sample. To have full control over volume and porepressure changes in the sample it has been necessary to have a close fitting impermeable membrane as a barrier between the pore-fluid in the sample and the water in the cell. Natural or synthetic rubbers are commonly used as membrane materials. A further requirement is that * Norwegian Geotechnical Institute.