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ENV-2E1Y Fluvial Geomorphology 2004 - 2005 Multiple Landlsides at Yuen Mo Village, Kowloon East during the rain storm of 29 - 31st May 1982 when over 530 mm of rain fell. The collapse occurred in the late morning of 30 th May and most of the huts in the village were destroyed or severely damaged. Three people were killed. At 16:15, the site was inspected by Emergency Duty Officer, N. K. Tovey who had previously inspected 4 other landslides in neighbouring villages, each one of which involved deaths. All remaining huts were condemed by Dr N.K. Tovey and a permanent evacuation order on all 120 inhabitants of Yuen Mo was issued. From that time Yuen Mo Village ceased to exist. Slopes and related topics Section 5 Slope Stability N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 Slope Stability and Related Topics 5. Slope Stability 5.1. Introduction Assuming we know the geometry of the slope and the underlying strata, the relevant material properties, and The stability of slopes and whether or not massive failure we also understand the water flow, then all methods of in the form of landslides occurs is dependent on several analysis begin with postulating a failure mechanism. factors as were described in the introduction to this course. These may be summarised as:- It is essential that we correctly identify the most critical mechanism, and this usually is a matter of experience. the geometry of the slope including the geometric In the past, some slopes have been analysed and given a configuration of the varying strata - determined by clean bill of health, but as a less than critical failure surveying methods, mechanism was identified failures have occurred on water flow within the slope - analysed using techniques "theoretically" stable slopes sometimes with potentially covered in section (2) of this course, disastrous consequences (e.g. the Tsing Yi, Hong Kong the material properties of the differing strata, including failures above the PEPCO oil storage depot following the the unit weight angle of friction and cohesion, which are rainstorm of 29th - 31st May 1982). in turn dependent on the previous consolidation history of the soil, 5.2. Types of failure additional loading by man. Failures in slopes may:- There are several methods by which the stability of a slope may be analysed, many are valid only under certain 1) be straight lines (particularly so in granular conditions. There is also a group of more general media) solutions which can be applicable in all cases, but 2) approximate to arcs of circles sometimes it is difficult to find a solution even with the 3) approximate to logarithmic spirals aid of a computer. 4) be a combination of straight lines, arcs of circles, and/or logarithmic spirals. Examples are shown in Fig. 5.1 Fig. 5.1 Examples of different methods for analysing stability of slopes. a) example applicable for a purely cohesive soil where c) Infinite slope where slope is of approximately slip surface is a circle; constant slope over a significant distance which is much greater than depth to bedrock. The failure b) example applicable for soil with both cohesion and surface is also parallel to this slope. Water flow can friction, but water flow must be absent. Failure is a be included if it is parallel to slope as can changes in straight line; strata; 55 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 d) general case for slope of general shape. Slip surface Strain is defined as a non-dimensional ratio which is the may be of any form and may be composite including displacement during shearing over the original length of arcs or circles and straight line portions. In the the sample. In a triaxial test, the sample is usually 75 example shown, there is a straight line section mm long and a deformation of about 1 - 2 mm is which is along the bedrock plane. Water flow can needed to achieve peak strength in a dense test be incorporated as can variation in strata and the representing 1 - 3% strain. For loose samples, the presence of tension cracks. deformation will be around 10 mm in a sample of comparable length. 5.3. Progressive failure. In a slope the soil mass is large, and it is not possible Most slope failures occur during or immediately after for the whole slope to deform (with the associated periods of heavy rain when the water table is high. Thus volume change instantaneously. Near the base of the slope failures on the North Norfolk Coast are more slope, the material can expand and bulge slightly and common in winter during times of high water table. allow small strains along the potential failure plane as Equally, movement of the Mam Tor Landslide in shown by the shaded region in Fig. 5.2. Derbyshire occurs during the winter months and usually only if more than 400 mm of rain falls in the critical The corresponding point on the stress - strain diagram period. In Hong Kong, landslides are rare in the winter is shown at point A on the rising part of the curve. months from November to March, and are very common Further around the failure zone, the points B and C in the summer months (May - August) and over 500 have low amount of strain on the stress - strain plot, landslides occurring in a single day have been reported. and the mobilised shear strength is thus small. Slopes may be triggered by rainfall, and may catastrophically fail if the rainstorm is prolonged (e.g. Po Shan Road, Hong Kong, 1972), but not infrequently, the failure is progressive with small amounts of movement until eventually the failure is catastrophic in a particular event (e.g. Aberfan, Tsing Yi). After massive and catastrophic failure, continued movement may take place (e.g. Mam Tor). Fig. 5.3 Failure is now more advanced. The most highly stressed region has just passed peak shear strength, while at B, the strength is approaching peak. Region C is still relatively lightly stresses. Bulging at toe might noticeable in aerial photographs and might be visible to naked eye. Once the lower part of the slope has deformed, the next Fig. 5.2 Region of high stress in a slope prior to part can deform (see Fig. 5.3. Here, the lowest part of failure. the failure zone is at the peak shear strength, while moderate strengths have been mobilised further along The stress-strain diagram indicates the approximate the failure arc. The stress points corresponding to states of stress at points along the potential failure zone. points B and C have moved further up the curve. Slight bulging of the toe would be discernible with accurate survey measurements. Finally (Fig. 5.4), after further deformation, bulging should become very evident at the base of the slope Unlike materials such as steel which show relatively while the stress at the point A will now be less than little deformation before failure, soils deform by a previously as it has past the peak strength while the considerable amount before the peak shear strength is stress at B is now at peak and that at C is rising rapidly. achieve in the case of dense sand or over consolidated clays. For loose sands and normally consolidated clays, All along the failure arc, the mobilising shear stress the deformation before the ultimate strength is reached will vary and it is the integrated value of the strength is large. along the whole failure surface which will determine 56 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 whether or not failure will occur. Such a failure which failure envelope on the Mohr - Coulomb develops in this fashion is called a progressive failure. envelope is a constant irrespective of normal stress (i.e. = 0). Of importance is the fact that there will be a time delay (albeit quite short in some cases) from the start of the This method can be used for any slope profile, failure to the time of catastrophic failure. Normally, but is more suited to simple shapes. Water flow evidence of failure may be detected from bulging of the must be absent, i.e. excess water pressures, toe (early stage) and the development of tension cracks although the method may be use for slopes at the top (later stage), and finally a settlement of the which are entirely submerged. Only single crest immediately prior to failure. strata must be present. 2) those methods where the potential failure surface approximates to a straight line. It is valid for solids with both friction and cohesion, but there must be no water flow. The analysis is possible for irregular shaped surfaces, but it is more usually used for simple shapes. It is not really suited if there is more than one stratum. 3) those methods for the analysis of slopes which are approximately infinite in extent compared to the depth of the soil material. The angle of the slope is approximately constant over a large distance. Differing strata may be present, but only parallel to the surface. The method can deal with water flow provided that it is parallel to the surface. Both frictional Fig. 5.4. Whole of potential failure surface is now and cohesive materials may be present. This highly stressed with region A well beyond method of analysis is known as the Infinite peak strength at the residual strength, region Slope Method. B at peak strength and region C approaching peak strength. Noticeable bulging at the toe 4) those methods which are applicable to the which should be seen by naked eye. Failure analysis of general slope stability. They are is imminent. valid for varying ground water flow conditions, for various modes of failure (straight - line, arcs of circles or various Though the circumstances leading up to the Aberfan combinations), for slopes with varying strata disaster on October 21st 1966 were contributory to the which may or may not be parallel to the failure disaster. There were many signs in the months and surface or slope surface, and for slopes in years before that a potential disaster that a disaster which tension cracks have developed from might occur, the consequences of the disaster could desiccation of the surface layers. have been avoided even at a late stage. Two people who were working at the top of the waste tip about 30 These methods are collectively known as the minutes before the disaster noted tension cracks and a Method of Slices, and there are several settlement at the top. In vain they attempted to raise the variants depending on the extent of alarm, but vandals had removed the wires of the approximations made. Generally speaking all communication telephones. assumptions are SAFE ASSUMPTIONS in that they underestimate the stability of the 5.4. Methods of analysis slope. There are many methods available for studying the In the case of the Infinite slope method, the failure stability of slopes, and for some there are several surface will always be parallel to the surface, and for a variants. In this course we shall consider 4 basic single stratum it can be shown (see section 5.7), that the methods:- stability is unaffected by the depth of the potential failure surface. 1) those methods for relatively shallow slopes in normally consolidated or lightly over For all other methods of analysis, the method first consolidated materials and in which the soil assumes a failure surface of appropriate shape and material may be considered to be purely analyses the stability to obtain a factor of safety (Fs) cohesive ( and undrained) situations (i.e. the 57 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 inherentstrengthof the soil Fs This cohesive force will be the resisting force, while strengthrequired for stability the weight of the slope acting through the centre of gravity of the potential siding segment will be the If the computed factor of safety is less than unity, the mobilising force. In this case since the cohesive force slope is clearly unstable and likely to fail. If the factor and the weight do not act at a point we must also of safety is greater than unity we cannot assume that the consider the moments of the forces in assessments of slope is stable as we may not have chosen the most equilibrium, i.e. since the centre of gravity is not below critical mode of failure (i.e. failure surface). It is thus the centre of the circle of failure, the weight will act as necessary to repeat the calculations with a different a pendulum and attempt to cause the segment to rotate. failure surface until the most critical one is found. This tendency to move is counteracted by the cohesive Usually, experience will give a guidance as to what force. types of failure are likely to be more critical than others. The assess the moments of the two key forces we need to determine the distances of their respective lines of In the case of the straight line failures (other than the action from the centre of the circle in a direction at right infinite slope cases), it is possible to assess the stability angles to the force. The weight acts at a distance X on a number of different failure planes each one at a from the point O, while the cohesion acts along a different angle to the horizontal. A graph such as Fig. tangent (i.e. at the radius of the slip surface). 5.5 is now plotted with the angle of the potential failure surface as the X - axis, and the computed Factor of Safety as the Y - axis. The value of Fs will be high for shallow angles, and fall as increases. After a critical angle, the value of s will rise again as continues to increase. In this example, the critical value of Fs is clearly the minimum of the curve. A similar approach may be used for the purely cohesive failures. In this case, the it is the radius of curvature of the failure arc which is used as the independent variable on the X - axis. Fig. 5.6 Failure of a slope in a purely cohesive medium. This failure surface is an arc of a circle. It is necessary to determine the distances X, R and L for analysis. Restoring Moment the factor of safety = ------------------------------- is given by:- Mobilising Moment cLR = ........5.1 WX Fig. 5.5. Variation of factor of safety with orientation Though this is a simple method of analysis there are of failure zone to horizontal in a straight line some practical problems which require some ingenuity failure. to solve. A question similar to this was set as an exam question a few years ago. 5.5 Method of Analysis - I - purely cohesive failures. One problem is to determine the weight of the sliding mass:- this is equal to the area of the slice multiplied Fig. 5.6 illustrates this type of failure. The failure by the unit weight. The area could be computed by surface is an arc of a circle with centre at point O. drawing the sliding mass on graph paper and counting There is no frictional component in the resistance to squares, or it can be derived from geometry (rather failure and pure cohesion is developed along the failure more complex!). In the case in question, since this was arc. If the cohesion is c kPa, then the total cohesive a 40 minutes question (rather than the 60 minute force will be c . . 1 (the 1 comes from unit distance question now set), the area of the wedge was given. at right angles to the plane of the paper). 58 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 This still left the question of the position of the centre b) use a ruler and approximately rotate it around of gravity, the position of the centre of the sliding following the shape of the curve and read off circle, and the length of the sliding arc to determine. the length. In the question, candidates were given a cardboard c) adopt the method used by one candidate who template of the exact shape of the sliding wedge, and a pulled out a strand of her hair and laid this to drawing pin on which to balance to wedge. So the follow the circle. Finally, the relevant length solution to the question began with attempting to was measured on a ruler!. balance the wedge on the drawing pin. Once the approximate centre of gravity had been found, the 5.6 Method of Analysis - II template was pricked through so that the diagram on the - Straight Line Failures question paper could be marked with the centre of gravity. Fig. 5.8 shows this type of failure and the key forces to be considered. First join ends of slip circle, then through middle draw line at right angles. Point D is determined from To solve such a problem we need the weight of the equation 5.2, and hence the centre of the circle can be sliding wedge. Once again this is derived from the area found. on a scale diagram (i.e. weight is area multiplied by unit weight). In all examples, the wedge is triangular, and the area may be obtained either from:- area = 0.5 x base x height (i.e. 0.5 b h in this case). or area = 0.5 a b sin C where a and b are the lengths of two sides of the triangle and C is the included angle. In this example, all the forces pass through a single point (i.e. directly below the centre of gravity of the wedge and on the sliding surface). Though reference is Fig. 5.7 Geometric construction to find centre of circle made to the centre of gravity, we do not actually have accurately when using first method of analysis. to determine its position in this method of analysis. The weight acting downwards will have two The centre of the sliding circle can be estimated by trial components, one acting down the slope, and the other and error using a pair of compasses, or alternatively a acting perpendicular to the slope. If the slope is just geometric theorem can be used. The extreme ends of stable then the frictional force will just balance this the slip circle are connected by a line which is then component of the weight acting down the slope and bisected at right angles (Fig. 5.7), such that AY = BY. attempting to cause failure. Equally, the component of These two distances can be measured from the diagram the weight acting at right angles will balanced by the as can the distance CY. normal force N. Finally, the geometric theorem states that DY x CY = AY x BY ......................5.2 Hence it is possible to determine the distance DY and then the diameter of the circle CD. Finally, the distance CD is halved (i.e. OC = OD) to get the centre of the circle The length of arc may be determined in one of three ways:- a) join the lines from the extreme ends of the slip circle to the centre of the circle and measure the angle. Convert this angle to radians () and multiply by the radius of the circle. i.e. L = R. Fig. 5.8 Straight line failure of a slope. The key forces are the weight (W), the normal 59 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 force (N), and the fractional force (F) which all 5.7.1 Dry Cohesionless Slope act through a single point. Four forces act on a block of slope (Fig. 5.9): The governing equations are thus (by resolving parallel and perpendicular to the slope):- W = weight of block of depth Z R = Reaction upwards F = W sin X1 & X2 are lateral forces on block. and N = W cos ........................5.3 It is readily shown that X1 and X2 have a common line For our factor of safety we note that:- of action [from Moments], and resolution parallel to the slope shows that they are equal and opposite. The inherent shear strength of soil effect of this is that we can effectively disregard these Fs = ------------------------------------------------- in subsequent analysis. force which would just resist failure (i..e mobilising force) Hence W = R but W = . bzcos the mobilising force = F = W sin | area of block Also the restraining force will be the inherent strength of the soil which can be derived from the Mohr - Coulomb envelope, i.e. cL + N tan ............................5.4 Note we MUST use forces here, and so we are dealing with the cohesive force (not the stress as defined by the intrinsic property of the material). Thus the factor of safety may be specified as:- c L W cos tan Fig. 5.9 Diagram of a typical slice on an infinite slope Fs ............5.5 showing relevant forces which act. The W sin reaction force may be further split into two components one parallel and the other For those who are Mathematically inclined:- perpendicular to the slope. Resolving perpendicular to the slope:- Note: with this geometry, it is possible to define the weight in terms of the slope angle and the failure Rcos = N = W cos = dzbcos2 angle , and also the length (L) of the failure surface in terms of and Equation 5.5 can thus N is the normal force so the normal stress be generalised, and it is possible to differentiate = dzcos2. dFs equation 5.5 to get and hence determine the d Resolving parallel to the slope minimum factor of safety from all possible failure angles directly. If you are mathematically inclined Rsin = F = Wsin = dzcos sin . you may wish to check this out. This may be plotted on a Mohr Diagram, remembering that for cohesionless materials c = 0. 5.7 Method of Analysis - III - Infinite Slope Method The angle the point subtends at the origin () is known as the angle of obliquity, This method of analysis assumes that the slope is of infinite extent, and that any water seepage is parallel to the slope. There are several cases which can be considered, some of which will be treated in detail below. 60 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 Fig. 5.10 Mohr - Coulomb Diagram for dry case stability. The point A represents the stress point on a plane at depth z below the surface. zcos. sin Fig. 5.11 Evaluation of the pore water pressure in the i.e. tan 1 d 1 z cos2 tan (tan ) full seepage case. The pore water pressure at d A is the vertical distance AB. i.e. in this case the angle of obliquity () equals the Thus pore water pressure equals w x AB. slope angle . But AC = z cos If is increased the point moves along the dashed line which is part of a circle which also passes through so. AB = AC cos = z cos2 the origin. and pore water pressure = w z cos2. Failure would occur if, or when reaches the value . However, the pore water pressure only affects normal stresses, so new normal stress equals:- Hence in this case failure occurs when = = , i.e. the slope angle equals the angle of internal friction. ' = - u = z cos2 - wz cos2 This latter is sometimes known as the angle of repose and is normally in the region of 30o - 40o for granular [Note: We must also change the unit weight from its media. Its value is higher for denser sands and lower for dry value to its saturated value.] loose sands. Thus ' = ( - w)z cos2 or 'z cos2, 5.7.2. Fully Wet Cohesionless Slope - Full Seepage apart from the change in unit weight, the shear stress remains the same. Along any equipotential, there is no change in excess pore water pressure. Thus the pressure at A can be evaluated in static head terms alone if the vertical distance AB corresponding to the equipotential AC is used (Fig. 5.11). Fig. 5.12 Mohr - Coulomb Diagram for full seepage case. The stress point A has moved to point A' as a result of the excess pore water pressure. This stress point is closer to the 61 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 failure line and the slope is thus less stable. The locus of the point A' is now an ellipse. the imposed shear stress will be:- (d + z) cos.sin The effect of the positive pore water pressure is to bring ( z d)cos. sin z d the stress point closer to the failure line, and thus the and tan ( z d z)cos2 ( z d z) tan slope is less stable. w w Once again failure occurs when zcos. sin For typical values of z and d = 1m and the same and tan ( )zcos ( ) tan 2 values for the unit weights as before:- w w 20 x 1 20 x 1 40 tan tan tan Typical values: - for a sand = 20 kNm-3 ( 20 x 1 20 x 1 10 x 1) 30 and w = 10 kNm-3. Failure once again occurs when = Hence tan = 2 tan In this case is approximately 0.75, in or 2 . other words the surface dry layer has increased the stability again Thus failure occurs when is approximately /2 and the slope approximately half as stable as in the dry case. 5.7.4 Case with damp soil - i.e. negative pore water pressures throughout. 5.7.3. Case with dry top layer and full seepage below If the whole section is damp with negative pore water pressures throughout, then the stress point in the Mohr The next case to consider is the one where there is full - Coulomb Diagram (Fig. 5.10) will move towards the seepage below a given depth d and material is dry right making the slope more stable (i.e. the reverse above. We shall neglect the effect of capillary action at direction to that with full seepage.. the interface (this has the effect of improving the stability so by neglecting it will be a SAFE APPROXIMATION). 5.7.5 Case with soil with cohesion. If the material forming the slope has cohesion as well as frictional properties, then some interesting things happen. If we return to the first case, we note that the slope is table provided that the stress point lies below the Mohr - Coulomb line. If cohesion is present, then the failure line is displaced upwards, and the slope becomes more stable. Indeed it is quite possible for a slope to plot as a stress point shown as B (Fig. 5.14 ) and be stable. However, the question of stability must be qualified since it was noted that failure on an infinite slope was Fig. 5.13 case with full seepage with a dry layer of soil independent of depth: i.e. at greater depth both the above. The dry layer increases the normal normal and frictional components increased at a stress faster than the shear stress and hence constant proportion so that the angle of obliquity the slope becomes more stable. remained constant, and hence the factor of safety. In the case of material with cohesion and the stress In this example we would consider a potential failure in point lying above the purely friction envelope, the the full seepage region (which is the more critical above statement is no longer true. Thus as the depth region). increases, the stress point will move along the line of constant obliquity and will eventually cross the failure the total normal stress will now be:- envelope when failure will occur. The conclusion that ( d + z) cos2 can be drawn from this is that infinite slopes with material having cohesion are stable but only up to a the water pressure will be: w cos2 given normal stress. 62 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 What does this mean in practice? when dealing with Mohr's Circles. However, the result is simple, the maximum depth that will remain stable is given by:- 2c depth of stable vertical slope = .......5.6 Since c is approximately 20 - 40 kPa for many soils, and is approximately 20 kN m-3, the maximum depth for a stable vertical slope will be 1 - 2 m. 5.7.6 An Infinite Slope which is completely submerged An infinite slope which is completely submerged will not have any water seepage (except in extreme conditions), and analysis will show that the stability is Fig. 5.14 Mohr - Coulomb Diagram for case with soil identical to that in the dry case. It is suggested that you having cohesion. Slope is more stable and check this out for yourself. Remember in this case you slope angles steeper than the angle of friction should use the submerged unit weight to work out both are possible but only for a limited slope the normal stress and shear stress and that excess pore height. water pressures are now zero. 5.7.7 Depth of Tension Cracks The normal stress is dependent only on the angle of the slope, the depth to the failure surface, and the unit Tension cracks will appear in cohesive materials when weight of the material. Since for a given slope it will desiccated. This will appear on flat surfaces, the most only be the depth that can vary, it is clear that any classic examples are the cracking that occurs on cohesion present will allow slopes of an angle steeper reservoir floors when the water level falls in summer. than the angle of friction only for a limited depth. We From the above, we can predict the maximum depth of may estimate this critical depth by drawing the diagram such cracks. If the cracks deepen, then minor failures as shown in Fig,. 5.14 and reading off the normal stress will occur and debris will fill the base of the crack such () on the X - axis that the maximum depth will never be much greater than that predicted for long. Hence the critical depth is given by:- The development of tension cracks can also be of z importance in the stability of finite slopes as will be cos2 considered in section 5.8. The maximum depth of such cracks can be predicted from the relationship in equation 5 6 and used in more detailed analysis of the stability of It is also possible to work out what z is directly without slopes. plotting the graph by trigonometry. Thus the vertical distance on the stress plot of the 5.7.8 The factor of safety for Infinite Slope Analysis. critical points is given by two relationships - one from the Mohr - Coulomb line and the other from the shear stress i.e.:- from Mohr Coulomb....... c tan ............ z cos sin from shear stress (c tan ) i. e. z cos sin A consequence of the above observation is to note that we can readily dig vertical trenches in clay when it is not possible to do so in dry sand. A question that immediately arises is what is the maximum depth for Fig. 5.15 Determination of factor of safety in the dry which a vertical slope in a cohesive material will be cohesionless case. stable? This analysis is beyond the scope of this course but will be covered indirectly in the Seismology Course 63 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 The factor of safety may be readily determined when using the infinite slope method of analysis. It is obtained directly from the Mohr - Coulomb Diagram. Thus Fig. 5.15 is a revised form of Fig. 5.10. The point A represents the stress point for a slope of angle while the point C represents the point on the failure envelope directly above the stress point. Thus the factor of safety is AB / BC. i.e. BC z cos 2 . tan tan Fs AB z cos . sin tan Fig. 5.16 General slope with slices drawn. Note that one Thus the factor of safety may be determined either slice boundary coincides with the intersection graphically using the Mohr - Coulomb Diagram or of the water table with the slip circle. Both entirely numerically using the above equation. the actual slip circle and the surface are approximated to a series of straight lines. For The factor of safety in all other cases may be similarly analysis purposes, it is convenient to number estimated. the slices separately. 5.8 General Solution to stability of Slopes 5.8.1 Introduction The general method for analysing the stability of slopes was developed by Fellenius and is usually referred to by the method of slices. There are several variants on the method with fewer or more simplifying assumptions. The Swedish Slice Method has the most such assumptions, but all are SAFE ASSUMPTIONS, i.e. they underestimate the factor of safety. All the methods begin by dividing the slope into a Fig. 5.17 Details of forces acting on a single slice. The number of vertical slices and analysing the stability of effect of tension cracks at the crest can be each. The overall stability is then given by a summation included as can the effects of varying shear of the stability effects on all the slices. strength properties. Once the factor of safety has been estimated along a As with many problems in Geotechnics, there is a trade single slip circle, further potential slip circles are off between accuracy and time as to the number of slices analysed to find the most critical one (or group of critical chosen. Typically, the number chosen should not be less slip circles. than 5, and numbers more than about 8 get tedious to compute. There are some computer methods to analyse the stability of slopes, but these require careful data 5.8.2 Method of Analysis input as the whole of the slope geometry must be specified, and there can be problems with convergence if A typical slope is shown in Fig. 5.16 with the various simple rules are not followed. slices included. In addition the approximate straight lines to the various curved sections of both the slip circle The method begins by drawing a realistic slip circle and and the surface as shown. then dividing the area of the slope above this circle into several slices. The slices may be of varying width, but In Fig. 5.17 the details of the forces acting on each slice the general rule is that the height no slice should be more are shown. These are (in the case without water):- than about 2 - 2.5 its width, nor should it be less than 0.4 times the width. Further, it simplifies calculations, if 1) the weight acting vertically downwards (W) slice boundaries are arranged to coincide where different 2) the normal force acting orthogonal to the local slip strata cut the slip circle and also where the water table surface (N) intersects the slip surface. 3) the frictional force acting parallel to the local slip circle. (F) 64 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 4) vertical inter-slice forces arising from friction 1) the area of each slice (so that we can work out between slices (X1 and X2). the weight, 5) horizontal inter-slice reaction forces between 2) the orientation of the local angle of the slip adjacent slices (E1 and E2). circle to the horizontal (). The inter-slice forces were encountered previously in the The analysis is most conveniently conducted in tabular Infinite Slope Method of Analysis, but in that case it was form such as that shown on the next page. easy to show that such forces on either side of each slice were equal and opposite and would thus cancel each a) We begin by measuring the local angle of the slip other out. In the general method, this is not the case as circle and enter this in column 1 of the table. the slices are of different sizes. However, overall there must be no net horizontal force or vertical force arising NOTE: some of the local angles of the slip circle from these inter-slice forces and various approximations will be negative in deep seated failures such as that are possible. shown in Fig. 5.16 (e.g. 1). A) The first approximation is to assume that the vertical forces on each slice balance (this is more b) We now need to estimate the weight of each slice likely than the horizontal ones). Overall they must and we do this as in other methods for slope do so, but not necessarily for a particularly slice. stability by determining the area of the slice. If By making this simplifying assumption to lead to an there is a single stratum, the we determine the area underestimate of the factor of safety and thus it is a of the whole slice whereas if there are several strata SAFE ASSUMPTION. we must sub-divide the slice so that the area of each stratum in each slice can be determined. Since the The computer method which you will use in the unit - weight of most materials is approximately the practical uses two methods, a rigorous method same, this refinement is not always necessary. The which does not make the approximation and a less estimated values of area are entered into column rigorous one which does. In execution using the [2]. The shape of slices may be approximated to rigorous method, it is necessary to first guess what either:- the imbalance of the two forces on the sides are. triangles [ area = 0.5 x base x height. It is usually The calculation is then completed at which time a convenient to use the vertical side of the slice revised estimated can be determined from the as the base of the triangle and the distance calculations. If the revised estimate and the initial from this slice edge to the opposite apex as the guess are close then the computed factor of safety is height in these estimates]. accepted. If the two are not close, then a second trapezia [ area is mean height of parallel sides multiplied trial is made using the computed values from the by distance between them]. first set of calculations. The processes is repeated through several iterations until convergence is c) The weight of the slice is then the area multiplied achieved. Unfortunately, if the slices are ill- by the unit weight. In the case of multiple strata, conditioned (i.e. they do not conform to the rules the each separate stratum can be approximated to a laid out above), convergence will not occur, and trapezium, and the resultant weight is the the solution can become unstable. summation of all areas multiplied by the respective unit weights. B) The second approximation is to assume that both the horizontal and vertical forces on the sides of the d) We have three basic forces controlling the slices cancel out. This is a more severe equilibrium of each slice [if we use approximation approximation, but is a SAFE APPROXIMATION B], and as with previous examples, the mobilising once again. force will be W sin while the normal force will be W cos We compute this values from the 5.8.3 Analysis of Slopes without water seepage data in columns [1] and [3] and enter the results in columns [5] and [4] respectively. In most of the discussion regarding the Method of Slices we shall assume that both the horizontal and vertical e) The cumulative effects of all slices may then be inter-slice forces cancel. We will return briefly to deal determined by summing the values in both columns with the more accurate solution using approximation A [4] and [5]. above later. Basically the analysis follows exactly the same form as NOTE: some of the values in column [5] will be that for a straight line failure. (We may incorporate the negative!!! effects of water pressure later). We treat each slice separately and from each we will need to determine:- f) Finally, we require (for the dry case), information on the cohesive force. To obtain this which measure the length 65 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 of the segment on the slip circle in each slice (column [6] column [8] from which the pore water force U may be ), and multiply these values by the cohesion to give the determined (column [9]). values in column [7]. Finally we sum all values in column [7]. Equation 5.5 which was used for determining the stability of a single wedge is then modified to incorporate all slices by a summation on each part of the equation:- i.e. Fs c W cos tan ....5.7 W sin This factor of safety will give the situation in the dry case ( i.e. without water seepage). Fig. 5.18 Flowlines and equipotentials in ground water flow in a slope 5.8.4 Analysis of Slopes with water seepage NOTE: we must work in terms of forces not stresses The analysis of slope stability in the wet case proceeds in (i.e. we multiply the pore water pressure by the area of the same fashion as the dry case, and once again , it is the base of the slice). most convenient to use a tabular form for analysing the slope. Columns [1] - [7] are completed as in the dry case. We need to assess the effects of water seepage. There is a more exact method involving the construction of a flow net and an approximate method. The approximate method will over estimate the water pressure and hence once again it will be a SAFE APPROXIMATION. We first need to sketch the water table (Fig. 5.18), then remembering that with the phreatic surface (water table), Fig. 5.19 Estimation of true water pressure (h) and the positions of the intersections of the equipotential approximate pressure (h'). drops with the water table will be spaced at points to given equal vertical head drops. This gives us the start of the flownet which we can then complete by sketching The approximation will always overestimate true value curvilinear squares in the normal way. and will thus be a safe approximation. In the example shown the water table is steeply curved and this Once we have done this we can then estimate the water accentuates the difference between the two values. pressure at any point on the slip circle by following the Normally the water table is less steep and the two values relevant equipotential line from the point on the slip usually differ by less than 10%. circle up to the water table and then evaluating this vertical head drop. (Fig. 5.19). This distance (h) will Finally, we modify the Normal force to account for the be less than the direct vertical distance (h') and so will pore water pressure and enter the value in column [10]. over-estimate pore water pressures. The water table is Column [10] is effectively column [4] - column [9] shown steeply curved in the diagram for clarity, but normally the water table is flatter and the difference The factor of safety in the wet case is then given by:- between the two estimates is much less [typically being around 10% or less. Fs c (W cos U )tan ........5.8 The values for the head of water (hw) , whether W sin accurate or approximate, are entered for each slice in Space for additional notes 66 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 TABLE 5.1: SLOPE STABILITY - METHOD OF SLICES Cohesion (c) = kPa Angle of Friction ( ) = Unit Weight ( ) = kN m-3 Slice No Area of Weight of W cos W sin c hw U N= Slice Slice Slice =hw gw W cos -U No (W) [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 Fs( wet) c ( W cos U )tan Fs( dry ) c W cos )tan = W sin W sin ========== ======== 67 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 5.8.5. Modification to allow for inter-slice horizontal and impose additional lateral stresses on the slope which forces - Approximation A. will make the slope less stable (see Fig. 5.20). Equations 5.6 and 5.7 give the factors of safety in both the dry and the wet case for the situation where the inter- slices forces are assumed to cancel out. Bishop proposed methods to avoid this approximation although the full analysis with no approximations is difficult to execute in practice, needs a computer program, and then can be beset with convergence problems if the correct procedure is not rigidly followed. However, the simplified method of Bishop does allow us to obtain a more accurate estimate by removing the restriction that the horizontal forces on each slice must balance. Fig. 5.20 Slope with a tension crack arising from desiccation. The presence of the crack The full derivation of the formula is beyond the scope of reduces the area of the sliding surface and this course, but it is sufficient to note that if we do know hence stability is reduced. Water filling what the factor of safety is it is possible to estimate what crack will impose lateral stress on the sides the horizontal inter-slice forces are and include them in of the crack causing a further reduction in our analysis. stability. This begs a question though. How do we know what the factor of safety is until we have estimated it using an Analysis of the slope in the dry case will follow the appropriate formula? same procedure as described above except that the failure surface terminates at the base of the crack. When The formula to use is:- water is present the analysis is more involved an beyond the scope of this course Fs (cb W tan ).M n 5.8.7 Effect of water filled tension crack on a straight W sin ...........5.9 line failure tan .tan Fig . 5.21 shows a potential straight line failure surface where M cos 1 with a tension crack both dry and filled. It is relatively Fs straightforward to analyse the reduction in strength in these two cases as shown below. Analysis of slope and bn is the width of the nth slice (not the length stability in the general case with dry cracks is easily parallel to the slip circle) accommodated in the analysis, but the effect of water filled cracks is more complex and beyond the scope of The term M involves Fs so the way we proceed is to this course except in general descriptive terms. first make a guess that the factor of safety is unity (i.e. the slope is just stable), we work through the analysis Equation 5.5 for the stability of a straight line failure is:- and obtain an estimate for Fs which is general will be c L W cos tan different from our original guess. We now replace the Fs .....................5.5 computed value of Fs as the denominator to the W sin expression for M and repeat. We will find that the difference in the two values on the second interation is much less and we continue until both the guess and the computed value are closely the same (usually agreement to 1% is sufficient), and this is then the correct value of the factor of safety. 5.8.6 Effect of Tension Cracks - general description. Tension cracks may, in dry weather, open to a depth of 2c/ (see section 5.7.7) and will affect the stability of a slope in two ways. Firstly, there will be a reduction in the length of the slip surface which has inherent strength, and secondly Fig. 5.21 Straight Line failure with water filled tension immediately after a drought, water can fill such a crack crack. 68 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 2 n is . 'W ( ) soc . 2dw . 1 Incorporating these modifications changes the factor of safety to:- 1 c L' ( W' cos . w d2 . sin )tan ….5.11 Fs 2 1 ( W' sin . w d2 . cos ) 2 5.8.8. Effect of Man Made Loading Man-made (or natural loading) on a slope can affect its stability. It can be incorporated by adding the load to the appropriate slices before analysis. The positioning of the load also affects the stability. In some cases it will increase the stability, in other cases it will decrease. Generally speaking, loads near the base of the slope will Fig. 5.22 The water pressure distribution with depth for tend to improve stability and those near the crest will the tension crack in Fig. 5.21 reduce the stability, but this is not always the case as it also depends on the position of the water table. With a dry tension crack, the equation is essentially the same except the length L' is now the relevant length, and the weight W' will be reduced to omit the small wedge to An example of a slope with a crack at the the right of the tension crack. top The formula now becomes:- Straight Line failure with crack at top which can become water filled c L' W' cos tan Fs .......... 5.10 [based on Exam question in January 1996] W' sin The geometry is shown in Fig. 5.23 In the case of the crack filled with water, the water pressure will be hydrostatic and thus the total water force Height of slope = 10 m, angle of slope = 60o, c = 15 from the crack acting horizontally will be: kPa; = 19o; = 16 kN m-3 1 wd2 2 Critical failure surface will be about ( + ) /2, i.e. about 40o Note that because the pressure is hydrostatic it has a triangular distraction and increases linearly wt. depth, We need the length of the failure surface (L), and also and the above expression is effectively the pressure at the weight of the sliding mass. mean depth multiplied by the depth. L = H / sin This force acts horizontally and thus perpendicular to the slope it will have a component of:- This is computed in column [2] of Table 5.2 1 Column [3] then computes the area of the wedge and . w d 2 . sin acting in a column [4] the total weight using the unit weight of 2 16 kN m-3. direction outwards from the slope and will thus reduce the effective normal force from W' . cos to the cohesive resistance is computed in colum [5] 1 ( W' . cos . w d 2 . sin ) column [6] computes the Normal Force 2 W cos tan = 0.3444 W cos At the same time there will also be a component of the the Shear Force (W sin ) is evaluated in column [7] water force acting parallel to and downwards along the slope equal to:- the factor of safety is calculated as ([5] + [6]) / [7] 1 . w d 2 . cos the procedure is repeated for other failure angles 2 around 40o. So the mobilising force increases from W' .sin The factor of safety is now plotted against failure surface to angle (Fig.5.24), or from the Data Sheets 69 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 c . L W . cos . tan Fs W . sin Fig. 5.23. Geometry of the Slope showing tension crack. This question appeared in the 1996 Exam Paper. failure failure area of weight of cohesion x Normal Shear Factor surface surface wedge wedge L Force Force of angle length Safety [1] [2] [3] [4] [5] [6] [7] [8] 35 17.4 42.5 680 261 198 390 1.177 40 15.6 30.7 491 234 129 316 1.148 45 14.1 21.0 336 212 82 238 1.235 38 16.2 35.1 562 243 152 346 1.142 Table 5.2 Analysis of factor of Safety at various Failure Angles without tension crack. Fig. 5.24 Plot of Factor of safety against failure surface angle Having obtained values for 35o, 400, and 450, it is clear that the minimum factor of safety representing the critical failure occurs between 35o and 40o, so a further angle of 38o is used and indeed has a slightly lower Fs. If time permitted, a further iteration would be done, this time between 38o and 40o. However, the error in using the value at 38o will be very small. 2. c 2 x 15 Tension crack depth = 1. 9 m 16 70 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 failure failure area of weight of cohesion x Normal Shear Factor of surface surface wedge wedge L Force Force Safety angle length [1] [2] [3] [4] [5] [6] [7] [8] 38 13.1 539 197 146 332 1.033 so the failure surface length (at 38o) is reduced by 1.9 / sin 38 = 3.1 m to 13.1 m and the weight lost beyond The slope is still stable (see Table 5.3), but is now the crack will be 0.5 x1.9 x 1.9.tan38 x unit weight = becoming somewhat critical as Fs is only just over 22.6. 1.000. Table 5.3 Revised Factor of Safety Calculations incorporating DRY tension crack. It is now necessary to out the effects of the crack being filled with water after heavy rain. This is done in the manner shown in section 5.8.7. i.e. the pore pressure in the crack is U 0. 5 w hc 2 = 0.5 x 10 x 1.92 = 18 kPa where hc is the depth of the tension crack. The normal force is now W cos tan - U sin tan and the Shear Force is ( W sin + U cos ) and the final, and most critical value of the factor of safety is now recalculated. Note that in this case it is only columns [5] to [8] that have to be recalculated as the values in the other columns remain the same. failure failure area of weight of cohesion Normal Shear Factor of surface surface wedge wedge xL Force Force Safety angle length [1] [2] [3] [4] [5] [6] [7] [8] ---- as above ---- 197 142 346 0.979 Table 5.4 Final calculation of Factor of Safety at critical angle – water now fills the crack. The factor of safety is now less than unity and failure is likely. 5.10 General Statement on Factors of As a guidance, the factor of safety required in Hong Safety. Kong is:- While all analyses compute a factor of safety and this 1.4 if the slope threatens residential buildings forms the basis of whether a slope is stable, a further discussion is important. 1.2 if it threatens lines of communication A factor of safety of unity indicates that a slope is just 1.1 if it threatens other man-made resources stable, while one which is above unity should be stable and one which is less than unity is critical. The choice of 1.0 ( or less) for other regions what factor of safety to use in a risk analysis is a matter of judgement, but there is a growing awareness that this Remember, that all approximations tend to under should take account of the consequences of a potential estimate the factor of safety and that is why there are slope failure. several slopes which have factors of safety around 0.9 and yet appear stable. What we can say is that their Normally, we estimate the factor of safety under the stability is critical and in adverse weather conditions worst conceivable conditions (e.g. a 1000 year storm). failure is likely. We would estimate the likely ground water level at this stage and carry out the analysis by one of the methods A further reason why the factor of safety is outline previously. We would then test alternative modes underestimated is that when we conduct shear strength of failure until we found the most critical one. If this was tests on the soil we will always draw an envelope to the above a given threshold then we should be satisfied weakest strengths measured rather than take a mean. In regarding its stability. this way we always work on the safe side, but this has the effect of underestimating the factor of safety further. 71 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 5.11 Predicting the form and position of potential failures. In early October 1996 the largest glacier in Europe in Iceland became the centre of attention when a volcano In the analysis we will always strive to find the failure erupted beneath the glacier causing a large amount of mechanism which gives the minimum factor of safety. the ice to melt with the threat of extensive and perhaps More often than not, we will find a band of possible slip catastrophic flooding. This section has been written at a surfaces which have approximately the same stability. time when the outcome is not known but we can consider Defining this critical band will indicate the position of possible modes of failure that would lead to flooding. the slip surface zone should failure occur. Fig. 5.24 shows a simplified possible cross section of the toe of the glacier some 150m long and 100 m high. 5.12 Summary of Slope Stability Behind this is a crevasse in which water from the melted glacier behind is accumulating and starting to rise in the Methods for analysing the stability of slopes all crevice. require the identification of a potential failure mechanism. Several such mechanisms should be The first question to address is what possible modes of tested to determine the most critical one. flooding are likely. Analysis can be done on current water level Four situations could lead to extensive flooding conditions, to determine the current stability, or more usually, a particularly severe rainfall event The water level in the crevasse may reach such a will be used as the basis for analysis (e.g. 1000 height that the flow of water beneath the toe glacier year storm). become critical causing a quicksand at the down steam end and excessive erosion and undermining of Many approximations can be made to simplify the the toe of the glacier. As the erosion takes place, the analysis and in all cases, the approximations resistance to flow will decrease causing an reduce the factor of safety, and hence are all increasing flow of water until the level of the lake SAFE ASSUMPTIONS. eventually dissipates. The choice of what factor of safety to use should The level of water continues to rise as more ice be determined on the basis of the consequences of melts (up valley of the toe] causing the toe glacier to failure following a model such as that used in be over-topped and flow of water over the surface. Hong Kong. Rarely is this done. This is unlikely to be catastrophic as the flow will eventually equal the rate of ice melting. Erosion of 5.13 Some comments about the stability of the surface of the toe glacier is likely causing further the Glacier in Iceland. release of water, but this should be controllable. Fig. 5.25 Schematic representation of toe of glacier with build of water in crevasse behind. 72 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology: 2002 - 2003 Section 5 Iceland is volcanic, and if the underlying part of the distribution as was should in section 2 in the lecture glacier is volcanic ash rather than rock, it is notes for flow beneath a weir. possible that the glacier will have carved out deep valleys. The same would be true in rock (e.g. U- To investigate whether the toe glacier is stable we need shaped valleys). If as a result these valleys are to work out the weight of the glacier. This we can do by steep, then exposed to a water level rise could dividing it into say 5 slices (as in the general slope trigger landslides which could suddenly displace a stability method), or by counting squares. large quantity of water causing a massive over- topping of the toe glacier and catastrophic flooding Let us assume that we have worked this out (remember (as happened to the dam at Vaiont in Italy in 1962). the unit weight of ice is 9.00 kN m-3 (see data sheets) Note, however, that since the slope is solely submerged, then that stability is the same as in air- Let this value be W i.e. it is not a seepage situation. If the underlying material is rock, then rock slides could also occur, The total resisting horizontal force provided by the depending on the dip of the strata - fissures may surface of the sand/silt would be have opened up while the ice was present, but with water filling the gaps, the effective normal force (W U).tan() would be reduced. and the factor of safety against sliding would be:- Uplift on the base of the toe glacier because of seepage pressures could reduce the effective normal ( W U ). tan( ) Fs stress and hence the ability of the toe glacier to H resist sliding following the build up of horizontal pressure on the side of the crevasse. This would be As previously, the situation is critical if Fs < 1. We can the most catastrophic failure mode. easily investigate how stability would be subsequently affected by changing the height of the water (h) and The horizontal force (H) from the hydrostatic pressure is reworking U and hence Fs. As in many previous proportional to depth below the surface of the water, so examples we would plot Fs against h. However, this the mean force will be time the value of Fs would continue to decrease as h increased. What we would be looking for is the value of 1 h where Fs becomes 1.0. H h2 2 The total area of the toe glacier (which was determined Since the length of the toe glacier is large compared by slicing the section) = 9650 m2, and since the unit with the depth of the sediment, and since the base is weight of ice = 9.00 kN m-2 - from data tables, the total nearly horizontal in this case, the hydrostatic uplift on weight of the toe ice is 86850 kN per metre width of the the base of the toe glacier will decline linearly from the glacier. above value at the crevasse to zero at the toe. Remember we have to work this out only once in the Thus the total uplift (U) on the base of the glacier will calculations (unlike the case of the slope stability as the be failure will occur at the base of the glacier, if any where, and the weight above remains constant. 1 U hd 2 As a first guess try a value of h = 50m. This gives a high factor of safety. Then try say 60, 70 m until Fs fall where d is the length of the toe glacier (150 m in this below 1,0. Finally perhaps, and if time permits add one case) intermediate point. If the base of the toe glacier was not straight or the Now plot the factor of safety against height of water depth of the sandy/silt was large compared to the length and read of value when Fs = 1.0. In this case it is 62.9 d, then it would be more accurate to plot the seepage m. flownet beneath the glacier to work out the pressure Height of water Horizontal Uplift force W-U Fs = (h) (m) force H U ( W U ) tan( ) (kN m-1) (kN m-1) H 50 12500 37500 49350 1.974 60 18000 45000 41850 1.163 70 24500 52500 34350 0.902 65 21125 48750 38100 0.902 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 Table 5.5 Calculation of Factor of Safety against failure Fig. 5.26 Factor of safety against height of water We can thus predict that failure of the ice block will occur when the water in the crevass reaches 62.9m. space for additional notes. 74 N. K. Tovey ENV-2E1Y: Fluvial Geomorphology 2004– 2005 Section 5 75

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