Notes on Topic 2 - Slope Stability PDF

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Topic 2 – Slope Stability 2.1 Introduction Landslides, slips, slumps, mudflows, rockfalls - these are just some of the terms used to describe movements of soils and rocks under the influence of gravity. These movements can at best be inconvenient, but in many parts of the world slope instability is widely recognised as an ever-present danger and the consequences can often be disastrous in terms of economic cost and loss of life. The topic of slope stability is covered in Knappett and Craig (KC), section 12.3. The main emphasis of KC, and also of the course, is to explain how slope stability is analysed using limit equilibrium methods. Additional information and case studies will be presented to highlight the main factors which affect slope stability. Slope formation and types of slope failure The figure below shows the processes by which slopes are formed, both natural slopes and those formed as part of civil engineering works.

Many systems of description and classification for the different types of slope instability have been proposed, but there is no agreement at present. However it would be difficult in any case to devise a system which would cater for the enormous range of slope movements.

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The main classes of slope failure - falls, slides and flows - are shown in the figure below. 1. Falls are characterised by movement away from existing discontinuities, such as joints and fissures, often assisted by water or ice pressure. Slides are where the mass remains intact while sliding along a definite failure surface. The two basic types of slides are; 2. Translational slides, which involve linear movement of soil blocks or a soil layer lying near to the sloping surface. These movements are usually fairly shallow and parallel to the surface. 3. Rotational slides, where the movement occurs along a curved shear surface in such a way that the slipping mass slumps down near the top of the slope and bulges up near the toe. These movements are characteristic of homogeneous soft rocks or cohesive soils. 4. Flows, the slipping mass is internally disrupted and moves partially or wholly as a fluid. Flows often occur in weak saturated soils when the pore pressure has increased sufficiently to produce a general loss of shear strength with no true shear surface developed.

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2.2 Principles of Slope Stability Analysis The forms taken by landslides are so varied, and therefore difficult to analyse, that we need to use simple sliding models. Most landslides progress by sliding along surfaces within the soil and rock. The surfaces along which this sliding is most likely to occur depends on;  The geometry of the slope (i.e. the slope angle),  The properties of the slope materials,  The presence of any weak zones or discontinuities in the slope and their orientation relative to the slope face. When designing a slope which is to be built, you must identify the critical slip surface i.e. the surface along which sliding is most likely to occur, and determine the factor of safety against sliding along that surface. Limit equilibrium analysis The most common methods of analysis are limit equilibrium methods, where sliding is considered to occur along an assumed or a known slip surface, as shown in the figure below. The forces acting on the slip surface are analysed to determine the factor of safety for that surface.

 The destabilising forces are caused by gravity i.e. the weight of the soil W.  The resisting forces are due to the shear strength of the soil or rock i.e. shear stress developed (or mobilised) along slip surface τd. The factor of safety Fs is the ratio of the shear stress developed along the slip surface to maintain stability τd to the available shear strength of the soil τf.

Fs =

τf τd

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 Fs < 1.0, slope is unstable   Fs ≥ 1.0, slope is stable 3

Topic 2 – Slope Stability

The Ultimate Limit State method adopted in EC7 uses partial factors on actions and resistances, but the principles are similar. If the actions and resistances are factored in accordance with EC7:DA1, then a factor of safety Fs = 1.0 would be acceptable for design. Under EC7, this number is sometimes called the Stability Ratio, but we will use the traditional term Factor of Safety, as does Knappett and Craig (KC). A limit equilibrium method of analysis considers either equilibrium of moments or equilibrium of forces. For a slope, the destabilising forces (from the soil weight) are considered by EC7 to be actions and the resisting forces (from the shear strength) are considered to be resistances. In terms of moment equilibrium, the factor of safety can then be expressed as follows;

Fs =

MR MA

(see KC , equation 12.15)

For a fuller explanation, see KC Section 2.3 (p.474). Finding the critical slip surface In practice, in order to find the critical slip surface, you will need to check the factor of safety for a number of potential slip surfaces.  For homogeneous soil, check circular slip surfaces.  For non-homogeneous soil, check circular and non-circular slip surfaces. Non-circular slips include plane or translational slips (infinite slope analysis) and compound slips (part circular and part planar). For homogeneous soil conditions, charts have been produced which enable the factor of safety (i.e. for the critical slip surface) to be found quickly.  For total stresses, cu & φu, use the Taylor and Janbu charts.  For effective stresses, c′ & φ′, use the Bishop and Morgenstern charts. However, most slope stability analysis is carried out using computer software, due to its ability to quickly check a large number of potential slip surfaces. This is essential when the ground conditions are nonhomogeneous. The use of computer software is explained later in the course.

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2.3 Factors Affecting Slope Stability A slope failure or landslide is usually caused by a combination of a potentially unstable structure and a trigger event. Groundwater Changes in the groundwater conditions is the most important single factor in triggering a slope failure, which usually occur during periods of heavy rainfall.  Rising water table increases pore water pressures, reducing effective stresses.  Seepage forces increase.  The unit weight of the soil increases as the degree of saturation increases.  Saturation produces softening and swelling of clays.  Therefore, drainage is crucial to slope stability. Toe removal  Quarrying or mining.  River or coastal erosion.

Surcharge loading  e.g. Fill placed for a new road.

Cyclic loading and vibration  Heavy road traffic  Earthquake vibration, leading to liquefaction of soil and flow slides. Strength reduction  Produced by weathering.  Slow creep causes restructuring of the soil.  Slow processes eventually reach critical points.

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Changes in slope angle  Uplift produced by tectonic forces, often combined with earthquake forces.  Another very slow process which eventually reaches a critical point. 2.4 Slope Stability Case Studies During the course, reference will be made to case studies which illustrate some of the factors described above (see references in section 2.11).  In KC 12.3 (p.473) you can read about the Holbeck Hall Hotel landslide, Scarborough, which occurred in 1993, and which is well known in the UK.  Another well publicised series of landslides are those which have affected the Scottish road network in recent years, particularly the A83 Rest-and-be-Thankful since 2004 (see Figure 2.1 below). Also shown below are figures from the Scottish Road Landslide Study (2005) which illustrates this type of debris flow.

Figure 2.1. Scottish debris flows landslides (Scottish Road Landslide Study, 2005)

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2.5 Translational Slips This is a useful technique for analysing shallow translational slides where the sliding surface is very long and parallel to the ground surface, as shown in Figure 2.2.

Figure 2.2. Plane translational slip (KC, Figure 12.13)

This type of analysis is often called an infinite slope. KC (p.483) derives two separate equations for the factor of safety, for effective stresses (eq. 12.26b) and total stresses (eq. 12.26c). However, this is a rather complicated approach. A single equation is often presented which can be used for both total and effective stresses, and this will be derived in class. The starting point for the derivation is the figure shown below.

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If the water table is at an elevation hw above the slip surface, the pore water pressure u can be expressed as; u = γwh wcos 2β The factor of safety is given by the expression;

c′ + (γH − γ whw ) cos2 β tanφ′ Fs = γH sin β cosβ If you compare this equation with the two equations in KC, you can show that they are identical if you make the correct substitutions for the shear strength parameters.  For total stresses, set c′ = cu , and φ′ = φu = 0.  For effective stresses, set c′ = 0 , and φ′ = φ′. It can be seen that for effective stresses, when c′ = 0 (e.g. for sands), the above expression becomes;

Fs = (1 − ru )

tanφ ′ tan β

where;

ru =

γ w hw γH

 For slopes with no ground water pressures (ru = 0), the limiting slope angle will be φ′.  When the water table is at ground level (ru ≅ 0.5), the factor of safety will be approximately halved (tanβ = 0.5tanφ).  ru is known as the pore pressure ratio, and it is a useful way of generalising the water table position when you are designing a slope, because you don’t always know where the water table will be located. It is also used in the Bishop and Morgenstern charts.

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2.6 Rotational Slips Rotation slips, either circular or non-circular, are common in clay soils. Recall our previous discussion on infinite slopes. When the value of Hcr (H for Fs = 1) approaches the height of the slope, the slope is considered to be finite. The methods for analysing a finite slope are shown below. Note that when you are designing a new slope, you need to assume the location and shape of the potential slip surface.

However, we can use our knowledge of slope stability to help locate the position of the critical slip surface. 1. Slope or face failure

 Occurs where there is a relatively weak layer in the upper part of the slope. CL419 Geotechnical Engineering 2

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2. Toe failure

 The most common location when the slope is relatively steep or where soil beneath the toe is strong. 3. Base failure

 Occurs in relatively flat slopes or in soft or weak soils, particularly is the soil is weak below the toe. The Holbeck Hall Hotel landslide (Scarborough, 1993) is a classic rotational slip, as shown in Figure 2.3 below.

Figure 2.3. Features of a rotational slip

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2.7 Total Stress Analysis and Taylor’s Chart This is the simplest type of circular slip analysis, and was the first to be carried out, and also led to the development of Taylor’s chart. It is relatively simple because the soil shear strength parameters are cu and φu, and the stability of the slope is not affected by the water table position. Furthermore, for the short term (i.e. undrained) stability of slopes in saturated clays, φu = 0, which simplifies the analysis even further. A full explanation of undrained analysis is given in KC 12.3 (p.473-4). It is similar to the one given below, but some of the terminology is slightly different. The φu = 0 method  This analysis can be applied to slopes in saturated clays under undrained conditions i.e. τf = cu.  Undrained conditions occur in clay slopes immediately after construction.  The analysis is written out on the next page.  You can see that the main difficulty is in determining the value of l. This is one reason why the method of slices was later developed.  The analysis was later extended to soils with shear strength cu , φu using a method called the friction-circle method (which will not be explained here). The equilibrium analysis is more complicated since it also includes the resultant of the frictional force along the slip surface.  These methods of analysis were used by Taylor (1937) to produce the first set of slope stability charts for homogeneous soils.

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Taylor’s Stability Coefficients Note that these coefficients can only be applied to slopes in saturated clay under undrained conditions i.e. short term stability (cu , φu). Taylor’s chart is shown in Figure 2.4. See also KC Figure 12.9, which shows an extended version for soils with cu increasing linearly with depth (the chart has been extended and improved many times over the years).

Figure 2.4. Taylor’s chart

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For a slope of height H, the stability number Ns for the slip surface along which the factor of safety is a minimum is;

Ns =

cu Fs γH

 You will recall this dimensionless parameter from our discussion on infinite slopes (when we derived the value of Hcr when F s = 1).  You can see from the chart that Ns depends on the slope angle β and the value of φu .  For φu = 0, Ns also depends on the depth factor D, where DH is the depth to a firm stratum (note that D is not the depth to the firm stratum). You can also see that the results from the chart are consistent with our discussion in section 2.6 concerning the position of the critical slip surface.  For β > 53° and φu > 3° , the critical slip circle is always a toe circle i.e. for soils with a relatively high shear strength.  For β < 53° and φu < 3° , the critical slip circle may be a toe, slope or base circle depending on the depth to the firm stratum i.e. base circles are more likely for soils with a relatively low shear strength.

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2.8 The Method of Slices Recall our previous discussion concerning the difficulty of determining the centre of gravity of the sliding mass for the φu = 0 method. This can be overcome by dividing the sliding mass into a series of narrow slices of width b. The centre of gravity of each slice acts through a line drawn through the centre. This can be seen on Figure 2.5.

Figure 2.5. The method of slices (KC Figure 12.11)

 This allows us to take moments about the slip circle centre o for each slice and then sum for each slice.  However, you can see that there are additional forces acting on each slice (E1 and E 2, X1 and X2 ) which are known as the inter-slice forces.  Therefore the problem is statically indeterminate and assumptions must be made regarding the inter-slice forces to obtain a solution. The two most commonly used solutions are developed below.  Note that the solutions can be applied to both total and effective stress conditions and they are first developed for a soil with general shear strength parameters c, φ.  For an undrained analysis, the shear strength parameters cu, φu are substituted.

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 For a drained analysis, the shear strength parameters c′, φ′ are substituted. In addition the effective stresses at the base of each slice must be determined by subtracting the pore water pressure from the total stress. General solution  Taking moments about the centre of the slip circle gives the following solution.

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Fellenius Solution  The solution assumes that the resultant of the inter-slice forces is zero for each slice, to give the following solution.

 See KC p.479 for further discussion.  Due to the assumptions made about the inter-slice forces, this solution over-estimates the factor of safety by up to 20%. So this solution errs on the safe side, but is not normally used except for hand calculations since more accurate solutions are available.  Since it is suitable for hand calculations, it is also suitable for use in exam questions!

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Bishop Simplified (or Routine) Solution  This solution assumes that the resultant forces on the sides of the slices are horizontal i.e. X1 – X 2 = 0, to give the following solution.

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 You can see that the factor of safety Fs appears on both sides of the equation. However, it can be solved using a trial and error procedure.  The method is laborious to solve by hand, but it is ideal for computer programs.  The pore water pressure can be related to the total vertical pressure by the dimensionless pore pressure ratio, ru.

ru =

u γh

 Therefore in Bishop’s equation, (W – ub) becomes W(1 – ru )  As mentioned previously, using r u is a good way of generalising the water table position, and was used by Bishop to develop a series of charts for effective stress slope stability analysis (these are not covered in the course).  Bishop and Morgenstern’s charts (1960) are discussed further in KC p.481. Method of slices – discussion Bishop’s solution can be rewritten in the slightly different form shown in KC Equation 12.22. By dividing the numerator and denominator in the square bracket by cosα , the denominator becomes;

 tan α tan φ  1 +  F s   If you were doing a hand calculation, you could use a starting value of Fs (Fellenius) x 1.2. Since the “1” tends to predominate, a high degree of accuracy will be obtained within a few iterations. However, since the calculations are repetitive and you need to check an acceptable number of potential slip surfaces, it is better and more usual to use a computer. Later, Bishop outlined a method of re-including the inter-slice forces. Despite the equations being much more complicated, the improvements in the Fs obtained were only about 1%.

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Why is Bishop’s Simplified method so accurate? Spencer (1967) resolved all inter-slice forces into a single force Z acting at angle θ to the horizontal.  For θ = 0, the solution is identical to Bishop’s Simplified.  As θ increases, the Fs hardly changes. The reason is that the Z force has very little effect on moment equilibrium (but has a greater effect on force equilibrium). However, the accuracy of Bishop’s method has not prevented many other solutions being proposed, as you will see when you use the slope stability software later in the course. Non-circular slips Notwithstanding the comment made above, there is also a need to be able to analyse non-circular slip surfaces. This is due to the complexity of ground conditions in many cases producing a non-homogeneous ground profile, particularly when you excavate to form a cutting. Again, ma...