 Welcome to lecture series on advanced geotechnical engineering course. In this lecture series we are going to introduce ourselves to module 5 and which is on stability of slopes. So this is the lecture 1 on stability of slopes in module 5. In this module we are going to discuss about the stability analysis of a slope and find critical slip surface or a potential failure surface and some special conditions like a sudden drawdown condition and effective stress and total stress analysis that is from long term and short term point of view and seismic displacements in marginally stable slopes that is to some extent on the seismic stability of slopes and reliability based design of slopes. And finally after having discussed different methods of analysis of slopes, the methods for enhancing the stability of unstable slopes, how these stability of slopes can be enhanced. So in this module 5 we are going to discuss about different analysis methods on slope stability for both infinite slopes and finite slopes and some special conditions like sudden drawdown condition, effective stress and total stress analysis, seismic stability of slopes and reliability based design of slopes and slope mitigation methods. So this lecture or we can say that module for couple of these lectures is divided into what are the types of slopes like we have said they are finite slopes and infinite slopes and what are the different types of failures like failure types and what are the causes of these slope failures, what combinations makes it to fail and what are the analysis methods which are available and then we will take some examples we will try to do by using different methods and compare the different analysis methods and concluding remarks based on this module what we are going to discuss. So after having discussed about the shear strength and you all agree that the shear strength has got two applications, broad applications, one is that earth pressure theories and other one is the stability analysis of slopes. In the stability analysis of slopes you need to know what is the shear strength mobilized by the soil and shear stress which is actually induced by the disturbing forces. So the ratio of shear strength available shear strength to the shear stress mobilized is referred as factor of safety. If this factor of safety is adequate then we can say that the slope is stable, if this factor of safety is inadequate then the slope is in the verge of failure. So as far as the stability analysis of slopes is concerned there are two types of slopes, one is infinite slope and other one is finite slopes. The infinite slopes are basically classified as slopes that extend over a long distance and the conditions remain identical along some surface or surfaces for quite some distance. So these slopes can be for examples like landfill slopes or slopes which are actually along hilly terrain with very very flat slope inclinations. So these are slopes they extend over a long distance and the conditions remain identical along some surface or surfaces for quite some distance. Finite slopes basically these are manmade slopes that connect the land at one elevation to the land at other elevation. They can be like approach embankments, they can be road embankments, there can be highway embankments, there can be ash pond dam embankments. So these are the most existent slopes the finite slopes and can exist also in nature as well as some hill slopes there can also be finite slopes and manmade slopes. So after having discussed about these slopes then these slopes are basically again natural and manmade. Natural the examples are hillside and valleys and coastal and river cliffs they are the natural slopes and as far as the manmade are concerned slopes in cuttings and embankments for highways and rail roads and earth and ash pond dams and temporary excavations and waste heaps which are nothing but the landfills and landscaping for the site development so they form the examples of manmade slopes. So if you look into the typical types of slope failures the typical failure which is termed as a circular failure and where you have identical radius at both the ends at entry point and exit point this is called the entry point of a slip surface and this is the exit point and with identical radius of rotation and this particular failure surface is referred as a circular failure surface and there can be failure surfaces which are truly non-circular they generally occur if you are having different stratifications of soils and particularly these both circular and non-circular they are actually called as rotational slip failures and there are also some translation slip failures that means that it actually commences with a circular failure and then it remains in the failure surface in contact with the film surface and then the translation and slip failure takes place. So the typical failure surface which is actually shown in this figure is called as translational slip failure and another type of failure which is compound slip failure or also called as a base failure in this case what will happen is that if the sub base or a sub grade soil which is not having any inadequate strength then there can be possibility that the base failure which actually extends and the failure surface remains tangential to the rigid stratum. So that means that these failure surfaces can occur on a surfaces where you actually have very soft soils and then the constructions on them may be in the form of embankments can lead to a compound slip failure surfaces. So consider in this example a block movement so here if you see before introducing the method of slope instability let us see this example where whenever a block of weight w is resting on a surface having certain friction static friction let us say mu s then for the force F the resting force is nothing but mu s n and disturbing force is nothing but F1 that means that disturbing force is nothing but F tends to become to F1 then there is a possibility that the block can move. Say let us consider in this case when F tends to become to F1 then the block actually tries to said to be you know when F1 you know over comes this force for the resting force then there is a possibility that the destabilizing activity has commenced that means that factor of safety here is nothing but resting force by driving force the resting force is indicated as FR and driving force is indicated by F suffix D. So as long as mu s n greater than F then the block is said to be stationary otherwise the block which is actually under instability and are in the process of movement. So the causes of slope failures are basically gravity so one is the predominant causes of the slopes of failures is gravity and seepage that is seepage seeping of water through the slope at quake erosion geological features and construction activities the construction activities are nothing but the construction at the downstream side of a slope or geological features are nothing but different stratifications of soils erosion when a slope is designed with certain slope inclination when erosion happens there can be possibility that the slope tends to become steeper and then will attain failure because the slope which has not been designed for that particular sloping inclination. So before discussing analysis methods let us see some typical slope failures so in this case a typical slope failure which actually reported in the region because of the seepage in the monsoon season can be seen here wherein you can see that the soil which is above the rock top is subjected to instability and here this is a typical slope failure wherein these highway slope is subjected to instability because of a slip and this is a typical slide which actually shows the landslide damage adjacent to a residential structure. So this is a landslide damage adjacent to a residential structure so that means that the safety of the structures lying are constructed close to the slopes need to be ensured and this is a typical slope failure beneath the building where it can be seen that the part of the building is remaining hanging and the slope is actually subjected to instability and then this is a typical coarse grained slope failure which is right bottom side of the slide which is shown here wherein the slope is subjected to failure because of the presence of some soft soil strata at a certain depth from the surface of the slope. So this is a typical highway slope failure in the hill slope failure and where it can be seen that the landslide which actually has caused as created a distress because of the instability and this is a typical sacrificial slope failure in highway embankment wherein it can be seen that the part of the highway embankment is subjected to slip which is predominantly can be seen in this slide. So this is the recent Uttarakhand 2013 slope instability failure which actually occurred so it can be seen that one of the stretches how the slope failure created and disrupted the transportation networks. This is a typical landslide in a urban area in Hong Kong where it can be seen that very close to the construction activity the building where the prevalence of the existing the building the part of the slope is subjected to seepage and then failure due to seepage and then resulted in a slope failure. So you can see that this is the portion of the area which is subjected to a slip and then this is the mass movement which is actually occurred up till this so this type of situations which are actually can endanger the buildings which are lying close to the slopes. So this is a typical schematic view of a building which is lying close to a particular slope and then if there is ingress of water either due to some utility pipes or due to some tension cracks or due to some raise of ground water table within the slope can cause failures and can lead to instabilities. So this is a typical view of a waste slide in a landfill which is reported by it that all in 2000 this failure which is actually occurred in March 16 1996 and this is a typical failure which occurred in a landfill where in the lateral displacement for reported to be about 275 meters and vertical displacement for reported to be up to 61 meters and 1.2 million meter cube of waste was reported to be you know displaced or subjected to movement. So this type of failures can occur in a manmade slopes which are with a material like municipal solid waste. So this is the another typical failure which is in a landfill and this was reported by Stark et al 2000 wherein you can see that a part of the landfill is subjected to a translation slip and can lead to the failure. So after having seen different sets of you know failures which are either in natural slopes or in manmade slopes let us try to understand about how this slope failure they basically depend upon what are the parameters. The first and foremost thing which comes into our mind is the soil type and the soil stratification and ground water the presence of ground water and seepage and slope configuration or geometry. So the soil type particularly if you are having a soil which is having some expansive soil type of minerals there is a possibility that the slopes which are actually in during wetting and drying seasons they can be subjected to instabilities. Particularly the soil stratification in the sense that the major portion of the soil can be with coarse grained soil but geologically if there is a soft soil which is lying underneath can lead to instabilities and presence of ground water or a raising of ground water table within the slope can lead to failures. In one way the raise of ground water table within the slope is also caused and termed as in a particularly in some loose field slopes and this is called as static liquefaction wherein the total stress will be equivalent to the pore water pressure and then the effective stress reduces to zero and the soil mass tends to become like a fluid and then undergoes a failure. And particularly the seepage induced failures which are actually possible in case of embankment dams or it can be slopes which are actually subjected to very high rainfall areas and slope geometries particularly when the slope geometries are constructed with cutting or with embankments with certain configuration and they can lead to failures. So the slope values are basically depending upon the soils type stratification ground water and seepage and slope geometry. One of the if you say the types of slope failure the if you are having a thin layer of weak soil and there is a soil war burden which is above that then this particular block is going to undergo a translational slide. So wherein this thin layer of weak soil which is actually not having adequate shear strength so can undergo the possible translational style failure that is the failure of a slope along the weak zone of a soil and the sliding mass travels long distances before coming to rest and it is common in coarse grain soils particularly in coarse grain soils and this is possible. So this is a typical a thin layer of weak soil how this can actually induce failure like a translation slide which is actually shown pictorially here. So this particular portion is enlarged and shown here. The rotational slide is one of the common failure in a homogeneous fine-grained soils it has its point of rotation in the space on an imaginary axis and parallel to the slope. So there are three types of you know slope failures basically one is the base slide or base failure that means that when the soft when the soil sub base soil is soft in nature and then the soil undergoes failure predominantly in the within the soft soil. So that is called the base slide and toe slide which is nothing but the failure which is actually passes through the toe and the slope slide which is nothing but failure which is actually occurs within the slope surface. So the base slide occurs by an arc engulfing the whole slope that is the predominant portion is in the soft soil this is the portion which is there in the soft soil because this soil is being stiff the failure surface moblages along the weak zones where the stresses or strengths are low and then this portion in which you have this entire zone is actually mobilized and undergoes a so called base failure. So a soft soil layer resting on a stiff layer of soil is prone to base slide similarly a soft soil resting on a clay surface and when it is loaded without any ground improvement there is a possibility that it can actually undergo a failure and toe slide the basically the failure surface passes through the toe of the slope. So in this picture with the slide which is actually shown here this is the failure surface and the failure surface predominantly passes through the toe of a slope and this is called the crust of the slope and another type of a rotation slide in a finite slope is called slope slide the failure surface passes through the slope. So this can be because of the low quality control or because of the virtue of the loading on the top of the slope or due to inadequate properties for the material which is used for constructing the slope. And in this particular slide a typical flow slide is shown the flow slide which actually occurs in certain type of sensitive soils you can see that this zone which is because of certain perturbance and this causes this type of failure. The occurs when internal and external conditions force a soil to behave as a viscous fluid and flows down spreading in all directions. Some sensitive soils when they are subjected to some disturbance and they actually behave in this particular manner and the typical failure which actually takes is called as a flow slide and multiple failure surface occur and changes continuously as the flow proceeds and basically this occurs in dry as well as some wet soils. And some block and wedge slide failures this is basically occurs when a soil mass is shattered along joints and seams and fissures and weak zones by forces emanating from adjacent soils or some excavations are dredging which actually happens in some clay soils under undrained conditions they actually also undergo some block failures. So this shattered mass moves as blocks and wedges down the slopes. So the slope masses moves in form of a blocks or wedges and this is also classified as a typical slope failure and which is called as rock falls and this can be triggered because of a seeping water can create an instability in the soil base and on which these rock fragments or rocks are resting and then can lead to a rock fall. So simple detachment of the rock mass from its parent body and can lead to rock falls. So this process is only gravity governed and which is very alarming in the sense that if you are having a transportation network along this certain area then there is a possibility that this rock falls can actually endanger the likes of people. So this lead to be addressed by an appropriate rock fall mitigation methods. So if you look into the causes of slope failure one cause we have actually addressed is that erosion in this particular slide wherein the original slope surface or configuration is shown here like this and here this particular slope surface is attained because of the erosion of the soil either because of the water agency or wind agency so because of this attainment of the steeper slope can lead to instability because the slope which is actually not been designed for that particular type of seeping relation can lead to failure. So one of the reasons of the slope failures is erosion and water and wind continuously erode slopes can make the flat slopes into steeper slopes and can lead to failures. So erosion changes the geometry of the slopes and resulting in a slope failure of our landslide. And particularly if the slopes are facing rivers or ponds the rivers or streams continuously cover their banks because of the wave forces and these undermine the natural manmade slopes and the slopes which are actually subjected to wave forces or the streams continuously in contact with the slope surfaces can lead to scours and the zone which is actually shown here from the water surface this is actually called as a scour zone and this can lead to movement and instability to two slopes. The another major cause of slope failure is the rainfall. If you are having a rainfall basically of rainfall intensities ranging from say 10 mm per minute to 80 mm per minute and 80 mm per minute is said to be very high intensity of rainfall. If this rainfall occurs over a certain period of duration the long periods of rainfall saturate and soften and erode these soils and can lead to slope instabilities. So the water enters through the existing cracks and may weaken the underlying soil and layers leading to failures basically it causes mudslides. So the rainfalls basically they trigger failures particularly because of the ingress of rainwater into the slopes and in a saturated state and they make these slopes in a saturate tend to saturate and soften and then erode the soils. So a typical slope failure due to rainfall is actually shown here. The another major cause of slope failures is the earthquakes. If you are having a structure which is resting on certain zone where earthquake actuation is expected then earthquakes induce the dynamic forces and particularly in addition to their weight in the vertical direction they can be possibility of horizontal forces and these forces can trigger the instabilities to the slopes. The earthquakes induce dynamic forces especially dynamic shear forces and reduce the shear strength and stiffness of the soil. So these things can lead to failures. These earthquakes particularly the pore water pressures in saturated soils, saturated coarse grain soils could raise, pore water pressures in saturated coarse grain soils could raise to a value equal to the total mean stress and cause the soils to behave like a viscous fluid. And this phenomenon is called dynamic liquefaction. So at the onset of earthquake the pore water pressure raises rapidly and becomes almost equivalent to a total stress and this lead to a dynamic liquefaction and the structures are formed on these soils would be collapse and then the structures founded on these soils would tend to collapse and the quickness in which the dynamic forces are induced prevents even coarse grain soils from draining the excess pore water pressures. So even the coarse grain soil is as you all know that has high permeability but even these soils they will not able to drain in a short duration of the earthquake. So because of that what will happen is that the seismic event can lead to failure under undrained conditions. See another slope cause of failure what we discussed is the geological features and many failures commonly result in unidentified geological features. So that means that if you are actually designing a slope an appropriate soil investigation is required up to the depth which is influencing the construction. So this can lead to a proper design of a slope. So many failures can commonly result in unidentified geological features. So a typical soil stratification zone where you have got a reasonably good soil and weak soil and good soil and weak soil and can lead to failure. So this is another type of slope failure. Then the another problem or a cause of slope failure is the external loading. Suppose if the slopes which are actually loaded either because in the form of a soil heaps or in the form of a construction which is close to the crust of the slope can lead to instability. That means that the loads placed on the crust of a slope add to gravitational load and may cause slope failures. That means that if a construction takes place or war loading at the crust of the slope takes place. Loading takes place far away from the crust. There can be we can say that the external loading will not have an influence on the slope. But if the war loading happens close to the crust of the slope then the loads placed on the crust of the slope add to gravitational load and may cause the slope failure. The construction activities is another cause of slope failure. So previously we have said that what are the different types of slope failures which are factors causing the slope failure. Now we are actually seeing different types of reasons for the slope failure. The another one is the construction activities. This is a construction activity where excavation actually has taken place in cutting. So in this situation if the slope inclination which is actually meant for excavation is steep then can lead to failure. The construction activities near the toe of an existing slope can cause failure because of the lateral the lateral assistance is removed here. So excavated slope failures due to construction activities is excavated slopes and fill slopes. So the slope failures due to construction activities is divided into two cases basically excavated slopes and fill slopes. In the fill slopes also if the slope is inclination is not properly designed can also lead to failure. So in the case of excavated slopes this is what we said is one of the reasons for the construction activities for the slope failure. When an excavation occurs the total stresses are reduced so because of the excavation the total stresses are released and the need to pore water pressures are generated. With time what happens is that the need to pore water pressure tends to dissipate and causing a decrease in the effective stress and consequently this lead to the lowering of the shear strength of the soil. So if the slope failure occurs they take place after construction is completed. So in case of excavated slopes we observe or tend to observe these failures once the construction activity is completed. The reason is that the total stresses are reduced because of the excavation and the need to pore water pressures are actually generated and with time these need to pore water pressures tends to dissipate and causing a decrease in the effective shear strength and which leads to the lowering of the shear strength and this is one of the reason for observing the slope failures particularly for excavated slopes after the completion of construction. In case of fill slopes basically they are common in embankment construction that is embankment is nothing but construction above the ground. If the foundation soil is saturated then positive pore water pressures are generated from the weight of the fill and the compaction process. So the effective stresses decrease and consequently the shear strength decrease and the slope failures in slope are likely to occur during or immediately after construction. So in the case of fill slopes basically if the foundation soil is soft in nature the pore water pressure tends to increase because of the weight of the fill material and also compaction process. So the effective stresses decrease and then consequently the shear strength decreases. So this lead to the slope failures in slopes are likely to occur during or immediately after construction. So we can actually list here after gray and laser 1982 the factors contributing to instability of soil slopes they are the factors that contribute to high shear stress. High shear stress is nothing but the shear stress which causes disturbing forces, removal of the lateral support either due to erosion, bank cutting by stream centrumers or human agencies like cuts and canals and pits or excavations and another factor which contributes to high stresses is the surcharge, natural agencies, weight of the snow, ice, rain water. Human agencies like fills, buildings constructed close to the slopes and can make the surcharge increase high and then lead to the high stresses and transitory stresses like earthquakes because of the earthquakes the transitory stresses increase tremendously and removal of the underlying support the sub aerial weathering like solutions by solutioning by groundwater. Suppose if there is a cavities which actually formed within the slopes and then can lead to instabilities like removal of underlying supports and the subterranean erosion that is piping which actually happens can also trigger the you know underlying support removal of underlying support and can lead to failure and human agencies like mining and removal of the you know soil from the excavations or activities which are close to the slopes can lead to removal of the underlying supports can lead to the contribute to high stresses. Lateral pressures the water in vertical cracks freezing water in cracks are root wedging. So root wedging or water in vertical cracks or freezing water in cracks can lead to increase in the lateral pressures and another way regarding again after grain laser 1982 factors that contributes to low shear strength. That means the initial state composition in inherently soil actually has got weak materials and texture basically the soil is having a loose soil metastable grain structures and the grass structures particularly if it actually has got some faults jointing bedding planes and warving of soil status can lead to you know low strengths and changes due to weathering and other physical chemical reactions like frost action and thermal expansion hydration of clay minerals and drying and cracking and leaching particularly this actually happens in laterate basal slopes and can lead to you know the slope low shear strength because of leaching phenomenon where the slopes tends to become unstable and this is predominantly occurs because of the weathering and physical chemical reactions and changes in intergranular forces due to pore waters, pore water pressure that is seepage pressure of percolating ground water or loss in capillary tension upon saturation. Suppose if you are actually having a capillarity which is with negative pore water pressure and if it is subjected to loss because of the upon saturation and buoyancy in saturated soils can lead to a low shear strength in a slope and changes in structure like fissuring in pre consolidated clays due to release of lateral restraint grain structure collapse upon disturbance this basically happens in sensitive clays where sensitivity is very high sensitivity is defined as q u are unconfined compressive strength in undisturbed to disturbed state. So in some soils or which are called as sensitive soils the collapse of the grain structure occurs upon disturbance and that lead to the failure. So these are the factors contribute to the low shear strength. So in the slope stability analysis now what we do is that we try to address upon the methods for analyzing these slopes and before addressing about the sudden drawdown condition and some measures for locating the critical failure surface. The slope stability analysis basically it is with the general assumptions the failure can be represented as a two dimensional problem because a slope or an embankment a railway embankment which is a finite slope which actually extends or long lengths can be treated as a plain strain structure. So hence two dimensional analysis is relevant but in certain cases now the three dimensional stability methods are also available. So the failure can be represented as a two dimensional problem the sliding mass moves as a rigid body and the deformation of sliding mass has a significant effects on the analysis. So the sliding mass moves as a rigid body and the deformation of sliding mass has no significant effects of the analysis and the properties of soil mass are isotropic and shear resistance along failure surface remains same independent of orientation of the failure surface. So the properties of the soil mass are isotropic in nature the shear strength along the failure surface remains same independent of orientation of the failure surface. So the basically the analysis is based on the limit equilibrium methods of course now there are also many other methods like based on finite element methods or finite difference methods. So before discussing the finite slope stability analysis methods let us look into the infinite slope stability analysis method particularly this infinite slope stability analysis means this we defined these slopes they can be natural or man made the examples of slopes which can be infinite slopes are given below here they can be ore or sand stock piling by dropping from a chute and the slope which actually takes shape the form of the slope which actually takes is a can be regarded as an infinite slope and embankment formed by end dumping from a truck is also regarded as an infinite slope and natural slopes formed by granular materials where the critical failure mechanism is shallow sliding or surface raveling time. So basically it occurs in some certain types of shear slopes wherein the slope surfaces remain parallel to the slope failure surfaces remain parallel to the slope surfaces and shallow in nature and can lead to failures. So the natural slopes formed by granular materials where the critical failure mechanism is shallow sliding or surface raveling which actually happens and the natural slopes formed in cogeo soils with a great extent or weak cogeo soil material on the ledge is also regarded as a infinite slope and slopes in residual soils where a relatively thin layer of weathered soil or like say a film soil rock. So this is an example which I was giving for a shale slope where the shale slope in the process of because of the saturation can though it actually has a film soil or a rock and beneath that if there is a soil which actually has a nature of weathering which actually undergoes underneath this film soil and where the slope failure is or the slope is actually also said to be an infinite slope. So in this particular slide a typical cross section of infinite slope is shown and the another example of infinite slope is also a slope which is actually formed for a manmade structure like for certain type of elements in a manmade structure like landfill where in the some of the elements are actually designed based on the infinite slope stability analysis theory particularly the covers or a landfill cover or lining system which is based on the infinite slope stability analysis. Of course when the structure is actually concerned when considered as a whole then one tend to one has to do the global stability analysis wherein considering entire cover system or lining system along with the waste body in a landfill. So in the analysis of infinite slopes in this particular slide a typical cross section is shown here and this is the slope surface which is inclined at an angle beta and this is the location of a failure surface which is at a certain depth below the slope surface that is the z depth which along the vertical direction so which is shown here and let hw is the depth of the groundwater table and is measured from the top surface of the ground surface. So if hw is equal to 0 that means that the water is at the ground surface where hw is equal to z that means that the water is at the failure surface or a surface over which the failure is occurring. And the beta is the slope inclination which is shown here and it is assumed that there is a failure surface which is actually occurring here. So consider a block b where small b is the horizontal distance so the perpendicular to this length is a per meter length or unit length is considered. So the area over which this block is sliding is b into 1 units that is b meter square units. The soil is assumed to be homogeneous and the stress and soil properties on every vertical plane are assumed to be identical and on any plane parallel to the slope the stresses and soil properties are identical that means that this block can be constructed here or block can be constructed here. So the stress and soil properties on every vertical plane are assumed to be identical on any plane perpendicular to slope stresses and soil properties are identical. So the failures in such slopes takes place due to sliding of the soil mass along a plane parallel to the slope at a certain depth in this case the depth is indicated as a jet. Now consider this particular free body diagram of that particular block where w is the self weight of the block and this line a, b, a, b, c, d is the block which is actually considered and b is the horizontal width and z is the depth and this is the portion along which the position where the failure surface is actually occurring. So the weight of segment a, b, c, d which is w is equal to gamma z into b into 1 that is nothing but the weight of the area of the parallel ground which is actually shown into the unit weight of the soil. Now the tangential stress tau down the slope can be obtained by this particular expression which is nothing but the tangential force divided by this area this particular length this is horizontal distance is b so this distance is b by cos beta because this inclination ad is inclined with horizontal with beta inclination so this along length of bc or ad is equal to b by cos beta so this component along this surface is taken over a horizontal distance b by cos beta into 1 so it is nothing but a shear force over that area where in tau is equal to gamma z b sin beta by b cos beta with that we can write tau is equal to gamma z sin beta cos beta. The normal stress sigma within the segment that is the normal stress sigma that is n which is actually acting here and over again this particular area which we can give as gamma z b cos beta divided by b cos b by cos beta into 1 so by simplifying the normal stress can be obtained as gamma z cos beta. Similarly with the presence of water like if water is from surface ad hw units vertically down that means that the water pressure which is nothing but gamma w into z minus hw in the pressure over that area which is actually given by pressure pore water pressure u on the slip surface which is nothing but z minus hw into gamma w into b cos beta by b by cos beta which actually simplified as z minus hw into gamma w cos square beta. Now this further actually simplified with normal effective stress which is nothing but sigma dash is equal to sigma which is nothing but the normal stress which is here and the pore water pressure here now which can be written as now gamma z cos square beta minus z minus hw that is within parenthesis gamma w cos square beta. So by simplifying we get this gamma z minus gamma w z plus gamma w hw cos square beta. So this is the normal stress sigma dash now the shear strength of the soil along the failure surface tau f at the base of the segment that is the resistance offered by the material at the interface of the soil above the block the failure surface and soil below the failure surface. So that is nothing but tau f is equal to c dash plus sigma dash tan phi dash. So now the factor of safety can be defined as nothing but tau f which is nothing but the shear strength available to the shear stress which is induced. So what we have done is that we have written tau f is equal to c dash plus sigma dash tan phi dash now here for substituting for sigma dash the gamma z minus gamma w z plus gamma w hw cos square beta. So that is substituted here divided by the shear stress which is obtained in the previous case because of the moment of the block is nothing but gamma z sin beta cos beta and that is actually obtained and substituted here and then for the general case the expression reads like this the factor of safety is equal to c dash plus tan phi dash cos square beta into gamma z minus gamma w z plus gamma w hw divided by gamma z sin beta cos beta. So this is the general expression now this expression when it is actually let us say that when you are having with water without ground water or when we have a saturated slope then different cases can be reduced but this particular expression which is shown in this slide factor of safety is equal to nothing but the shear strength available to shear stress that is tau f by tau and this is expressed as c dash plus tan phi dash into cos square beta into gamma z minus gamma w z plus gamma w hw divided by gamma z sin beta cos beta. Now in this particular slide the analysis of infinite slopes which is actually shown and the special case where in the previous expression if you put this c dash is equal to 0, hw is equal to z in case of a dry cohesionless soil the expression gets simplified to factor safety is equal to tan phi dash by tan beta. So basically this is factor safety in infinite slope with a cohesionless soil basically independent of the depth of the failure surface. So if you look into this for a critical case the factor of safety is equal to 1 that means that the slope inclination equal to the friction angle this is nothing but you know the also called as the angle of repose that means that when the slope inclination is equal to the angle of internal friction of the soil then that is the called as angle of repose and this particular expression is actually independent of the depth of the failure plane. So this can be explained here on a Mohr circle plot where normal stress is on the x axis and shear stress on the y axis wherein this is the Mohr failure envelope with friction angle phi dash and the slope inclination which is actually shown here. So for beta less than phi dash as long as beta less than phi dash the slope is actually stable independent of the depth of the slope even the depth of the slope is 2000 kilometer or more than that and this can lead to ensure the stability to a slope. So this is the slope is said to be stable as long as beta less than phi dash and if beta is equal to phi dash the slope is on the edge of the failure or in the critical factor of safety which is actually called as 1 the slope is said to be just stable when beta greater than phi dash that means that when this line lies above this then it is actually called as the slope would have already failed at all depths that means that it would have said to this particular sloping inclination. So here if you look into this why when beta less than phi dash the slope is said to be stable and independent of the depth of the slope so at a particular normal stress if we consider and if you notice here that for a given normal stress this is the shear stress mobilization but the soil actually has got a shear strength which is much higher than the shear stress mobilized so because of this unless the failure the sloping inclination meets this failure and allow there is no possibility of a failure surface. So as long as this sloping inclination is actually less than this internal friction angle of a soil then the slope is actually said to be stable. In the case of a analysis of infinite slope but this is a case of a saturated cohesion less slope where C dash is equal to 0 Hw is equal to 0 when you substitute in this general expression then factor of safety is given by gamma dash tan phi dash divided by gamma tan beta that means that factor of safety of a saturated cohesion less slope is about half of the slope without saturation. So we have said that in case of a saturated slope the factor of safety is 50 percent of a slope without saturation and the case C when we substitute Hw is equal to 0 in the just expression what we derived for infinite slope to be analysis we get for a C phi soil factor of safety is equal to C dash plus tan phi dash into cos square beta into gamma dash z divided by gamma z sin beta cos beta where beta is the sloping inclination phi dash is the angle of internal friction of soil and C dash is the effective cohesion of a soil. So in this particular lecture what we studied is that we have understood that what are the different factors which actually can cause slope failure and what are the different types of slope failures and what are the different types of slopes like infinite slopes and finite slopes and we have addressed about the infinite slope stability analysis which is based on the block which is actually considered and then we have deduced a general expression then afterwards we have considered a three cases case A which is a dry slope and case B is a saturated cohesionless slope and which is a saturated cohesionless infinite slope and in case of a cohesionless C that is where C phi soil is there in this case when you look into this where C phi soil when you consider with a factor of safety is equal to 1. So here for a C phi soil there is a limiting depth of the stability. So this particular expression when we put factor safety is equal to 1 and Z tends to hc, hc is given as hc is equal to C dash secant square beta into gamma within parenthesis tan beta minus gamma dash by gamma into tan phi dash. So this C dash by gamma hc is referred here as stability number tan beta minus gamma dash by gamma into tan phi dash by secant square beta. So only for C phi soils there is a limiting depth of stability. So this is particularly represented here wherein for this when you represent this in a tau sigma plot the depth at which tau sigma tau is equal to tau sigma is actually called as the critical depth. So that is what we have discussed that when Z tends to zc. So here when it meets the failure envelope the shear stress is equivalent to the shear strength of a soil and lead to the failure here. So for beta 1 less than phi dash which is actually here and tau is less than tau f but when beta 2 is less than beta 2 greater than phi dash then at a certain point it meets this failure envelope and that point at which the tau tends to tau f is actually called as the critical depth. So in this lecture we have actually discussed about the infinite slope stability analysis. In the forthcoming lectures of in this module we will discuss about the finite slope stability analysis methods.