 Welcome to lecture number 2 of module 5 stability analysis of slopes in the course advanced geotechnical engineering. So the title of this lecture is lecture 2 which is the part of module 5 stability of slopes. So in the previous lecture we understood about different types of slope failures and causative factors for slope failures and we also discussed about infinite slope stability analysis methods and we have also discussed that there are two types of slopes predominantly they are infinite slopes and finite slopes. Then the finite slopes are the one which are man made slopes to the maximum extent and some hill slopes also can be called as a finite slope. Now when we look into the analysis of finite slopes there are different possible failure surfaces. The failure surfaces can be planar in nature or if it is a homogeneous soil then there can be possibility that you can get a circular failure surface and if you are having a different stratification of soils there can be possibility that we can get non-circular failure surfaces. So in this slide three different failure surfaces are shown one is defined as planar failure surface this occurs along a specific plane or weakness. So where the potential weak plane will exist and this failure surface occurs along a specific plane or a weak plane of weakness. This basically excavations into stratified deposits where strata are dipping towards the excavation that is the layers of soil dipping towards the excavation and in earthen dams along the sloping course of weak material not likely to occur in homogeneous soils. So these planar failure surfaces they occur along a specific plane of weakness and the circular failure surface this is for soils exhibiting cohesion C or cohesion and friction angle and no specific planes of weakness are great strength. So this actually takes in the form of a circular failure surface and non-circular failure surface when the distribution of shearing resistance within an earthen dam is non-uniform and failure can occur along surfaces more complex than a circle. So this is said that non-circular failure surface mostly occurs in layered soils. So in the typical cross section of a finite slope with one vertical and horizontal where the small n is the number if n is equal to 2 it indicates that the slope is having one vertical to horizontal sloping inclination that is about 26 degrees with horizontal. And if n is equal to 1 that indicates that the slope is actually having a 45 degree sloping inclination with the horizontal and the horizontal surface indicates that the back slope which is 0 degrees but in practice there can be a possibility that the slopes can also have a particularly especially for natural slopes in nature there can be with natural sloping inclination of back sloping inclination ranging from 10 to 20 degrees or so. So in this for a cross section of a finite slope a typical planar failure surface is depicted here and then a two-bedge type of failure surface like as it has been told in earthen dam suppose a sloping core and if this happens to the potential weakness plane then the two-bedge failure mechanism can come and this is the typical circular failure surface and this is also a circular failure surface where the majority of the failure surface extending into the soft soil which is underneath the say for example beneath this level and this is a tow slope failure where it can be seen that the entry and the exit point is from the tow of the slope. So this is called the entry point commencement of the entry point of the failure surface and this is called the exit failure surface. So in a given slope there can be number of failure surfaces but one need to determine the failure surface which actually gives the critical factor of safety or the least factor of safety and that particular failure surface is called as the potential failure surface. Now in this particular slide there can be typical rotational slides which are shown one is a cylindrical shaped failure surface you can see that this is a circular arc but the failure surface which actually shown is mostly this is common for plane strain structures there can be a settlement at the certain point away from the crest of the slope and a failure surface which actually occurs with the soil movement which is actually shown here and this is a typical rotational slide and which is called as a spoon shaped failure surface. So you can see that this failure mode or a slide which is actually called as a spoon shaped slide. Now we actually yesterday we introduced ourselves to that there is a effective or total stress parameters. We know that the total stress parameters may be can be adopted for the short term conditions and effective stress parameters can be adopted for the long term conditions or getting the long term stability of a slope. Total stress parameters can be obtained for the short term stability assessing the short term stability of a slope. So the short term basically low permeable soil basically in clays at the end of the construction the soil is almost still undrained. So here the adoption of total stress analysis making use of undrained shear strength U is adopted. So at the end of the construction of the soil which is clay is almost still undrained hence the total stress analysis making use of undrained shear strength C U is adopted. Some free draining materials sands and gravels if you look into it. The drainage takes place immediately hence effective stress parameters C dash and phi dash are used. Suppose if you have got a free draining materials sands and gravels when the drainage takes place immediately so there is a need for adopting effective stress parameters that is C dash and phi dash. Long term relatively after a long period of time let us say that an embankment is constructed on a soft soil after having waited for a certain duration the fully drained condition might have been attained. So the fully drained stage will have been reached hence the effective stress parameters C dash and phi dash can be used. So what we broadly say is that the effective stress parameters are used for long term conditions are also used for you know the assessing the stability of a slope with free draining materials like sands and gravels. In case if there is a short term condition with low permeability then the adoption of total stress parameters is more relevant. So the total stress strength is used for short term conditions in clay soils whereas the effective stress strength is used for long term conditions in all kinds of soils or any condition where the pore pressure is known. So this is after John Boo 1973 stated that the total stress strength is used for short term conditions in clay soils whereas the effective stress strength is used in long term conditions in all kinds of soils or any conditions where the pore water pressure is known. So this pore water pressure may result due to ingress of rain water into the slope that we have discussed one of the causative factors of the slope instability is rainfall. So because of this once the ingress of the rain water is captured in the form of periodic surfaces we can actually get the pore water pressure having known then we can actually estimate the total stress parameters and then the long term conditions can be used. The analysis of the finite slope particularly with the factor of safety the factor of safety for finite slopes depends basically on the assumed location of the center of rotation of the slip surface. So as it was shown in the typical failure surfaces particularly for circular arc failure surface wherein it all depends upon the location of center of rotation in suppose if a slope analysis which is done in two dimensions that is basically for a plane strain structure in x and y direction then assumed location of the center of rotation for the slip surface and the radius of the failure surface basically what radius it actually exists and type of failure that is toe failure base failure or slope failure. So before attempting the analysis methods before attempting the analysis methods let us look into the various definitions of factor of safety here the definitions are given with respect to limit equilibrium point of view force equilibrium point of view and moment equilibrium point of view the topmost figure if you see there is a failure surface which is indicated and there is a shear strength which is actually shown and there is a shear stress which is actually shown the shear stress is mobilized due to the disturbing forces and the shear strength is mobilized to as a resisting force. So as a limit equilibrium condition for total stress we can write factor of safety as shear strength available divided by the shear stress in case of effective stress which can be written as factor of safety is equal to C plus sigma dash tan phi dash by the shear stress which is mobilized. As far as the force equilibrium is concerned for a typical wedge failure which is having a planar failure surface extending a unit length perpendicular to the plane of this figure wherein we can actually write the force equilibrium as a factor of safety is equal to sum of resisting forces divided by sum of driving forces or disturbing forces. So here if you look into it the resisting forces are nothing but the resistance offered by the soil that is nothing but the shear strength and driving forces is nothing but the because of the weight of the slice which is actually acting vertically downwards and by resolving this along the potential plane of failure that is the planar failure where which is actually taking place and that component will work out to be W sin alpha. So by writing factor of safety is equal to Su by W sin alpha. So we can write this as Cl, Cl is nothing but cohesion which is mobilized along the, it is assumed that the cohesion is mobilized uniformly and over a length L, L is nothing but the length of the plane of surface that is measured from this point to this point into plus n that is n is nothing but the normal force tan phi divided by W sin alpha where total length of the sliding plane which is actually shown here which is nothing but this is the length of the sliding plane. Then similarly for moment equilibrium the factor of safety is defined as resisting moment by driving moment. So here the summation of resisting moments divided by summation of driving moments. So when we try to get the resisting moment by taking the shear strength along this particular arc, let us assume that we are having Su1, Su2, Su3 and Su4. Then we can say that Su1 into dl1, Su2 into dl2, Su3 into dl3 plus Su4 into dl4 into R that is the, this is the force which is acting over this particular length. We have taken each arc length as dl1 into one unit that is the unit perpendicular to the plane of this figure into R we will get the force into moment that is the resisting moment divided by W which is nothing but the entire mass is actually assumed as the CG of this area is assumed to act here and the at the center of gravity here this W into this horizontal distance from the center of rotation that is x. So that is actually called as Wx. So factor of safety is equal to in the integration form if you show that R is equal to 0 to L Su into dl. So this indicates that you know the factor of safety can be defined from the force point of view or limit equilibrium point of view or moment equilibrium point of view. Now let us try to review the different stability analysis methods. Basically all limit equilibrium methods utilize the more column expression to determine the shear strength tau f along the sliding surface. So prime of AC the more column expression is used to determine the shear strength tau f along this sliding surface and then as we defined in the previous slide the factor of safety is given by Su by tau which is for total stress conditions and for effective stress conditions F is equal to or Fs is equal to C dash plus sigma dash tan phi dash by tau. See basically the available shear strength tau f depends on the type of soil. So if it is say a particular type of soil and the effective normal stress whereas the mobilized shear stress tau depends upon the external forces acting on the soil mass that is the self weight of the soil which are called as geostatic conditions and as well as any external loading if it is there on the crest of the slope that also adds to the destabilizing or disturbing force. So here different methods are summarized here but we are going to discuss about some selected methods. We have many methods as you can see from this slide. The first method is ordinary method of slices and then it is followed by Bishop's simplified method is there. So in all these methods the zone or area within the failure surface is assumed to be divided into the number of slices basically vertical slices and then the slice equilibrium is considered either from the for the four circular point of view or moment equilibrium point of view and based on that the deductions for the factor of safety have been obtained. So for example in the case of ordinary method of slices the circular failure surface is assumed and moment equilibrium is considered and in this method the both normal forces acting inter-slice forces that means that normal forces acting on the perpendicular to the vertical face of the slide as well as the tangential forces are neglected. In the case of Bishop's simplified method it is for circular failure surfaces and whereas moment equilibrium is satisfied but it considers E that is the normal force perpendicular to the vertical surface of the slide but on both sides of the slice but neglects the tangential forces, the tangential forces are assumed to be 0. In the case of John Bush method it is mainly predominantly for non-circular failure surfaces and this also can be used for circular and non-circular failure surfaces but here this method is based on predominantly on the force equilibrium. So the moment equilibrium is not satisfied the force equilibrium is satisfied. So it considers the normal forces perpendicular to the inter-slice forces but it again like Bishop's method it neglects the tangential forces that is one of the components of the inter-slice forces. Then we have methods by Spencer method wherein it considers the T and U with some constant inclination where T is equal to a relationship between tangential force and the normal force E which was given as with a constant inclination T is equal to tan theta into E and then Morgan Stern's method wherein it can be used for both circular and non-circular failure surfaces and it satisfies both moment and force equilibriums and the assumption for T and E which is defined by a function X where T is equal to function X into a constant delta into E. So if you look into this the Sermas method and Morgan Stern's price method it satisfies both moment equilibrium and force equilibrium methods but the majority of the applications the Bishop's simplified method is used or to some extent the John Booth's method is also used. So this is the typical section of the unit width assumed for the analysis as the slope is assumed to be like a plane strain structure. So a unit width is considered for the analysis like a per meter width of the analysis is considered for doing a two dimensional stability analysis. Of course now there are the methods which are actually available for performing the three dimensional slope stability analysis. Now before discussing about the ordinary method of slices let us look into the evolution of different types of methods for undrained conditions and drained conditions and then we will introduce for the ordinary method of slices and Bishop's simplified method and the other method like Morgan Stern price method. In the case of circular arc analysis for undrained condition or it is also called as pi suffix u that is angle of internal friction undrained condition is equal to zero analysis. Basically this is for a analysis basically performed in terms of total stress analysis and applies to short term condition only for a cutting or embankment assuming the soil profile to come price of fully saturated clay. And the on the right hand side a cross section of a slope or an embankment which is actually shown here and this is the slope inclination which is required to be determined. Here what slope inclination you need to be provided so that the adequate factor of safety need to be ensured that is the point of the importance. Now here it is assumed that the potential failure surface which is indicated with this yellow broken line can be seen here and this is the undrained coefficient is assumed to be mobilized along the failure arc and W is the CG of this weight of the entire area which is subscribed in this zone A, B, C, D in this zone. So this requires determination of the, determination of area of this portion and then by knowing the area into the one meter length perpendicular to plane of the spigar we can calculate by knowing also the unit weight of the soil for use for embankment or unit weight of soil in the cutting we can determine what is the weight and with respect to the moment of rotation which is considered we can also determine what is the horizontal distance from the center of rotation, horizontal distance that is D is nothing but the distance from the CG of the area where the weight W is acting to the center of rotation. So for this by taking moment equilibrium where by taking moment about the moment of all resisting forces about the center of rotation to moment of the all the driving forces about the center of rotation. So here there is no this W is because of the self weight of the soil so factor of safety is equal to MR by MD where MR is nothing but Cu into LA, LA is nothing but the length of the entire arc into that R, Cu LA into R is nothing but the resisting moment divided by WD, WD is nothing but the driving moment or the disturbing moment. So here one of the disadvantages of this method is that accuracy in which actually you will determine the weight W or area and the determination of D which is actually involved. So the calculation of factor of safety if you are having let us say an external load on the within the failure zone. If the external load which is either due to distributed load or due to if there is a distributed load then one need to consider the you know the if this is identified as the potential failure surface then in this zone the load is distributed over this length into the length which is perpendicular and at the CG of that load is actually located. But in this case let us assume that there is a boundary wall which is located at a certain distance where when we consider about this intensity of the load we can actually now take the disturbing moment is nothing but W2 into D2, W1 that is due to the self-heated soil into D1. So the resisting moment is nothing but still the Cu into L into R. So the factor of safety in this case is nothing but Cu LR divided by within brackets W1 D1 plus W2 D2. Now from the undrained analysis Taylor 1948 has developed a method wherein the potential failure surface is given so that the least factor of safety can be determined. So this development is sourced from the undrained analysis which is which we have discussed just now wherein here it is actually considered that this entire the soil portion below this depth is considered as D and then this height is say H and here this is the slope inclination which is beta and beta is equal to 90 degrees means it is a vertical cut and this inclination is 2 theta and this inclination with horizontal is alpha and this is the cohesion which is actually mobilized and this is the weight and then this is the D from the which is horizontal distance measured from the CG of the weight from the weight of this entire portion from the center of rotation. So D is nothing but the depth factor what we call which is the nomenclature usual for developing stability charts wherein here the factor of safety we know that we have just discussed in the previous slide where factor of safety is equal to C L R W by W D where factor of safety is the lowest factor of safety obtained from the circular arc analysis and the factor of safety or the self-weight of the area or portion involved in the failure which is the active zone what it is called is function of gamma H and geometry of the failure surface. The geometry of failure surface can be characterized by three angles alpha beta and theta and so what it has been taken is that the by rearranging the terms here C by factor of safety is written and which is written as C R is equal to gamma H into function of alpha beta and theta the CR is nothing but the required cohesion of a soil just to maintain a stable slope and function of alpha beta and theta is a pure number and basically that is actually designated as a stability number. So the Taylor's stability number is actually given by CR is equal to CR by gamma H. So CR is nothing but required cohesion to just maintain a stable slope otherwise it can also be written as C by gamma H by FS into CR when you substitute for C by FS we can write by C by FS into gamma H. So for a factor of safety is equal to 1 it actually induces that NS is equal to C by gamma H. So the required cohesion CR is nothing but the required cohesion to just maintain a stable slope. So here with reference to angles alpha beta theta what has been done is that the weight portion which is actually considered that is the weight which is subscribed as the function of gamma H and geometry of failure surface are considered in the form here in the form as a to represent the geometry of the failure surface. So based on that deliberations the Taylor actually has given stability charts which are known as popularly known as Taylor's curves wherein on the x axis we have sloping inclination it ranges from 0 degrees to 90 degrees and on the y axis it is the stability number that is NS is equal to C by gamma H. So here it can be seen that one up to beta is equal to less than 53 degrees it is actually depend upon the the D which is nothing but this failure surface is assumed to pass through below the base but when we when the beta greater than 53 it is found to independent of D. So it is only depend upon the sloping inclination which is when it is more than 53 degrees. So it can be seen therefore more than this it it is the constant the stability number will be constant. So so here this is used for u is equal to 0 and mostly for undrained conditions this is used. Now the undrained analysis of the by using this Taylor's method basically for beta less than 53 degrees as we discussed in the previous slide the stability number is a function of beta and D by H. So that means that here there can be a possibility of the base failure for gentle slopes the critical failure surface goes below the toe and always restricted above the strong layer hence this depends upon the its location. For beta greater than 53 degrees when the sloping inclination is greater than 53 degrees the stability number is only function of sloping inclination that is beta all critical slip surfaces are passed through the toe. So this is basically because the for such steep slopes the critical failure surface passes to the toe of the slope and does not go below the toe. Let us assume that say for vertical cut beta is equal to 90 degrees NS is equal to 0.26 from the short term condition. So it can be seen here for beta is equal to 90 degrees the NS value is about 0.26. So when we calculate back when by substituting this particular expression wherein the critical height is nothing but where the factor of safety is equal to 1 the critical height is defined as a height at which the factor of safety tending to 1 by doing this we can actually get Hc as 3.85 C by gamma this is actually obtained from the Taylor stability number whereas one performs the conventional analysis by using at pressure 3d it is can be obtained as C is equal to 4 C by gamma which is nothing but by considering the earth pressure equation like sigma a is equal to k a gamma H minus 2 C root k a and wherein if you consider both negative pressures that is the depth of the tension crack that is where the negative pressure is existing and equivalent portion below the this particular zone and when you take the equilibrium of that we can actually get the 4 C by gamma that is nothing but the critical height or the critical height of a vertical cut which is actually called and this is the height at which by attaining this height the soil supposed to fail. So undrained analysis stability charts given by Taylor method the position of the critical surface may be limited by two factors one is the depth of the start term in which the sliding can occur and the possible distance from the toe of the rupture surface of the toe of the slope. So the possible distance from the toe of the failure surface from the toe of the slope that is nothing but from the Taylor actually has given charts for determining this particular value n, nH in terms of nH it has been given like this is you know the distance in terms of n multiplied by H, H is nothing but the slope height and this is the failure surface which is obtained for slopes which are actually less than 53 degrees also. So the stability numbers for homogeneous slopes for phi is equal to 0 say for example by knowing d by H that is the depth factor and beta the nS and n can be obtained from this chart. So here n is nothing but by knowing this we can see that where it actually cuts and that is the n and then by taking this horizontal projection out onto the stability number we can get the stability number nS. So for d by H is equal to 1 that is d by H is equal to 1 beta greater than 53 degrees the n is equal to 0 that means that the slope actually the failure surface passes through the toe of the slope. So and some more important points as far as the Taylor's method is concerned they are actually summarized in this slide it is necessary to ignore the possibility of the tension cracks otherwise geometrically similar failure surface do not occur on slopes having different heights. So the possibility of the tension cracks is ignored here and Taylor's stability numbers were determined from the analysis of total stress conditions only that is basically for undrained or short-term conditions only and total Taylor's method is practically restricted to problems involving undrained saturated glaze or too much common cases where the pore pressures is everywhere 0 that means that the pore water pressures are everywhere 0. In case if you are actually getting a tension crack let us assume that the tension crack which is can be determined let us say from the by using the earth pressure fundamentals where sigma a is equal to k a gamma h minus k a gamma z minus 2 c root k a when for a clay soil when phi is equal to 0 we can say that c is equal to c u and k is equal to 1 and at point where the pressure tends to become 0 we can say that z naught is equal to 2 c by 2 c u by gamma that is nothing but the depth of the tension crack. Now when we consider the tension crack and let us assume that if the tension crack is filled with water and it can actually cause a destabilizing force and the portion within this tension crack zone that is the circular arc which is actually passing beyond this point is not considered in the analysis that means that in the factor of safety determined by this particular method is nothing but c u into L ac from here to here that is the length of this arc into the radius divided by w into d in plus of gamma w z naught square that is this force into the lever arm that is say L the L is nothing but from this distance to the vertical distance from this from the horizontal vertical distance L is nothing but the vertical distance measured from the center of rotation to the location of the horizontal force Pw. So like this when we have the tension crack this has need to be accounted. Now let us look into the ordinary method of slices in this method the potential failure surface is assumed to be circular arc with center o and radius r. The soil mass basically the soil mass abcd above a trail surface ac is divided by vertical planes into a series of slices as we have discussed that one of the limitations of the undrained slope stability analysis with phi is equal to 0 analysis is that determination of area and then you know determination of the weight and then d. Here what has been done is that the portion within the soil mass abc which is undergoing failure surface is divided into number of slices. The number of slices depends upon the by convenience it is actually divided and these slices are divided such a way that they have some uniform horizontal distance not necessarily uniform but mostly the uniform horizontal distance and the base of each slice assume to be straight line and so the circular arc is assumed to be as a straight line and factor of safety is defined as the ratio of the available shear strength to the shear strength tow m which is which must be mobilized to maintain the condition of limiting equilibrium. So here what actually we can say is that the ordinary method of slices satisfies both the moment equilibrium for circular surface but neglects the inter slice normal that is here the normal forces on these slices and tangential forces so this is a typical slice so the free body diagram of a slice in case of ordinary method of slices where there is a vertical force which is shown that is this is the W is the weight of that a particular slice if there are n number of slices and if this is the ith number of slice then the weight of ith slice is Wi and this is the resisting force that is s and this is the normal force that is n dash so it satisfies moment equilibrium condition and neglects inter slice normal and shear forces normal and shear forces shear forces in this direction one acting downwards when acting upwards and gives the most most conservative factor of safety so this gives the most conservative factor of safety basically useful for demonstration. So ordinary method of slices the moment equilibrium for slip surface but neglects both inter slice and normal shear forces the advantage of this method if you look into it is simple in solving the factor of safety since the equation doesn't require any iteration process. So this is described in the form of a figure which is actually shown here where this is a typical slope and this is ABC this is the failure surface which is assumed to be a one of the potential failure surfaces there can be innumerable number of failure surfaces and but ABCD in this particular case is assumed as a ABC is assumed as a failure surface and D is the point of the crest and O is the center of rotation and R is the radius of rotation and if you divide the slices of certain horizontal length let us say is B and if you consider the free body diagram of this particular this thing and it actually has the normal forces e1 and e2 both are different and x1 and x2 the shear forces when we consider when you don't consider these forces then e1 minus e2 is equal to 0 x1 minus x2 is equal to 0 and then here this e1 is equal to 0 and e2 is equal to 0 x1 is equal to 0 and x2 is equal to 0 and this is the you know from the when the vertical this is the weight is actually assumed to be weight of each slice is assumed to act at the center of this slice of width B horizontal distance and the arc is assumed to be as a straight line here and this is the tangential force and n dash is acting on the base of the slice and UL is actually acting a pore product pressure acting on the base of the slice so n this is n dash UL which actually gives this normal force and this angle which is subtended from the this n dash directly extends to the center of rotation that means you can see here and this angle is called as the alpha so as we traverse from this side to this side the angle alpha changes that can be noted from here. So the method of slices from the limited equilibrium condition point of view it can be given as tau m tau factor safety as tau f by tau m the factor safety is taken to be the same for each slice implying that there must be mutual support between the slides and force must act between the slices and the total weight of the slice W is equal to gamma bh so if you consider the slice of width B and the height is actually measured at the center that is this height which is actually regarded as the height of the slice this is the height of the slice at the center where weight is actually acting and then the weight is equal to gamma into B into H and if so let us assume that in determination of this we are having in a given slice there are three layers of soils then we have to take gamma 1 H 1 gamma 2 H 2 gamma 2 gamma 3 H 3 into B the V is nothing but the width of the slice the total normal force n is equal to sigma into L sigma is the normal stress acting over the length L into perpendicular to plane of that figure what we consider is 1 meter so it is the force is nothing but sigma into L into 1 so this includes n dash is equal to sigma dash L and U is equal to U L and U is the pore water pressure at the center of the base and L is the length of the base and the shear force on the base is nothing but T is equal to tau m into L the tau m into L the L is the length along the the failure surface in a given slice of having width B the total normal forces on slice even in E 2 and the shear forces on the slice x1 and x2 so the method of slices considering the moment about O the sum of the moments of the shear force T on the failure arc AC must be equivalent to the moment of the weight of the soil mass ABCD so this we can write as sigma T into R is equal to sigma WR sin alpha so by writing by using T is equal to tau m into L is equal to tau m is equal to tau f by factor of safety in L and we can write sigma TR as sigma tau f by factor of safety in L is equal to sigma W sin alpha so factor safety is given by by rearranging these terms we can write factor safety is equal to tau f into L divided by which are sigma that is nothing but the resisting resisting portion divided by the the driving that is disturbing one W sin alpha so factor safety is nothing but sigma tau f into L divided by sigma W sin alpha this is from the limited equilibrium condition from for an analysis in terms of effective stress we can write factor safety is equal to sigma C plus sigma dash tan phi into L so by writing sigma dash into L as N dash we can write C dash into LA plus tan phi dash into sigma N dash divided by W sin alpha so equation one is exact but approximations are introduced in determining the forces N dash so in factor safety is equal to C dash into LA plus tan phi dash in case if you are having a cohesion less soil slope then we get to factor safety is nothing but when C dash is equal to 0 the first term will get cancelled and where we have tan phi dash into sigma dash N dash divided by W sin alpha so this can be used for both undrained the short term stability as well as for the long term stability by substituting the relevant characteristic strength parameters. The felilius or Swedish solution is actually given it is assumed that for each slice the resultant of the interslice forces is 0 and the solution involves resolving the forces on each slice normal to the base where here we give N dash is equal to W cos alpha minus U alpha and this is actually given by rewriting the equation one which is shown in the previous slide as this particular one C dash LA plus tan phi dash into sigma dash N dash where N which is actually rewritten this equation in terms of so this is nothing but C dash LA plus tan phi dash into sigma W cos alpha minus UL divided by W sin alpha sigma. So this expression is actually or this with this modification this is called as the felilius or Swedish method of slices both are ordinary method of slices and felilius method of sizes one and the same but this is the minor difference which is actually there C dash into LA plus tan phi dash into sigma of W cos alpha minus UL divided by W sin alpha which is summation. So the how we can actually do the analysis by using the felilius method of slices means in a given portion we need to assume the potential failure surface and then divide the slices of the horizontal having horizontal width and then number the slices like the numbering is done from one order 1 2 3 4 5 6 and with a given center of rotation and for a given potential surface this is the entry point and this is the exit point and where in from the each center of the slices is actually identified and the angles alpha 1 alpha 2 alpha 3 alpha 4 alpha 5 alpha 7 are determined. So for an analysis in terms of total stress the parameters C U and phi U can be used and the expression will be like this and for a phi U is equal to 0 analysis which is nothing but factor safety nothing but C U LA divided by W sin alpha. So this particular Swedish method of slices was extended by Bishop in this solution basically assumed that the resultant forces on the slides of slices are horizontal where x1 minus x2 is equal to 0 is considered. So the resultant forces on the sides of the slices are horizontal and assumed that these x1 minus x2 is equal to 0 by with this assumption the equilibrium of the shear force on the base of any slice is given as T is equal to 1 by factor safety into C dash into L plus n dash tan phi dash. So resolving the forces in vertical direction we can get W is equal to n dash cos alpha plus UL cos alpha plus C dash into L sin alpha by factor safety plus n dash tan phi dash sin alpha by factor safety after some rearrangement and using L is equal to b secant alpha because that inclination is alpha and horizontal distance is b L is the length along the straight line length along the slice. So by simplifying we get the factor safety expression as 1 by sigma of W sin alpha in a summation C dash b plus W minus UB within brackets tan phi dash into secant alpha divided by 1 plus tan alpha tan phi dash by factor safety. So this expression wherein you can notice that the factor of safety is existent in the both the methods. So the generally how it is done is that this is done by iteration methods and the factor safety which is obtained by performing some couple of iterations will yield to a factor safety. So for this here what it is done is that the factor safety is determined first primarily by Swedish method of slices and that is used as the initial value and then the number of iterations are specified or performed based on the logic which is actually set in the software which is actually used for determining this or manually with a couple of iterations it can be determined by using this expression. In Bishop 1955 it is simplified method of slices he showed that how non-zero values of the resultant x1 minus x2 could be introduced into analysis but refinement is only a marginal effect on the factor safety. So the pore water pressure also can be related to total field pressure where it is given as the pore water pressure coefficient which is nothing but U by gamma H, U is nothing but the pore water pressure at any point divided by gamma H, the H is the let us say the height of the slice. So it is pore water pressure is defined in terms of a total field pressure at any point by means of dimensionless pore water pressure ratio which is called as RU. When RU is equal to 0.5 it is said that the slope is completely saturated and for any slice RU is equal to U by W by B by rewriting this one in the previous equation what we have given we can write or express factor safety as 1 by sigma W sin alpha into sigma C dash B plus W into 1 minus RU tan phi dash into secant alpha 1 plus tan alpha into tan phi by factor safety. So in the Bishop simplified method as is actually shown here it satisfies the equilibrium of the factor of safety and satisfy the vertical force equilibrium for n and moment equilibrium for factor of safety. By determining factor of safety the moment equilibrium is satisfied and vertical force equilibrium is satisfied with respect to n and consider the inter slice forces even in E2 and more common in practice applied mostly for circular failure surfaces. The salient features are actually given here and this is the typical free body diagram of a slice in case of Bishop simplified method where the weight of the slice and the shear strength which is actually mobilized along the value surface and dash is the normal force acting on the failure surface normal to the failure surface and even in E2 are the inter slice forces on a particular slice force. So the Bishop simplified method consider the inter slice normal forces but neglects the inter slice forces it further satisfies the vertical force equilibrium to determine the effective base normal force n dash and further we also said that the John Booce simplified method. In John Booce simplified method is basically based on a complete slip surface and the factor of safety is determined by the horizontal force equilibrium. Basically here both horizontal and vertical force equilibrium are satisfied and it does not satisfy the moment equilibrium. John Booce simplified method does not satisfy the moment equilibrium and considers the inter slice forces even in E2 like Bishop only and it is commonly used for composite shear surfaces. So basically for layered soils or stratified soils or when you are having a non-homogeneous soils the John Booce method is advocated for its use and basically it is used for composite slip surfaces or non-circular slip surfaces and the factor of safety is determined by horizontal force equilibrium only and moment equilibrium is not satisfied and in this slide the free body diagram of a slice which is actually considered in the John Booce simplified method is shown wherein which is actually very similar to simplified method wherein really the difference is that it does not satisfy moment equilibrium only the force equilibrium that is vertical and horizontal are satisfied. So in this lecture we try to introduce ourselves to different failure surfaces and we will also look into different types of methods which are actually used for analysis. We have seen the phi is equal to u0 method and Taylor's stability chart method which is the deduction of from the extension of unrained slope analysis method and from there we also discussed about the ordinary method of slices and felonious method of slices and then Bishop's simplified method which is mostly used for commonly used for circular failure surfaces and for composite slip surfaces or non-circular slip surfaces in stratified soils the John Booce simplified method is also introduced. So in the next lecture what we do is that we will try to look into some examples wherein we can actually see how the problems can be solved by using typical calculations with the manual calculations as well as the in this particularly we will actually try to see some demonstrate some problems by using some relevant packages for academic purposes.