 I have been discussing about the slope instability, discussed a lot about the infinite slopes different conditions starting from drain conditions to the slopes having seepage you know in the downward direction, upward direction parallel to the slope under partial submergence and the complete submergence of mostly dry sandy materials which was extended to the these situations where the water table is present and we have done detailed analysis to find out the factor of safety. These are the infinite slopes alright, now in most of the situations which we as geotechnical engineers deal with the slopes are of finite nature. So now all words I will be discussing about the analysis of finite slopes, you remember we had talked about what is the difference between the infinite slopes and the finite slopes and there we had talked about the material properties as a function of depth vary, they do not remain constant, state of stress does not remain constant infinite slopes as compared to the infinite slope where they remain constant, peculiar finite slopes could be embankments which you are making by compacting the soil mass for railway tracks, for highways, roadways, runways and so on, runways are normally not done on the embankments. But the finite slopes could be because of excavations mostly. So here we talk about two situations, one is the planar failure surface and the second one would be circular yes circular failure surface. So let us begin with our discussion on planar failure surface, these type of situations occur due to excavations and excavations the stratified soil deposit, what is the second characteristic? This is going to be the weakest plane parallel to the strata and this could be a combination of 2, 3 planar surfaces also, may be a combination of 2, 2, 3 planar surfaces alright. One of the examples of this type of situation would be if I consider a slope this is a natural soil mass alright, this is of inclination I and the backfill here is of I prime, this happens to be the slip surface, the planar slip surface alright of height H, so height is taken up to the top most point over here. The inclination of the slip surface is assumed to be as what theta always, now my question to you would be draw the free body diagram of the forces which are going to act on this, the easiest possible way would be drop a perpendicular from A and designate this as H, now you can compute W1 easily, W2 can also become computed, so basically I am writing this as W, so W is known clear, what are the other forces which are going to act on this, given a chance the entire soil mass or the slope would have a tendency to slide down and this is what is being protected by C and phi which is getting mobilized in the system. So, this is the first time I am writing C and phi which are getting mobilized, mobilization means the entire C and phi of the material is not going to be used against instability alright, certain fraction of this is going to be used, that means if I say that C mobilized is equal to C over factor of safety associated with cohesion and similarly if I say phi m is the mobilized friction angle which is equal to tan inverse tan phi over f of phi, so I am using two factor of safeties, one is the factor of safety for cohesion, another one is the factor of safety for friction, ideally fc should be equal to f phi okay, so we are going to talk about these type of situations subsequently, just to begin with let us show the force of acting over here, so this is the component of Cm, now Cm is going to act on surface BC, so multiplied by length of the BC the surface on which it is acting okay, so this is BC, what are the other forces acting on the system, there has to be a reaction, so this is the reaction let us say and what is the relationship between the action and the perpendicular if I draw over here, this is going to be phi m, now this friction angle mobilized is what is that we have computed from here, so factor of safety can be obtained, factor of safety can be utilized to find out what fraction of cohesion and friction is getting mobilized, now once this is done we know the W's, W can be computed okay, the only thing which we have to keep in mind is I can utilize this H and basically this is a function of H correct and H can be related to capital H and what else, I s theta, I prime will not come to the picture, so ultimately what is that we want to prove, we want to prove that this theta critical value is the one where the failure is going to take place, so if you compute this W will be equal to half into gamma into H into LBC, this is the one equation which I can obtain and the second equation could be, is there relationship between H and capital H, think, so H is also related to capital H in what way, H can be written as AB into sin of I minus theta, is this correct, check it out, if this angle is I minus theta, so this H is equal to AB into sin of I minus theta and what is capital H, AB into sin I, so these two expressions are going to help us in finding out the weight, is this correct, so what we have done, now we can relate H to H as H upon H equal to sin I minus theta over sin I and this can be expressed as H equal to, BC is known, I can draw this section on a graph paper, I can find out the length, otherwise also I can compute the weight of the block by plotting it on a piece of graph paper, that can also be done by graphically or by analytically you can obtain it like this, fine, now further what we have to do, let us complete the force diagram, so how the force diagram will look like, yes try to complete this, we have a component of the weight and then we have a reaction and this Cm into LBC is the cohesion force which is getting mobilized on the plane and this then becomes the R value, is this angle known, what is the value of this angle, we have discussed several times active condition this minus phi m prove it or check it out whether it is correct or not and what about the other included angle between the Cm value and the weight, Cm and the weight, what is the included angle, this is theta, this is 90 minus theta, what about the third angle, I can use this relationship to obtain what relationship between W, this is of ABC which we have obtained, so WABC divided by sin 90 plus phi m equal to what is the principle unknown, how much Cm is getting mobilized, I do not know, so I have to obtain C value phi value, I do not know even factor of safety sometime, so I have to assume and then go ahead, so this will be equal to Cm into LBC divided by sin of theta minus phi m, what is that I am going to get from here, I am going to get the value of Cm from here, so once you have got the value of Cm, how are you going to use this, any idea, this Cm if I correlate it with the factor of safety or stability number both ways, I will prefer to use the stability number over here, so what will be the value of stability number, Cm over gamma H, this is okay, just quickly compute what is the factor which we are going to get, this will be equal to 1 by 2 because of half gamma H square which is coming over there, so this will be equal to yes sin of I minus theta into sin of theta minus phi m, this whole thing divided by sin I into cos of phi m, alright, so this is the expression which we have obtained, now what is that you want to do by obtaining this function, you want maximum stability or you want minimum stability both the ways I can try, so suppose if I want to maximize this function, so what I will have to do, I will have to differentiate this with what, what is the principle unknown here, the principle unknown is the theta value, so that means this has to be maximized with respect to theta, so if you differentiate what is that you are going to get, theta will be equal to I plus phi m divided by 2, prove this, this is okay and this theta is normally written as theta failure, that means this is where the failure is going to take place and at failure what we can do is we can also assume that the value of phi is equal to phi m, the most critical condition that means the factor of safety becomes 1, so theta f is the slope of the critical failure surface or critical slip line is basically a geometrical you know manipulation which has been used to obtain the stability number here, I might be having different types of combination of the planar surface, so here this is 1, I can create a situation where you have a slope like this which again goes further and so on okay, this could be the typical critical surface, so what I am going to add is 1 ABC followed by another ABC, if it is a multi-layered system what is going to change the value of Cm and phi m, so there are 2 ways of doing this analysis either you average out the C and phi for different layers by using their thickness or if you want to be very precise then the slip surface should be passing through the each section and then ultimately what you have to do, you have to do piecewise computation of Cm into LBC which is acting on the base followed by the weights, choice is yours.