 I have discussed about slope stability analysis and I was talking about two situations, one is the infinite slopes where we did lot of analysis for different types of infinite slope conditions, then the second situation was finite slopes and I talked a bit about in my previous module how to analyze the slopes which are finite in nature. And this is where I introduce the concept of factor of safety, you must have noticed and that factor of safety we had defined as factor of safety associated with the friction angle and the factor of safety associated with the cohesion and we have denoted this as F phi and F c. And the problem which I solved in the class I showed you that the best possible optimal solution would be when F c is equal to F phi and that becomes the condition where you have the equal weightage for the cohesion and friction to get mobilized for the soil mass. Another interesting example of this concept of mobilization of cohesion and friction is what is known as friction circle method. So, I will introduce today the concept of friction circle and basically friction circle method is applicable to the cohesive soils, purely cohesive soils you know we call it again as a total stress analysis. It is understood that the phi u is equal to 0, undrained cohesion is equal to 0, but then we will extend this concept to a general C phi soil. We assume that the failure surface is a circle, part of circle alright, we define this as arc of the circle. I introduce this concept of arc and the chord in the previous lecture also, you remember we have shown the mobilization of cohesion along the chord of the slope. So, I will explain it again today. So, in short suppose you have a slope here, this is the ground surface, this is although the ground surface and we have been talking about different modes of failures of this slope alright. So, there could be a situation where the slope might fail like this, typical phase failure alright. The failure is of this block, there could be a situation where the failure might occur like this, the slip surface is passing through the base of the slope and there could be even deep seated failures also alright. Now I am sure you will realize that these type of situations are going to be valid for higher C values. So, that means there could be a transition from 1, 2 to 3 as C increases. There is another interesting situation when we talk about the slope stability of finite slopes could be you know many times people ask you a question, suppose if I rest a foundation over here, there is some facility which I want to construct. First question as a designer which comes to the mind is whether this is going to be a slope instability problem or whether this is going to be a bearing capacity problem related to the foundation of the system alright. So, a good designer would be checking for both the situations depending upon what what is the critical criteria which would differentiate between a shear failure which could be for the bearing resistance or which could be for the slope instability. So, suppose if B is the foundation with and if this is located from the edge of the slope at a distance of x. So, what is going to happen if you keep on decreasing the value of x the foundation comes and sits at the edge of the slope one situation and what is going to happen then we assume semi infinite soil mass, but then you are defining that. So, that means if I say that this is a hypothetical surface and this is a continuum if this foundation is going to sit over here what is going to happen I have removed this much portion. So, only this portion is contributing it is not going to be a bearing capacity problem the slope is going to be unstable because of this type of situation. So, that means x becomes a very critical parameter. However, if x keeps on increasing it becomes 3 to 5 to 10 times to 20 times the value of B what is going to happen this keeps on shifting on the right hand side and this becomes a typical bearing capacity problem fine. Now extending this surface I want to analyze let us say the stability of the slope because foundation designs and analysis you will be doing in another course not in this course this is a introductory course for soil mechanics 2 or geotechnical engineering 2 where we talk about the properties of the soil mass and how these properties particularly the shear strength parameters are utilized for designing the systems like this. So, I am going to restrict my discussion only on the stability of the slopes which happens to be the topic for discussion ok. So, we are assuming that this is the typical slip surface circular it might be passing through any of the points either 1, 2 or 3 in general what we do is we normally use graphical methods and we define a point or the axis of rotation. So, point of rotation means this is the point about which the slip is going to take place look at the motion of my hands. Now this is the point about which the material is going to slip off ok have you come across this problem somewhere in 10 plus 2 physics yes your J e yes correct yeah have you come across this situation there is a bowl and in this bowl there is a small ball simple harmonic motion yes that is right. So, suppose if I drop a ball like this and this ball would be rolling down and then following a simple harmonic motion and then what happens stops over here similar problem see physics of the material remains same the material changes this material and the ball material is different than what we are discussing over here the mechanism remains same ok. So, when this is the axis of rotation I can complete the figure like this. So, this becomes the included angle ok. So, this angle is normally taken as beta angle rest of the things are same I might be having a slope of height h you know this angle is some you may say some theta normally we take beta you are right, but in this case it does not matter ok. What are the forces which are going to act the weight of the block and what is this block A B and C this B is the width of the footing yes you are right. So, this is the width of the footing. So, this is the point C through which the slip surface initiates passes through the A point alright same thing I can do for any other situations also 1 2 and 3. So, in practice you know the chances are the failure might take place through any of these surfaces you never know correct. So, most of the slope stability analysis problems become iterative in nature that means, you select a slip surface follow the procedure which I am going to discuss over here alright and get the factor of safety terms I did this analysis in the previous discussion you remember when we were talking about the Taylor's chart Taylor stability chart correct that was also 5 equal to 0 total stress analysis. So, I might be having several situations like this you know 1000s 100s of slip surfaces very closely spaced and what is going to change what is going to change is only the center of rotation correct. So, if you plot this factor of safety associated with this point of rotation for which there is a unique slip surface what I am going to get I am going to get a plane in which you know you will be having different values of factor of safety associated with different slip surfaces. So, your generation is very lucky that you need not to do these things or these analysis manually, but when we were students we used to take some 10 slip surfaces we used to analyze them by following the method which I am going to discuss in today's lecture. Now, there is everything is software based they are very good very potential software which are available in the market I will be talking about that also, but before I do that just let me give you some concepts and try to follow the concepts. So, that you can use the software which are available commercially in a better manner. So, one of these slip surfaces I have selected over here as AC alright. So, in simple words or if I want to show this, this is how it will look like. So, this is A, B, C and what we did is last time we considered this as the straight line I am sorry for AC happens to be the chord and the curved AC we differentiate it like this happens to be the arc this is the axis of rotation. So, what is going to change if I take another slip surface just now I showed the axis of rotation in two dimensional plane is given by x, z or x, y. So, its location will change and beta will change yes. So, you are right. So, basically the slip surface is a function of the type of material yes and what else axis of rotation its location basically. So, this is the location of axis of rotation R this is R alright what else geometry of the slope theta H and of course the included angle beta fine. So, these are the parameters which would influence the location of the slip surface. Now, what is going to happen on the slip surface? So, suppose if I take an element over here of let us say very small length C and what I have done I have discretized the entire arc in small small segments which are linear in nature linear means these are lines yes. So, circle can be considered to be constituted of several infinitesimal sections of linear sections. What are the forces which are acting on this try to draw the free body diagram? So, at this surface which is let us say A and B what are the forces which are acting on this surface yes. So, the one is going to be C m into L correct what about the next force normal stress and when there is a normal stress what else is going to come there will be a shear stress now what should I write here phi or something else this cannot be the total friction angle which is getting mobilized this is going to be phi m that is right. So, this angle is going to be phi m force we have what is that we are assuming we are assuming that only certain fraction of cohesion and friction is getting mobilized in the system. So, you remember the factor of safety term if you have talked about. So, suppose if I say tau failure upon factor of safety is what this will be equal to tau mobilized. So, this will be equal to Cu undrained shear strength total stress analysis correct plus sigma tan phi u yes upon factor of safety f. So, this can be written as Cu upon f plus sigma tan phi over f this f is normally denoted as f c this f is normally denoted as f phi correct and what did we assume last time f c is equal to f phi equal to factor of safety for optimal solution is this fine. So, this becomes C m plus sigma tan of phi m. So, what is the relationship between tan phi m and tan phi u phi m tan phi m equal to tan phi over f phi. So, this becomes tan of phi m. Now, this C m is getting mobilized in the form of C m at a very infinite S m l length of the arc and this friction angle phi m is the included angle between sigma and the shear stress which is getting mobilized C m is coming because of the component of the cohesion which is getting mobilized on that length particularly. So, this is the basics of the whole thing. Now, what we do is we assume that the total cohesion is getting mobilized somewhere at a distance of this is the friction circle method let us say capital R not capital R I will say R 1 capital R will be reaction which I will be using subsequently ok. Now, this is the force component and what we are saying is or assuming is that C is acting parallel to AC. So, C force is parallel to AC is this part ok and what will be the value of C this will be equal to C m into yes LC capital LC yes. So, LC is equal to what AB and this is the chord. So, that is how we are depicting it. So, what we assume is that the total cohesion is getting mobilized and its direction is known and the magnitude is going to be equal to this fine and it is acting at a perpendicular distance of R 1. Can I obtain now a relationship between R 1 and R and you know AC AC and AC chord R suppose if I take moment about this axis what is going to happen C into R 1 capital C into R 1 will be equal to all the C m LCs are going to get summed up length of AC is this fine. So, that means C value I can substitute as C m into length of chord. So, length of chord I am writing is as AC into R 1 this is equal to sigma C m into LC into AC prime or AC R. So, C m gets cancelled out because C m is constant. So, what is the value of R 1 you are getting all this summation of small small chords is going to be what is the value yes compute it. So, no sorry this is going to be C m into I have not taken the moment. So, you have to take the moment about this also multiplied by R value alright. So, this is R. So, this becomes what AC R divided by AC into R. So, that means by using this concept we have obtained the point of application of capital C also hope this part is clear is this ok. So, this R 1 is nothing but the point of application capital C AC chord R can be obtained AC chord is known R is known I can obtain R 1 normally this type of analysis is done on a piece of graph paper. So, when you are plotting a section for a given slip surface what you know is the weight agreed draw the force triangle yes. So, force triangle would be W what are the attributes of W direction is known magnitude is also known correct very good. So, direction is known W direction is known and magnitude is also known what else is required to complete the force diagram the second component is capital C you are right. So, capital C is direction is known magnitude is unknown why I do not know how much cohesion is getting mobilized that is the reason what is the third force which is going to balance the two the reaction yes is the magnitude of R is known direction is known that we have to find out. So, what is going to happen W direction is known magnitude is known C direction is known but magnitude is unknown R we have to obtain its direction should be known and its magnitude should be known. So, as if what you are doing is you are you know plotting let us say W the direction is known magnitude is not known. So, if I superimpose on this the resultant force with the known direction and magnitude the triangle can be completed is this fine how will you obtain now R that is a big question now this is what actually friction circle method talks about. So, friction circle method talks about this in such a sense that application of capital R first of all it is going to be tangent to a circle which is known as friction circle alright. So, suppose if I draw a circle of radius R m this component of R is going to be you know tangent and passing through this in such a manner that if you draw a perpendicular from on any point it is going to pass through the center. So, if this becomes my normal stress if this becomes the normal stress what is the angle of inclination between R and n phi m ok. Now what we assume is that R m is equal to some multiplier multiplied by R sin of phi m. So, this multiplier actually comes in the form of a constant. Now what is this constant? There are charts which are available to obtain the value of k normally k is dependent upon beta and this relationship is something like this you know exponentially increasing starting from 20 degree onwards. So, you can refer to a book and you can obtain the value of k and k is normally placed between 1 to 1.2 alright and beta could be 120 degree also. So, we will obtain k value from here multiplied over here R is known for a geometry phi m what you will be doing with phi m remember phi is always assumed we assumed in the previous analysis also what we did in the previous module or the lecture. So, we assume f phi and f phi you know how to obtain it and then you get f c f c value and see whether both of them are same or not. This method is known as friction circle method I know there is lot of clutter on the board, but then if you follow these steps you can still understand this. So, just to repeat what we have done is we have made a force triangle w direction magnitude are known c we have assumed to be parallel to the chord ac its magnitude is going to be c mobilized into lc lc is known but cm is not known. So, for cm we have to obtain cm that means knowing the direction of c we have to know the magnitude r can be obtained and once the r is obtained then how will be obtaining us yes a good question. So, we have to do like this f phi the moment phi m is known what you will be doing you can draw r value which is passing through this drop a perpendicular from this point complete this friction circle get the value of r m r m will be equal to k into r sin phi m k we will obtain from here r m is known correct and then once the r m is known can you not obtain the value of r capital R we can obtain capital R also. So, this is how we normally do this analysis please follow a book to read the complete methodology I hope you can solve this. So, when you are doing this type of problems stick to the basics take a graph paper draw the geometry of the slope compute the area of cross section of the slope and along with the slip surface get the point o get the w and rest is all known to you sometimes this angle beta is also defined as the central angle. So, what will the relationship between central angle and the length of arc that you know is this not. So, 2 pi r upon 360 multiplied by the central angle will give you the arc length that is it. So, you have to just maybe analyze this type of situation. So, in other words weight is known what we can obtain is length of the arc is known length of the chord is known. So, this point is known you know r 1 point of application of r 1 r 1 is known. So, this is where the C is acting compute the value of C m and the moment you have C m f c will be equal to yes f c will be equal to total C over C mobilized and this C is nothing, but your undrained cohesion.