 So, let us start discussion on the circular failure surface. So, historically the circular failure surfaces are basically they depict the rotational failure of the slopes alright. So, these are this is the planar. Now, we are going to talk about a rotational failure. So, this is the slope and the failure is going to be along this surface, which is a part of circle or it could be a circle depending upon the material properties. So, the genesis of this analysis is 1916 is so old. Mostly this is subscribed to the people who are working in the sudden and Swedish group that is why we know we call this method also as the Swedish circle method by Pettersen ok. So, this is attributed to the Swedish geotechnical society this method. These are the societies of different countries like in India we have Indian geotechnical society which monitors and controls the activities of geotechnical engineering professions professionals in the country and the profession also. The basic philosophy here is now if I consider a failure surface like this a b c what are the governing forces on this block a b c one is the weight another one is the cohesion which is acting all along the surface. So, that means the cohesion which is acting on the surface b c all right. We show this direction as c in the form of force. So, I will use a capital C here as the force the direction of c is always taken as parallel to the chord b c this is the arc and this is the chord. So, the direction of c is assumed to be parallel to b c. So, one unknown is less the direction is known magnitude is not known. Now, suppose if I ask you to complete the triangle force triangle we have c prime and we have w what would be the third force the reaction which is acting at the base now there is something very interesting. Suppose w decreases or increases what is going to happen. So, basically w directly is proportional to the c prime or the c prime is what is c prime the cohesion which is getting mobilized on the surface fine. So, that means I can always say that w controls the magnitude of c the direction is known lower the value of w lower c is required. So, you remember what we did what we did when we were discussing about the tension cracks which are occurring in pure cohesive materials. This is the tension crack let us say from the lateral side there is no force which is coming on the block this block that means w is going to be balanced by the cohesion which is getting mobilized on the surface. So, if w tends to 0 c also tends to 0 correct if w tends to 0 c also tends to 0 that means a smaller the weight a smaller amount of cohesion is required or gets mobilized to stabilize the whole thing. So, what we can say is c upon w is a term which remains a constant any idea why we are using the term c upon w look at the fact stability number. So, stability number is defined as c over gamma h gamma h is associated with the weight of the destabilized block alright. So, can I write this expression as c m into length and what is length? Length itself is a function of h height of the slope and when you say c m what will be the c m value c m will become c u over factor of safety associated with cohesion because this becomes c m. So, c m is getting mobilized along the slip surface which is the function of h, h is the height of the total slope and what is w the w is a function of gamma h square. So, that means what we are going to get is a function I said. So, that means I am going to get stability number as what is the stability number c u over factor of safety against cohesion multiplied by gamma into h that is it. Incidentally this c m is also same as what we have written over there the interpretation was different the c mobilized is going to be the total cohesion under undrained conditions divided by the factor of safety why undrained conditions because the failures are going to be very quick instantaneous failures undrained conditions alright when these are undrained conditions we have to follow total stress analysis I am sure you are realizing how to interpret c pi parameter which we have got from different types of triaxial test and then how to include them in totality to depict a situation of failure normally these type of problems are defined as the short term stability problems alright this is part clear. So, what we are trying to say is c u upon f c into gamma h is a constant which is known as stability number for undrained situations where you have pure cohesive failures. Now yeah one more thing which you can maybe would like to understand is that this function will be equal to c m into length of the cord along which it is acting. So, length of the BC as a cord alright. So, this is the cord this hypothesis is attributed to Taylor name of the person who has proposed this and we call them as Taylor's stability chart first of all they will be applicable for total stress condition and truly speaking these charts have been derived based on the friction circle method which I will be talking about later on. You have done this friction bearings in your engine mechanics course is another application of this concept. So, by definition the way the Taylor's chart have been defined we use the term NS this stability number is defined as NS and this is normally written as equal to c u over f into gamma h. It is understood that f is basically associated with the factor of safety of cohesion and we take this f as the minimum value. So, that the factor of safety term or the stability number gets maximized as per Taylor this NS is a function of the slope angle beta and the phi u value. So, the stability charts which will be talking about they look like you know a relationship between this is the factor of safety or the stability number this is the beta value these are the values of f phi u which direction phi u will change suppose friction angle increases from top to bottom or bottom to top which one is correct that you have to think of. So, suppose friction angle is 0 is the top most line or the bottom most line for the same value of beta which slope is more stable stability number yes that means even the steeper slopes can stabilize can stand alone without any support if the friction angle is more. If the friction angle is less you require less steeper slopes. So, for phi u equal to 0 there is a special condition we will be discussing about this and normally the phi u increases in this direction and for phi u case it would depend upon there is something known as depth factor. So, suppose if this is a slope and this is a hard strata height of the slope is h inclination of the slope is beta one of the failure mechanisms which is going to be most critical would be you know the failure like this a slip surface which is circular particularly pure cohesive soils and if this is h we define this as d into h the depth factor. So, d is the depth factor I will show you these graphs so that you can use them for doing the analysis. So, these stability charts were proposed in 1937 by Taylor and how do they look like I am going to project it over here yes. So, this is the first stability chart you know there is a embankment or there is a slope and this is a critical circle which is passing through the toe of the slope there could be different cases the slip surface may not pass through the toe which is case one there could be an out crop. So, what you are observing here is the slip surface passes in such a manner through the foundation of the embankment or base of the embankment or the slope and then there is some out crop. So, this out crop is depicted as nx into h where h is the height of the slope beta is the phase value or phase angle. So, there are three cases the first case is passing through the toe second is touching the hard strata intersecting the phase of the slope this is what is known as phase failure this is what is known as toe failure and the case two is depicted as the base failure for the same height edge. Now, stability numbers are defined based on zone A and zone B sort of analysis alright. So, if you look at this line which is the dotted line which is passing like this is starting from about 25 degree the right hand side zone is zone A left hand side is zone B zone A is the critical circle passing through the toe and zone B has three cases the case one which we discussed critical circle through toe full line case two critical circle below the toe as the dotted line and the third is case three where we have a very strong strata what I have depicted there as d into h which again is the dashed line alright x axis is slope angle y axis is stability number ns the friction angle under undrained condition increases from top to bottom and for the first case you will realize you know it depends upon the depth factor. So, as the depth factor increases what is going to happen if the if the depth factor increases in this graph for phi equal to 0 you will have different lines in this direction the depth factor will increase. So, let me define this as depth factor there are some more cases of stability number which have been discussed over here phi prime equal to 0 corresponds to undrained or the total stress analysis depth factor is here treated as ND which I have taken as df into h will be the total d of the or what we call it as the depth of the deepest point of the slip surface you have the stability number we have ND value and then for Nx we can use the dotted lines for a given slope angle and then we can compute the factor of safety. So, you can use these graphs for analyzing the stability number for different types of slope conditions. Now, we will solve one problem to showcase to you how the analysis is done. So, one example problem would be suppose if I take a 60 degree sloping surface in embankment 6.5 meter the soil properties are 18 kilo Newton per meter cube 28 degree and C is equal to 20 kPa. Find the factor of safety with respect to strength shear strength and use Taylor's chart suppose for the sake of simplicity alright subsequently you will have to understand which charts to be used. So, depending upon the material property you might have to apply your intelligentsia to select the right chart and go ahead with the analysis. The general principle of analysis is like this that we always assume you know a value of F5 say 1.5, 1.6 this is a starting point. So, once you assume the value of F5 you can compute phi m yes tan phi prime tan inverse tan of 28 degree divided by 1.6. Now this comes out to be approximately 18.4. So, what we have obtained is we have obtained the value of friction angle which is getting mobilized out of 28 degree friction angle which is available the mobilized value is only 18.4. Now what should be done we can use Taylor's chart is beta known yes what else is required phi m we have obtained can you interpolate between the two lines beta is known here on this graph. So, beta is known for 60 degree go up 18.4 you have to come somewhere here can you obtain the stability number n is equal to 0.1 007 I mean you have to be very careful while you are using the numbers and usually you should go up to the 4 decimal place. So, it is always better to use the empirical relationships analytical solutions substitute the values and obtain it rather than seeing the graphs. But nowadays you need not to bother much because most of the softwares which are commercially available include these parameters which are inbuilt alright. So, if I know the value of n which is equal to Cu over fc into gamma H. So, what I can obtain from here I can obtain the value of Cu which is nothing but cm the mobilized value. So, this will be equal to 0.1 007 into gamma is 18 into H, H which is 6.5. So, this turns out to be 11.78 kPa what will the factor of safety for fm the total value of C is known as is this correct. So, this fm is basically fc factor of safety for cohesion and this is mobilized. So, this will be equal to 20 over 11.78 and this comes out to be 1.698 what this indicates this indicates that the starting point of f5 as 1.6 is not equal to fc. But it was not a bad assumption we started with f5 as 1.6 and we landed up with fc value as 1.698 what I should be doing then of course this depends upon the value of f5 which you assume to start with. Now, suppose if you assume f5 as 1.5 you might have to go for 2, 3 iterations to satisfy this condition that f5 is equal to fc. So, until this condition is obtained we keep on doing iterations. One of the ways to plot this is one of the ways to optimize this would be if I plot f5 versus fc ideally these 2 values have to be same correct. So, you keep on assuming the value of f5 and compute fc you get a point over here. The second trial what we will have to do is 1.698 and 1.6. So, I will do this whole analysis by assuming f5 equal to 1.65 let us say. So, from 1.65 what is the value of fc we are going to compute. So, you keep on computing this and wherever this 45 degree line is cutting this curve this is where fc is equal to f5. So, this was the first trial which we did this whole exercise is to be repeated until or unless your f5 becomes equal to fc. This is what in the most simplest form the application of Taylor's method is. Now, if you solve this problem you will be getting the right answer as 1.671 please try this yourself.