 The slip circle method of analysis of finite slopes, I will be talking about now the slip circle method which is the simplest possible form of analysis of the slopes for finding out the stability. Slip circle method, simplest possible method of course total stress analysis undrained analysis ok. Two things are important to understand here. Saturated conditions of soils number one and this is also known as conventional method of analysis alright. Total stress analysis, saturated soil conditions, strength of the soil is supposed to be because of mobilization of cohesion only, pure cohesion. Skemptons method normally this is known as Skemptons 1948. These are very old classical methods which are still being followed by the geotechnical engineering fraternity. So, what we do here is we consider the slope, the axis of rotation we have to show and let us say this is the slip surface alright. So, this is let us say r is the radius. Suppose if I consider an element over here ok in such a manner that this is suppose this is the ith element or sometimes we also known as slice. The entire system has been considered to be made up of several slices alright. So, later on you will be studying the method of slices also. We are discretizing the entire slope in such a manner that the slope constitutes of several infinitesimal sized strips, slices, elements alright. So, suppose one of the elements which I consider of this slice is is slice is wi. The way we define this as suppose if I take a perpendicular from here the point of application of wi with respect to this is xi ok. So, if I connect this point from here this becomes alpha i for that particular surface. Now pure cohesion. So, this is the cohesion which is getting mobilized yes. So, this is the centroid you may say correct of the slice. Now there are two ways of defining the factor of safety. One is in the form of the forces which are acting on the system. So, the first form of the factor of safety is in terms of the forces sigma of resisting forces which are trying to resist the movement clear. What are those forces? So, the tendency of the slope is to slide like this fail like this. What is resisting? Cm mobilized shear strength. So, shear strength happens to be a resisting force divided by summation of disturbing forces yes. Sometimes we also call them as stabilizing forces. Disturbing forces would be destabilizing forces. Rest is all simple mechanics. The another way of defining this would be factor of safety in terms of the moments which you have been doing in your energy mechanics course. You take the moments of the forces and then try to see whether the body is in equilibrium or not. So, these class of problems are also known as limiting equilibrium problems. So, I can take the moments of these forces and then I can say sigma moments of resisting forces are divided by sigma of destabilizing forces or the disturbing forces both are correct. But normally we do not use the word destabilizing forces we say overturning moments correct. So, this MR will be yes what will be this MR resisting moment correct. Now this C is acting on that length in terms of the force. So, if I want to take the moment what will be the moment of resisting forces this will be equal to r into Cm into length and summation of all this. And what about your disturbing for moments the disturbing moment is coming because of the wi. So, that means md will be equal to wi into xi is this clear. So, what is the factor of safety this will be equal to r into sigma Cm into l divided by wi into xi. So, this will be equal to Cm into l summation sin remains there yeah because these are all small small segments which you are taking and you are integrating over or summing over. So, this will be equal to I am sorry this should be summation here. So, this will be equal to wi into xi and this r I can take down. So, what is the value of xi upon r this will be a correct. So, this is xi ok and then r term. So, what this term would be this would be a function of sin term sin of alpha i correct xi over r is going to be a function of a sort of a sin term. Now how are you going to do this this alpha can also be defined as an angle of inclination over here. So, is this ok. So, that means w and this line through which is passing through this will also be alpha i. So, this is one way of defining the whole thing. So, I have got the factor of safety term the beauty of this method is that even if suppose there is a layered system I have to just ensure that the slip surface passes through suppose this is soil 1 2 3 and so on. So, I have to just consider the slip surface let say like this I can go by parts first slip surface second slip surface third slip surface and so on and I can obtain the same relationship only thing is everything will get added up over there. In these methods you must have noticed we are not taking into account the influence of the forces which are acting from the sides of the slices what we have done we have just taken the weight we have mobilized the cohesion ok and we are ignoring the effect of side forces first as well as the side friction simplified method very simplified method that is why this is what is known as simplified method mostly valid for cohesive soils, but when you are dealing with the frictional material you cannot do so then you have to compute the forces which are acting on the sides little thrust you know shear forces which are acting on the system and so on and of course the normal stress will also come yes there could be another situation when you are dealing with pure cohesive soils what is going to happen what is the biggest problem tension cracks yes you are right. So, that means what is going to happen here is pure cohesive material we are talking about Cu case undrained analysis. So, what is going to happen here you have a component of you know this is the tension crack we have defined this as Z naught if you remember this is approximately taken as 1.3 times C over gamma depth of tension crack C by gamma multiplied by some factor 2 C by gamma is also correct, but then this is a more accurate value people have obtained this number by using the limit analysis that was my PhD thesis by the way ok. So, now what is going to happen we have done this type of analysis earlier also this is the shear surface the tension crack develops the length on which the cohesion is getting mobilized is reducing and hence the factor of safety would be decreasing or reducing or increasing if you think about it more than that the situation which is going to be critical is when the rains come what is going to happen you know this is all getting filled up with water and I think we have discussed this earlier also this is what is going to be the pressure from water during rains this is the weight suppose this is the axis of rotation now because of introduction of tension crack what has happened this is a destabilizing force shear strength is a stabilizing force what about the water pressure look at the direction of movement it is opposing the stabilizing force so basically water pressure also becomes a destabilizing force correct. So, what is going to happen your factor of safety terms are going to reduce so I can always do it like this that suppose this is the value of water pressure and I can assume this to be acting at a depth of let us say distance not depth distance of z value. So, this component will be p w into z which gets added up to destabilizing force fine this is another situation which we have created the third situation could be which is another interesting problem or situation could be you have partial submergence partial submergence means suppose at the downstream side the water is somewhere here so what water does this height of water is going to exert a pressure on the slope what is the direction of this pressure it is a destabilizing force or stabilizing force correct. So, the weight is acting like this cohesion was acting to stabilize this is destabilizing force fine this water pressure is now getting added up to the cohesion. So, what is going to happen because of the presence of the water table the stability is going to get enhanced is this part clear I am sure that now you will have to use your concept of finding out the weights of these two sections how are these two sections getting created one is above the water table and second is below the water table this is going to be submerged and this could be partially saturated variably saturated dry depending upon the material and the water table characteristics and the capillary action which we have talked about earlier. So, you have the component w you have w 1 let us say w 2 you will be having the lever arms from this point of rotation pressure and I can make it more complicated by tension crack also getting developed over here. So, there are three components now two blocks of weight w 1 w 2 cohesion not getting mobilized beyond the surface there could be a water pressure coming from this and this is more realistic situation. So, suppose if I keep on filling the water on the left hand side what is going to happen this stabilizing force is going to increase correct, but then the slope is going to be fully submerged analyze this situation is this part ok. So, this method you know which we were talking about here the simplest possible method is also known as Bishop's simplified method where we ignore the side forces and if I do not ignore the side forces this becomes the rigorous method Bishop's rigorous method which method will give you more factor of safety see rigorous method includes the influence of side forces clear. So, you are not nullifying them you are not ignoring them. So, factor of safety which you are going to get from rigorous method is going to be more than the factor of safety which you are going to get from simplified Bishop's method. Most of these methods are going to be trial and error why simple logic factor of safety is a function of C mobilized and C mobilized becomes what this is truly speaking a form of C prime over factor of safety. That means what we are doing now on the left hand side also we have a factor of safety term on the right hand side also we have a factor of safety term you got the point on both the sides we have factor of safety terms. So, that means now you have to do iterations trial and error. So, as I said during our days we used to take slip surfaces plot them on the graph paper analyze them it used to take days to solve one problem, but now the world has changed there are several softwares which are commercially available which can be utilized. So, with this I am going to stop discussion on slope stability which was the last topic to be covered on you know application of shear strength theory for geotechnical structures. I wish to show you two websites of the commercially available softwares which you should try to you know browse learn from them so that you can analyze the slope stability problems. So, if you type on internet geo studio you will go to geo studio student edition and then you can see different features over here there are several features here like slope w, seep w, sigma, quake, temp, citron, air. So, slope w is for analysis of slopes, seepage analysis, stress analysis, earthquake analysis, temperature analysis, contaminant transport analysis, air migration analysis for unsaturated soils alright. So, if you type over here or if you click over here you will be taken to the slope w. And then you will see that there is a very good video available over here what it says is that slope stability analysis is the leading slope stability software for soils and rock slopes slope w can effectively analyze both simple and complex problems for a variety of slip surfaces shapes. What we have done so far is we have assumed to be circular slip surfaces which in real life are not valid for water pressure analysis, soil properties and loading conditions. So, this is the trend right now people are following slope geo slope for pore water pressure analysis rapid drawdown condition, material modeling and so on. You can click over here different types of solutions for dams and levees, reinforced ball and slopes, excavations and open pit mines, roads, bridges, embankments, environmental protection, groundwater, ground freezing, climate change and so on. So, it is a very comprehensive package and we are into the module slope w alright. So, whenever you get time please go through this and you can also see the different types of how the anchors have been designed and you know mechanically stabilized earth have been designed and so on. Please explore this package. The second one which I wanted to bring to your notice is what is known as soil vision. So, if you type on soil vision database systems for saturated unsaturated soils, you will be taken to this website and you can see here different types of modules which are available alright. Series of modules which they have added and you will find several analysis over here different types of analysis and if you go to the slope stability by using soil vision package then you can go to the different sites here and you can SV slope you know different types of analysis you can perform. So, these are the packages which are worth exploring. They have a very good package SV slope three dimensional seepage and stability analysis for different type of dams other than dams how to create three dimensional dykes and so on. So, my recommendation would be adopt one of them and see what facilities these systems provide to you alright. So, with this I finished my course CE 330 geotechnical engineering 2 and I have tried to cover the most important and critical aspects of the materials soils in their different states and how to utilize these materials in different states as the engineering materials or how to do engineering with these materials. Thank you.