 We have been talking about this strength of the materials and this is where I gave you some idea about the state of equilibrium in the soil mass and why do we require the tools to analyze the state of stress in the soil mass and what I did is I took a point and I was trying to analyze the state of stress acting at this point which is lying on a plane along which the failure would occur. And in the process I had derived the relationship between sigma square and tau square as A square and this is the equation of the circle which is known as Mohr circle. Now the interpretation of the results is like this that if I put the sigma and tau scales as equal alright. So sigma scale and tau scale are equal this is very important to remember. The whole analysis depends upon the fact that sigma and tau scales have to be same. So this is the circle which you get and as we discussed in the previous lecture this is the center of the circle. Now this is what is known as sigma 1 and this is sigma 3. We call this sigma 1 as principal stress principal major stress and sigma 3 is this point we call it as principal minor stress. So in short this Mohr circle defines the state of the material at which the system remains in equilibrium. I hope you can realize that point P the state of stress at a point P is not going to be possible it is a virtual situation because before the point P is achieved the material has already failed. Similarly if I say that the point P1 which is lying within the Mohr circle that place also the material is not going to be in a failure state. So in short the Mohr circle defines the state of stress acting at a given point which is sitting on a plane along which the failure is about to take place. So for all practical situations the point is still in equilibrium alright. Now if you remember what we did is we wanted to know sigma and tau the question is why and I cited few examples in the previous lecture a simple situation would be let us say those of you have gone to hilly terrains this is a huge let us say Himalayan range and for the sake of you know creating infrastructure what I will do is I will cut this slope over here I will create a bench remove this much of the soil and this is where the road or the pavements are going to sit. So this becomes a pavement alright on which the road is going to be sitting I have to realize that the situation is quite critical it is very easy to draw something like this on the board and then say that I have constructed a road over here. So whenever you get a chance to go to the hilly terrain you will realize that most of the infrastructure is being developed on the hills by cutting and leveling the ground or the hillocks I will take you through a real-life situation in which I was involved to give you more idea about how these concepts which are so simple preliminary can be utilized to analyze these situations. The larger picture here was that we wanted to know state of stress as a function of sigma x, sigma z, tau zx and alpha alright it could be a reverse problem also. If I know this I would like to know this this is also a situation and that is where this Tamagno he was talking about tau zx thing comes in handy or the forward situation would be if I know this and I would like to compute sigma and tau both ways possible. So in all this situation what we have to do is we have to first draw the Mohr circle which represents a state of stress and once this has been done I can analyze the way I want to analyze it alright. So there are few characteristics of the Mohr circle which you should write down and try to understand the Mohr circle represents the state of stress at a point and this point I hope is understood is lying on a plane and that happens to the plane along which the shear is going to take place or the slippage is going to occur. At a point at equilibrium and applies to any material I am sure when you are doing strength of materials you must have not realized that this concept can be utilized to any material now in this case we are going to utilize this concept for the geomaterials like rocks and soils and the mechanical engineers would be utilizing this for composites steel different type of girders different types of I beams and what not alright any material any material and not only to the soil alright. So last time I wrote an expression where we had defined sigma and tau as sigma x plus sigma z by 2 plus sigma x minus sigma z by 2 into cos of 2 alpha and shear stress is sigma x minus sigma z by 2 sin of 2 alpha I can interplay the stresses in such a manner that these expressions can be written now in the form of sigma 1 sigma 3. So I can always say that sigma x please understand this concept this will be very useful for rest of the course and sigma z sigma x could be greater than sigma z simple situation would be if I consider a point somewhere here in the soil mass now this is sigma v this is sigma x or sigma h I normally define sigma h as sigma x and sigma v as sigma z alright this is a state of stress and of course to complete this if I am dealing in terms of sigma x and sigma z there will be shear stresses which are going to act this is what is known as state of stress existing in the material at a given point there could also be a situation where sigma x becomes less than sigma z can you guess when this type of situation might occur. So the way I read it is the lateral stresses are more than the vertical stresses in the first case in the second case the lateral stresses are less than the vertical stresses. So these are two situations which you might be analyzing in day to day civil engineering practices alright. Now the commonsense says if the vertical stress is more than horizontal stress I still require some confinement over here this becomes a retention system which will analyze. So this becomes a retaining wall you must have seen when you move around the hilly terrain what do they do is they try to stabilize the entire slope by putting a retention scheme. So this becomes a retaining wall there could also be situation where the reverse happens lateral stresses are more than the vertical stresses they are defining the gravity the lateral stresses are so high that they are defining the gravity tectonic motion two plates coming and hitting each other and formation of mountains correct. So it is very interesting to start with this simple concept if I can project and if I can solve the state of stress which exists in the geomaterials for different real life situations clear. So this is a typical case of let us say tectonic motion formation of hillock plates coming and hitting each other lateral stresses are more than the vertical stresses there is a reverse situation. So I can still use these equations what I will have to do is I will have to just substitute the values of sigma x and sigma z to sigma 1 and sigma 3 this I can do analytically and when I am dealing with sigma 1 and sigma 3 your tau xz vanishes why because sigma 1 sigma 3 always acts on the plane where the shear stress is 0 look at this diagram sigma 1 and sigma 3 is acting on the planes where the shear stress component is 0 and that is the reason we call them as principal major and minor stresses. So these are the planes on which shear will never act or it will be 0 alright but suppose if I consider a point over here and if I define this as sigma tau now this is a state of stress which is acting at some plane and we will discuss about this the moment this becomes critical the failure takes place this is part clear now what I have done is I have written here that it might be applicable to any material and not only to the soil this could be rocks this could be steel plates it could be anything so I am sure in your strength of material course you must analyze the situations where suppose if I give you a metal platen if I ask you to stretch it or compress it normally metals have to be stretched so suppose if I say this is a state of stress and find out how the fracture is going to take place this is okay there is a failure plane or the fracture plane I could reverse the situation also and I could make it more complicated by saying that I need a situation where there is a hole punched plate in your steel structures you might be using it for gassets rivets is it not for the gasset plate when you connect with the I beams or the sections you do punching so that you can put a rivet and you can connect the two and suppose if I ask you what is now happened to the material if I stretch it very complicated looking problem very simple to solve once you have the Mohr circle with you remember this is divide of the material you have not put the material property over here now the question is if I want to really make it valid for a material like soil or for any material what I should be doing I have to define the shear strength of the material did you get this point and that is the reason when we start talking about the shear strength of soil we first of all try to understand the state of stress and superimpose on this the material properties is this fine okay now there is another characteristic of so I can replace the whole thing and I can say that sigma and tau would become just replace sigma x by sigma 1 and sigma z by sigma 3 later on we will study that this type of a situation where the horizontal stresses are more than the vertical stresses becomes a passive state of the material passive of pressure and as long as the lateral stresses are lesser than the vertical stresses this becomes the active state of earth pressure state EP corresponds to earth pressure so if I just interchange these things I will be getting sigma as sigma 1 plus sigma 3 by 2 Mohr circle sigma 1 minus sigma 3 by 2 cos 2 alpha and tau will be equal to sigma 1 minus sigma 3 by 2 sin 2 alpha alright so this becomes the generalized law now when we deal with the Mohr Coulomb circles or the Mohr circles to be precise not Coulomb Coulomb have not brought yet in the picture we define a point which is known as pole hope you must have done it in a strength of materials have you done it or not no no issues you remember or you have not been exposed to this no issues basically the characteristic of the pole is that this is the point on the Mohr circle through which all the planes will pass fine this is the fundamental nature of this pole the reverse is also possible alright any plane is starting from a known state of stress wherever it intersects the Mohr circle that becomes the pole fine both ways now we will talk about this a lot so what are the properties of the pole the simplest possible theorem is any straight line drawn through the pole will intersect the Mohr circle at a point which represents the state of stress on a plane inclined at the same orientation in a space as the line please read this a bit and then we will go further go through this statement and try to understand what I have written so pole is a point on the Mohr circle through which all the planes will pass these are the planes on which the state of stress is known remember what I did I took out a element from here alright and then I said we can continue on this itself we know the state of stress here tau also might be acting if these are the principle stresses tau will be 0 we have sigma 1 sigma 3 and then we are trying to find out what is the state of stress at this point when I zoomed it it became like this this is the point oh let us say this is sigma this is tau and then we did some simple analysis so property the pole any straight line drawn through the pole will intersect the Mohr circle at a point which represents the state of stress on a plane inclined at the same orientation the space as the line is let us talk about the reverse what we call it as a lemma if the state of stress that is sigma and tau on some plane in a space is known a line parallel to this line parallel to this plane can be drawn through sigma and tau on the Mohr circle the pole is the point where this cuts the Mohr circle where this cuts the Mohr circle in simplest possible form at this point if I show the element this is sigma 1 this is sigma 3 alright and we are assuming that sigma 1 is equivalent to sigma z and sigma 3 is equivalent to sigma x okay sigma 1 is the plane is the stress which is acting on the horizontal plane this is the horizontal plane correct sigma 3 is acting on a vertical plane the state of stress at this point is sigma, tau that means sigma 1, 0 by virtue of being a principle stress shear stress is 0 clear at this point the state of stress is sigma 3, 0 where are these points located on the Mohr circle I have shown them as this point and this point where this state of stress is acting sigma 1 where it is acting horizontal line so starting from the state of stress which is sigma 1, 0 if I draw a line which is horizontal because sigma 1, 0 is acting on a horizontal plane clear now this is going to cut the Mohr circle at sigma 3 so by the first definition this becomes the pole reverse the situation sigma 3, 0 is this point and on which plane it is acting vertical plane clear so if you draw a particular plane passing through sigma 3, 0 how it looks like this is how it will look like please excuse me for my poor drawing but anyway and this is going to be a tangent so by definition the state of stress at a given point is known the plane passing through that intersecting the Mohr circle is going to give you the pole so pole is a unique thing and what we have done we have cross verified either you start from sigma 1, 0 take a horizontal line cutting the Mohr circle at point P or reverse sigma 3, 0 is known draw a plane point of tangency this is the pole so what you have done in short is we have identified the pole and this pole has lot of interesting peculiar characteristics read this if the state of stress on the plane in the space is known a line parallel to this plane when it is drawn through sigma tau will you know on the Mohr circle this will also result in the state of stress of the material reverse what we discussed just now starting from the pole if I draw a horizontal plane is going to go and cut over here the Mohr circle clear so this becomes your sigma 1, 0 is starting from this pole if I draw a plane which is perpendicular is going to cut the Mohr circle at this point itself this becomes your sigma 3, 0 is this part clear please understand this thing clearly because henceforth I will not be discussing this but I will be utilizing this whole thing and what you will observe is 80% of the course is going to be a discussion only on the state of stress see last one what we did is first we started from the known state of stress the known state of stress is sigma 1, 0 acting on a horizontal plane clear where is sigma 1, 0 acting at this point draw a plane which is parallel to this plane cutting the Mohr circle point becomes pole fine state of stress here is sigma 3, 0 acting on a vertical plane draw a vertical plane it cuts at this point this becomes pole the reverse process if I draw a line passing through the pole wherever it cuts the circle it gives a state of stress on that plane so starting from pole if I draw a horizontal plane it cuts over here the state of stress on this plane is going to be sigma 1, 0 starting from this point if I draw a vertical plane it cuts at this point itself and hence it gives sigma 3, 0 state of stress fine so I am sure you must be realizing slowly and slowly what we are doing starting from the simple models now we are approaching complexities and trying to answer the real life situations what I have to do is only in this model which is divide of the material properties I am just going to input the material properties and what material properties I need I need hs and characteristics that we will discuss later.