 So, now I will be talking about the state of stress in soils. At this point what is happening, oh, the end effect of all these forces at this point is there is going to be a normal stress, I write normal stress as sigma and there is going to be a shear stress, I write it as tau. So, the question number one is when we say state of stress in soils, can we find out sigma and tau or as a function of external forces, is this okay? And this sigma and tau is always on a plane, why? Because I am interested in understanding the detachment of a block from the parent body and this plane gets created because of the criticality which we are talking about in the soil mass and the condition is each and every point within the soil mass is going to be critical simultaneously, so sigma tau acting at alpha, so plane indicates alpha value and then there has to be a point, so point O external forces and alpha, is the statement of the problem clear? I can modify this as sigma x, sigma z, now you might say why not 3 dimensional, again my answer would be let us do first 2 dimensional cases which are more prevalent in engineering because most of your structures are 2 dimensional in nature, though they appear to be 3 dimensional, why it is so, embankments, soil mass, they are all extending up to infinity in the third direction, is that correct? So we are taking a slice of this and hence a 3 dimensional problem is being converted into a 2 dimensional problem and once you become an expert in solving 2 dimensional problems you can extend them to 3 dimensional problems easily, that is not a very difficult thing. So sigma x, sigma z what else is going to come, tau xz, very nice, what else alpha, so what I have done all these forces, external forces I have converted them into stresses, in geomechanics, engineering sciences and engineering technology we do not talk about the force ever, we always talk about the stresses, though the forces are acting on a system we convert them into stresses, alright and then we put them over here like this. So what I am interested in is, I am interested in finding out sigma and tau as a function of sigma x, sigma z, tau zx alpha, this is the state of stress which is existing at a given point and this point lies on the plane at which the failure is going to take place, okay, let us start this process, I will isolate this whole system now and go for the free body diagram of this, it is a point and point is lying on a plane, correct. So suppose if I say that this is the point O and this is the plane 1, 1 and on this plane the normal stress and shear stress are acting, clear and then we have combination of sigma x and sigma v, so what I have done, all f1, f2, fn I have combined in sigma x and sigma v, so truly speaking sigma x is your horizontal stresses divided by area of cross section and this is also equal to vertical stress divided by area of cross section. So henceforth we will not talk about this v and h, as I said we normally do not talk about the forces, we talk about the stresses. Now if I take the projection of this plane and if I assume that this plane extends up to unity in the third direction, clear and this is of unit length and this is also unity, so area of cross section is 1, so area of cross section of this is 1 square 1, if this has the angle of alpha this is going to be, the horizontal force is going to be now sigma x multiplied by sin alpha and this is sigma v multiplied by cos alpha. Can you do simple equilibrium and try to prove that tau will be equal to sigma x-sigma z by 2 into sin 2 alpha and sigma will be equal to sigma x plus sigma z by 2 plus sigma x-sigma z by 2 into cos of 2 alpha. This is what is known as the equilibrium equations and these are known as 2 alpha equations. I hope you can realize that if tau is this and sigma is this truly speaking this is the form and if equation tau square plus sigma square equal to a square and this is nothing but a circle that means these equations are the generalized form of the circle which is known as the Mohr's circle, okay. What we have done in the last 3 minutes and Mohr's circle onwards you are aware of everything in the mechanics you have done, so now soil does not appear to be a foreign material, now we can handle it easily because strength of material is already done and you start with Mohr's circles. So I have brought down everything to the Mohr's circles and I will be using this as a tool to define the state of stress which is getting induced in the material causing failure because of the external forces, distresses and what is the assumption? It is a rigid body. So we are not talking about any type of deformation, settlements, compression now, it is the 100% shear failure. This is a simplified solution which we have obtained for this type of everybody diagram here. I have taken this as the radius so you can interpret it the way you want and then this is tau and sigma, so what he is suggesting is I have not taken into account this thing, I am just coming to that, good that you have hinted on this. So this equation is valid provided your sigma x is sigma 1, you know what is sigma x equal to sigma 1 and sigma z equal to sigma 3. So if this is the condition which I am assuming over here where this happens to be the major principal plane, principal stress and this happens to be the minor principal stress, now I think your question is answered and by virtue of being this sigma 1, sigma 3, sigma x, sigma z tau z x tends to 0. So what I have done, a complicated problem I have further resolved into a situation where sigma x, sigma z happens to be the minor and major principal stresses that means there is no shear stress which is going to act over here. So your point is correct, if I want to generalize this thing I will have to show a shear stress component here and I have to show a shear stress component over here and this is what is going to be tau xz, we need not to enter into these complications at all, I will tell you why because what I have done is I have created a subset out of this and if I can define this circle where sigma 1, sigma 3 correspond to tau and sigma, this is going to be a solution to a problem which I can generalize further on. So this is the situation where you have this is as sigma 1 and this is as sigma 3, so what we do is we plot it as a function of tau and sigma on the tau sigma axis, this is a center O, so by virtue of this it becomes sigma 1 plus sigma 3 by 2 the coordinate of the center and the radius is sigma 1 minus sigma 3 by 2, so this is the coordinate of the center and this is the radius.