 Now, the question is plot it on this PQ plane and see what happens. So, depending upon whether I am doing drain analysis or a drain analysis, where is the KF line? If I am working with let us say NC material or sands, what is going to happen? The friction angle is this is going to be the K line, this is part okay. Everything just below this is going to be K a line and this K a corresponds to active earth pressure. Now, somewhere here, now tell me one thing, where would be K equal to 1 line? If this term becomes K equal to 1, tan beta becomes 0. What happens to the beta line? Beta value 0, correct? So, what is the meaning of this? This becomes K equal to 1 line and what is K equal to 1? Isotropic compression, fine. The question is, where would be K greater than 1? So, this is the KF line, the same material can fail under active state of stress. What you are realizing is under K equal to 1, the material is never going to fail because of pure compression. Here here, this will be K greater than 1 and this is what is known as passive earth pressure line. So, you have to transform the state of stress from first quadrant to the fourth quadrant. Now, we will analyze these situations, clear, is this part clear? Each one of them is a failure line, this except for this because this happens to be the isotropic compression where the failure is not going to take place, okay, is this concept clear? This failure line corresponds to the situation that this is the material, clear? This is a situation where you have sigma v is greater than sigma h. This is a situation where you have sigma h greater than sigma v. Now, when I am dealing with earth pressure theories, it is always beneficial to assume that KF condition is corresponding to K0 line and this is the elastic equilibrium, we call this as earth pressure at rest. So, in nature what is happening, the rivers are bringing these loads of sediments, they are getting deposited in the ocean bed, clear, all right, natural synergy between the particles. So, that means this is the ocean bed particle, first layer of the particle gets deposited, second layer and there is no impact of next coming mass of the particles on the previous layers because this is all sedimentation going on. So, you have studied critical velocity, Stokes law, no energy is getting imparted from the next batch of the particles on to the previous deposited particles, K0 condition, elastic condition, at rest condition, system is at rest. Now, if you come and shake it earthquake, the at rest condition gets disturbed and then the chances are that there will be a transformation from K0 line to Ka line is starting from here I can achieve the failure like this, I can achieve the failure like this also and that is all governed by the combination of sigma H and sigma V. So, with this preface and with this information in mind, let us do some mathematics. So, suppose if I ask you plot hydrostatic condition on a, where is the hydrostatic condition? Now all of you know that hydrostatic condition is going to be at this point, so this is the point A which corresponds to sigma H equal to sigma V, is this okay? From this point if I start shearing it, the sample, there are several ways, where is my failure line? This is the KF line and the mirror image of this line would be somewhere here because the Mohr's circle is a circle, clear? What we did is for the sake of convenience, we took only half of it, so we took only the positive portion of the Mohr's column envelope, truly speaking this is going to be a mirror image, both sides you have an envelope, fine? Suppose if I put a condition that stresses are changed in such a manner that delta sigma V is equal to delta sigma H, try to understand what is the significance of this. Starting from the hydrostatic condition, if I start increasing sigma H and sigma V in such a manner that both sigma H and sigma V are same all the time, the increment has to be depicted in the form of a stress path which is going to show whether the sample is going to fail or not, so can you draw it, draw it now. I have written something over here, so you can get the increments of Q and P and plot in terms of beta, like you take this box okay, triaxial sample and I am keeping all sigma as sigma V increment same, isotropic compression, hope you will realize that this is how you will move, this is a stress path, are you going to achieve the failure? This line is not going to cut failure line anywhere, so what should I do to fail the sample? Now suppose if I change the situation, now if I say stress path number 2 is minus delta sigma V equal to minus delta sigma H, please for God's sake do not cancel the negative signs, because negative sign indicates something, it is not mathematics, pulling apart system from both the sides, so how do you break something by tearing it apart, is it not? So suppose if I pull it from both the sides and from these 2 sides also, the chances of the system is going to fail, why, look at this, it is okay and this is definitely going to cut the failure envelope at somewhere and what is going to give me, is going to give me the tensile strength of the material, fine. So I have shown you now 2 states of material, the way you wanted to utilize them by following the 2 paths, now let us mix slightly more complicated situations, there is no fun having these type of situations in engineering practice, now suppose if I say this point A was for the case when sigma V equal to sigma H, now suppose if I say sigma H is not equal to sigma V, what is going to happen, you are violating the hydro study condition, clear, so point A is going to jump to point B somewhere over here, I can create a situation where I would say these are the linear relationship, material is non-linear, clear, I will put a condition sigma H is a function of let us say delta sigma V and this function could be non-linear, so all these lines which I am drawing at the straight line, they will become non-linear curves, so what it indicates is if I put a condition sigma H is not equal to sigma V, this becomes my starting point and then can I superimpose this condition or not, yes you can, so starting from this point is going to be easier for me to fail the sample as compared to this point, how different ways, this is one of the ways, this is one of the ways, this is one of the ways, this is one of the ways, this is one of the ways, this is one of the ways, so on, so what I have done, I have created different types of stress paths, if I follow the state of stress along them, I can fail the sample, now impact is what do we want to do, we do not want to fail the sample, so it is a reverse process, if I understand under what type of loading combination the material is going to fail, I can be conservative while designing it, longer the path, the longer efforts are required to fail it and the probability of the failure is going to be lesser as compared to situation where the path is going to be shorter, so look at this situation, it is very easy for me to fail the sample like this, mathematically you can obtain from here, you have delta Q by delta P equal to tan beta, so the slope of these lines is nothing but the beta angle, clear and then you can compute the beta, if the K is known, number 1, if sigma 1, sigma 3 is known, again you can compute the beta value by substituting delta sigma 1, delta sigma 3 and you can simply compute this, this is fine, good, that is what I want to hear, now the last part of the triaxial testing would be, suppose if I give you the P, P prime plane and Q plane and this is the KF line and this is the KF line, yes, so I will now elaborate upon your this concept, this concept and the combination of these two, fine, so starting from the hydrostatic condition, if I draw a line like this, tell me what this will correspond to, it is understood that the betas are different alright, so that you can compute, do not bother about that, stress path itself indicates the slope of the combination of delta sigma and delta sigma v, so suppose if I plot a line like this, what is the interpretation, this is what is known as compression, how the element will look like, overchap, if initially it was like this, when you are doing pure compression, clear, what is going to happen, sample is going to get deformed, so this is how it will look like, volume might remain constant as long as you are doing a CU testing, undrained testing, clear, the moment you do drain, it is not possible, I hope you understood this part, then your a1 v1 equal to a2 v2 plus delta of volume of water which comes out of the sample, the equation which I wrote, is this fine, now this is being done under sigma v, sigma h is constant, what we call this as a one dimensional compression or loading, so this is a typical loading curve, loading stress path, sorry, what is the reverse of this, suppose if I project it on this side, what is going to be, what is the reverse of loading, unloading, under what circumstances, delta sigma h is equal to 0, delta sigma v, sorry, I mean you cannot put both the things, so you have to say that under the constant loading or whatever, so I will assume this as delta sigma v is equal to 0 and delta sigma h we will keep as it is, so here delta sigma v will be less than this and let us say delta sigma h vary, now this becomes the unloading or excavation, now if k is greater than 1, what we have done is we have created 2 more situations, so if I draw a line like this and the reverse of this would be, this is what is defined as active earth pressure and this becomes the passive earth pressure, if I look at the sample at this point, this is the initial sample because delta sigma h is more than delta sigma v, what is going to happen, the sample is going to get squeezed in the lateral direction and it will expand in the vertical direction, good, so this visualization is important, sometimes this is also known as axial expansion, these concepts you are going to use from next chapter onwards or a better word is dilation, if you remember, clear, we talked about dilation also, the state of stress is like this that the delta sigma h is higher, delta sigma v is 0 under constant vertical stress because the location of the point remains same, the horizontal stresses are much more than the vertical stresses, so what is going to happen, the system is defying the axial confinement alright and this is how the dilation process would be, in this case delta sigma v is equal to 0 and what about delta sigma h, here delta sigma h was greater than 0, this will be less than 0, so these are the four mechanisms which we are going to use quite frequently in the whole geomechanics henceforth, we will get rid of now the material also, I do not want to talk about c phi and all those things, why, because I am using q p, clear, so material is being depicted now p q, state of stress is being depicted now by the stress vector and I know starting from this point how much amount of stresses are required to achieve the failure under this type of loading, how much amount of stresses are required under this type of loading and the combination of the two, this and this and of course this also, that is always valid because this is the relationship between sigma h and sigma v, what you have to understand is what is the state of k, so in other words if k is greater than 1 or k is less than 1 or k is equal to 1 what really happens, exactly, so that is what is getting balanced by this ratio, so I need not to bother about all sigma 1 sigma 3 now, so look at this, this becomes my k p and this becomes my k a, that is it, I have further simplified everything, I need not to worry about delta sigma i sigma delta sigma v sigma 1 sigma 3, nothing because k takes care of the relationship between sigma h and sigma v, the way it is getting balanced this is what the important thing is because this shows the formation, this shows the OCR, this shows the rate of shearing, this shows the type of material, this shows the type of drainage conditions, this shows the type of whatever, keeping sigma h constant, you are sitting in a fluid, so only vertical stresses are being changed, not the horizontal stresses, so I have to study pressure, this is all what I wanted to discuss about the triaxial testing and the shear strength theory of the soils, henceforth what we will be doing is, we will be utilizing these concepts of shear strength theory to deal with the stability of different types of walls, what we call them as retaining walls, sheet piles, bracing, skirts and so on, the basic concepts I have clear, now rest is all application of these concepts, the second part of the course which I will be discussing quite in details would be the stability of slopes, so there again we will be utilizing the shear strength parameters, the concept of shear strength theory to prove whether the slopes are stable or not, so in short geotechnical engineering 2 deals with 3 major components of the discussion, one discussion was on determination of shear strength and second is application of shear strength parameters and when we talk about application of shear strength parameters, we will be talking about the stability of retention schemes and slopes. Next module of this goes into the foundation engineering, so foundation engineering is nothing but application of shear strength theory to deal with the stability of the foundations, so this is how the whole scheme of discussion is fine.