 The first major part of this course I have already completed that was the shear strength of soils and I spent about 13 lectures discussing the importance of shear strength, how to determine this, what are the tests that are to be conducted and how to interpret the results and at many places I have given you the importance of parameter selection, so that they can be utilized in real life projects which I hope would be very useful and helpful for all of you who are into the field of consulting and advanced research in the realm of geomechanics. So, today onwards I am going to start the discussion on the second topic, major topic of the course that is the plastic equilibrium in soils, so needless to say it is understood that this is the state of the stress which is causing the plastic equilibrium in the soils and as a geotechnical engineer we would like to understand what causes this state of stress to develop in the system and what are the applications of this type of concept which is quite prominent in the practice, so when we talk about the plastic equilibrium in soils the first question is where are we going to apply this concept, so the applications are basically this is the application of how to use the parameters, so designing of retaining walls particularly I normally call this as retention schemes and when we talk about the retaining walls and the retention schemes the most prominent one which we are going to discuss would be the sheet piles, we will be talking about the bracing and before we talk about the bracing we will be discussing about the cuts in the soils, these are also known as excavations, we will be talking about different types of sheet piles or the retention systems, this would be anchored sheet piles alright and you may say non-anchored also, so this is the first subtopic of the plastic equilibrium in soils, having introduced the concepts of plastic equilibrium in soils today I will move on to the analysis of retaining walls from the next or next to next lecture onwards, the second major subtopic or the application would be analysis of slopes and this is where we will be talking about two types of slopes, the finite slopes, the infinite slopes, the basic objective is to analyze the slopes so that the failure does not occur, so rather than talking about the stability analysis normally this thing is also called as stability of alright, so it is always better to talk about the instability analysis, so instability is you know included, so more focus is to study how the instability occurs, so that as an engineer and as a designer I can stop the instability to occur, then we will go for different types of analysis methods and I would like to spend enough time on slope stability analysis, it is a major topic which one should be studying during the undergraduate, the third major application would be the bearing capacity of soils, of the soils or you may say geomaterials because later on some of you might explore the possibility of studying the rock mechanics and then soils becomes a misnomer and this can be applied to the rocks also, however as I said sometime back this is beyond the scope of this course and there is a specialized course which you might be doing in the fourth year that is the foundation engineering, however one thing is common in all these problems and what is common is that the objectives are common and these objectives are I want to find out what is the cause of failure and how this failure gets stopped clear, so that means what we call them as stabilizing forces, the second one is destabilizing forces alright, so this is the principle objective, in all these problems or the class of problems what we try to do is we just try to analyze the system for two types of forces, one is the forces which are stabilizing a situation or the failure and the another one is destabilizing forces, so most of the time it so happens that destabilizing forces are gravity and so destabilizing forces will be the gravity number one, it could be rain, so during the monsoons in the newspaper you keep your hearing that in the western guards you know landslides have occurred, buildings have collapsed, dams are failing whatever during rains, so one of the destabilizing forces is rains, it could be snow also, we are lucky fortunately that we are not in the, we are in the temperate climate and where we do not normally come across snow forces much except for the northern reaches and the north eastern part of the country, otherwise this is a major issue in most of the entire world and of course the earthquake alright, man made alright, so this could be what we call it as a human interventions, so one of the good examples would be earthquake comes, systems fail, nice because they give you a little force, human interventions as you said vehicular movement yes on the heli terrain you are chopping off the hills and then you are moving the vehicles, vibrations get induced and because of that destabilizing forces generate, now you must be realizing that destabilizing forces are quite you know in isolation, they do not have a big company, so truly speaking the destabilizing forces are what any guess, what could be the destabilizing force which would resist the failure, very good, who has answered this, excellent, nice, so you have understood the materials very well, so this is the shear strength, so it was worth spending 14, 13 lectures on understanding the material why, because this is a lone fighter understand, so the entire destabilizing forces in the system get mobilized because of its shear strength and we have done lot of justice with shear strength, we have tried to understand what is shear strength, how to germinate, how to interpret it and all those things, so now we are well equipped to move ahead and solve all these real life complicated problems alright or what else could be the destabilizing forces, one is shear strength, shear strength could be very weak, so remember you go to the offshore environment or the mud flats where most of the construction is happening in bombasticity right now, you remember my statement that most of the best possible land has already been utilized by us or our grandfathers, now what is left for you guys and the generations to come is all marshy land, which cannot be rehabilitated so easily, that means it is understood that the shear strength is almost either negligible or extremely poor and hence they fall in the category of challenging problematic type of soil deposits, so truly speaking the fun is to deal with a situation where I can create destabilizing forces in the system, despite the fact that shear strength is almost tending to 0, a beautiful example of this is marine deposits where you cannot even stand, you cannot even stand, forget about walking because if you try to stand over there you sink alright, so this is the objective and this is what the art of practice of geomechanics is, what I have used the word as art of practicing the geomechanics, clear, in the worst possible situation also normally the criminal lawyers do not raise their hands and they accept the client fully knowing the fact that this guy is a criminal and he is going to be hang, so this is a very similar situation which we deal with clear, so shear strength almost tending to 0 marine deposits but look at this, the economy of the country and the world depends upon whatever structures you have created in the offshore environment not onshore that is where the money is fine, so these are general concepts which I have talked about the applications in so many areas and objectives, so for that matter one of the objectives of stabilizing forces enhancement could be let us say ground stabilization or ground modification or soil improvement starting from shear strength 0 they will bring it up to a value where Cu becomes greater than let us say finite alright and this finite value is obtained because of ground modification because of soil improvement, so if this algorithm is clear to you now what we will do is we will start describing the plastic equilibrium in soils, so until now whatever you have studied was all elastic state of the material, though I have introduced this concepts of K and Kp and in the previous lecture if you remember we have tried to relate Sigma H and Sigma V with K parameter clear, so what we define as is if I define K parameter as Sigma H upon Sigma V I hope you understand the connotation this is the ground level I have taken a point somewhere here at a depth of Z this is point P the state of stress at this point is Sigma V and Sigma Z alright what I have done is I have defined this also Sigma 1 equal to Sigma 3 tan square 45 plus 5 by 2 if you remember plus 2c cos of no 2c tan 45 plus 5 by 2 this we derived sometime back using the Mohr circle we drew the Mohr circle and then we took Sigma 1 Sigma 3 and then we did some geometry and from there we describe this clear and this is where I introduce the concept of roll reversal of Sigma 1 and Sigma 3, so in today's lecture I am going to talk about this much more in detail, so please be focused and try to understand this concept once for all once you understood this concept nothing can be due, now this parameter is known as n Phi and this is also equal to K P which is also equal to 1 plus sin Phi over 1 minus sin Phi the way I am using this Phi terms over here is this is generic term depending upon the situation this could become Phi prime this could become Phi Cu this could become Phi undrained Phi drained and all clear that is implicit the reverse of K P is known as K A and this is defined as tan square 45 minus 5 by 2 and this can also be written as 1 minus sin Phi over 1 plus sin Phi, so truly speaking this equation itself is defining the state of plastic equilibrium in the soil, so only thing what I have to do is if I write an equation let us say K is K P is K A is this understood, so this term K P is defined as the passive earth pressure coefficient and K is defined as active earth pressure coefficient, sometimes they also call it as coefficient of active earth pressure coefficient of passive earth pressure I hope you can realize this K can be termed as K naught this becomes coefficient of earth pressure at rest what is at rest this is the elastic state, so if this is the elastic state and we are saying at rest now if I use this relationship the lateral strain equal to 1 upon E sigma H minus mu sigma H plus sigma V I hope you have understood how this equation has been involved in the two dimensional space, so this corresponding to epsilon H radial strains, now when this tends to 0 at rest condition no movement natural deposition of the sediments, rivers are bringing sediments they are getting deposited in the offshore environment just under Stokes law conditions, no transmission of energy from the next load to the previous loads of the sediments clear elastic condition, no shaking no movement nothing elastic situation what it indicates is during the elastic state there is no strain in the horizontal direction confinement at that time what happens sigma H will be equal to mu times sigma H plus sigma V that means I can write this as sigma H 1 minus mu equal to mu into sigma V, in other words I can write this sigma H equal to mu over 1 minus mu into sigma V, so this thing I have derived and what it gives me is I am trying to now relate the state of stress existing in the material at its elastic condition elastic equilibrium Poisson ratio is involved over here and then what I can do sigma H upon sigma V I can write this as equal to k, so very intelligently what we have done we have derived a relationship for the k parameter using the Poisson's ratio this relationship is normally utilized to obtain the k parameter, there was a further simplification which came from simplification which came from a person known as Jackie and this simplification states that k naught equal to 1 minus sin phi, so all these terms and the equation which we are using right now though they are derived from the elastic equilibrium they are all valid for the plastic state of the material. Now let me define what is the plastic state of the material is this part okay and of course if you remember in the previous lecture we discussed this k naught is a function of you know what Rd, Ocr, type of soil etc., etc., water content and so on this is okay. Now the question is how would you depict this state remember we derived this equation A by this construction, so this is the tau sigma sigma plane the Mohr-Coulomb envelope at this point you have sigma V and sigma Z depth is constant that means sigma V equal to gamma into Z hydrostatic condition on this plane if I draw a Mohr circle we have already analyzed this situation the pole is at this point clear this is equal to sigma 1 which is equal to gamma into Z equal to sigma V perfectly alright this point is sigma 3 okay and then what we did is so this is the Mohr-Coulomb envelope fine the material remains same only the state of stress is going to change at this point if I draw a perpendicular this becomes a center what is the inclination of the failure plane any clues very good those of you who have not understood please understand it go back to your hostels and try to follow all these things to avoid disasters fine. So this is the failure plane at this point the failure is occurring what is this angle very good 90 plus 5 excellent clear what is this angle a plane passing through the pole cutting the Mohr circle is the failure plane at which the state of stress corresponding to the failure is occurring perfectly alright so this is a failure plane now what it indicates is as long as your sigma V is greater than sigma H alright this is going to be an active earth pressure condition alright why because K is going to be less than 1 look at this so that means in this case your K is going to prevail so what we have done is we have proved the state of the stress acting in the soil mass in the plastic lubrium though I have not defined yet what is plastic lubrium I will define it and what I have said is that this is equal to sigma H upon sigma V sigma V by virtue of the point located at a depth remains constant that means sigma H equal to K A into sigma V now this discussion we had some time back about the role reversal of sigma 1 sigma 3 clear or let us say sigma H and sigma V suppose if I am interested in knowing what is going to happen when this condition gets violated and K A tends to become K P starting from K naught value yesterday we discussed this so if you realize what I did is this was the KF line KF line and somewhere here if I have K naught line if this is the state of the equilibrium which I am going to achieve from K naught I can go to K A or I can go to K P stress parts now keeping this sigma 1 constant because this is gamma Z if K happens to be more than K naught what is going to happen your sigma H term is going to be more than sigma V where how are you going to plot this now this is what is going to get plotted like good I could manage so this is the circle which is bigger in size number 1 number 2 I have maintained the sigma V value wherever this cuts the X axis this becomes my sigma H those sigma H again becomes sigma 1 now fine so the way I read this is this is sigma I this is sigma V sigma H is equal to sigma V into K term which is multiplied by K P now the question is where is the failure plane and what is the inclination of the failure plane determine it find out the pole first so what is going to happen is the more and more complications in the problems come henceforth we have to identify the poles that is it and that is what I told you at that time once you know where the poles are life is simple we just sit down in your design office know the material properties you have obtained C phi and other things get the more coulomb envelope plot it over here obtain where the failures are going to take place go back to the basics where sigma H is acting on horizontal plane or vertical plane sigma horizontal is acting on vertical plane sigma this horizontal was acting on vertical plane the failure took place over here the failure is going to be over here now can I prove that the pole is going to be this point prove this so this is your P active and this is going to be P passive this I did it when I was teaching the derivation of poles now what you are realizing is where is the failure plane any point any plane passing through the pole intersecting the Mohr circle alright is this correct so this becomes your failure plane under passive earth pressure condition what is the inclination of the failure plane no mug it up I mean I think it is easy to understand is it not so again draw a tangent from here let it cut over here clear so this angle comes out to be 45-5 by 2 very good will realize that there is some contradiction which we have been talking about anyway as for undergraduate things are concerned you go ahead with this so what we have done something 45-5 by 2 something 45-5 by 2 what is this plane passing through the pole plane passing through the pole inclination of the failures are going to be different under 2 conditions active and passive condition that is what is known as the state of plastic plop in the soil and these 2 planes are going to be conjugate to each other that means the plane 1 1 and the plane 2 2 are going to be conjugate to each other with the insert section between both of them as what 90 degree so if this plane is inclined at an angle of 45-5 by 2 and if this plane is inclined at an angle of 45-5 by 2 I think geometrically things are compatible what happens is when the state of plastic equilibrium develops in the soil mass all right the failure planes get developed at either 45-5 by 2 with respect to horizontal where the principal stress is acting or 45-5 by 2 from the plane where the principal stress is acting this is what is known as state of plastic equilibrium in the soil yes how do not do not do not do not include the plasticity part it is a plastic equilibrium just hold on hold on hold on hold on 5 minutes I am coming to that so first of all what I have done I have defined the state of stress clear okay good I will answer your question truly is there is no plasticity coming in the picture but I will explain it to you sometimes in books you will find that this is also written as flow factor somebody may ask you during your interviews the material has a tendency to flow that means what we do is very conveniently we put C equal to 0 pure frictional material C is standing to 0 and the material is ready to flow frictional materials flow go to any silos where the grains are being stored all right so sugar, rice, wheat, urea particularly fertilizers they are all dropped from a hopper so normally what they do is they convey and everything to a hopper belt which is a conveyor system and then they drop it from the top and this is how the heaps get formed you know the value of 5 value the pose angle and if you see the silos they are normally designed in this manner keeping in view the friction angle of the material this is where the application is and this is how the term comes the flow factor how much material can flow all right in a dry state now I will extend this concept further and this will answer your question so by definition the state of plastic equilibrium in the soils is when each and every point in the soil mass is at the verge of failure following this state of stress due to gravity only understand none of these forces are going to come over here these are always superimposed so gravity only plays the important role remember what we have done is the entire derivation is based on the gravitational stresses the stresses which are getting induced because of the gravity clear rest of the forces might play a spoil sport or they may help you in stabilizing the system that is the engineering but the basics are this so that means the state of stress in the material remains either in active state or passive state so next time on when you go to these monuments you know just come out of the monument and see look at the walls the way they were constructed and I am sure if you are matching the line of wall along with yourself you will observe that the walls have moved out or they have moved in clear or those of you who love hiking and trekking you must have realized that when you go on the top of the hills you will find depressions there at the top of the hill clear and at the base they will be sort of a bulging so all these things are because of the state of stress which is acting in the system because of gravity only and we are trying to decode all these types of mechanism which we which prevail in the materials which are made up out of which are in the structures which are made up of soils alright so what we have done until now is we have just defined the state of plastic equilibrium starting from this state of stress let us say if this is the soil mass alright ground level remember when we were talking about the granular material long long back we considered to rigid boundaries in the soil mass hypothetical so I am assuming that these are the 2 hypothetical planes which are existing in the soil mass which is semi infinite on both the sides what we studied during the compressibility of the material was if it is a granular material and if the walls happen to be rigid and if I compacted what is going to happen no deflection in the sides only compression is going to take place but if the walls are flexible and if I apply the load the chances of the material will flow out clear I had done this action also if you apply the loading over here granular material will flow out like this provided the walls are flexible so now I am assuming the first case we have analyzed compressibility and consolidation in the CE323 by keeping a rigid ball assuming that epsilon h equal to 0 fine and not allowing any lateral strains developing in the system 2 conditions exist stop writing so one is AA another one is BB look at the motion of the planes now suppose as long as sigma v is constant there are 2 possibilities sigma h will either decrease or it will increase agreed if sigma h increases what is going to happen these 2 systems will get they will come closer to each other when sigma h increases alright that means what we call this motion as if delta sigma h is greater than 0 these 2 will have a tendency to move closer to each other this is what is known as passive earth pressure so somebody was asking in the last lecture how this happens gravity does this at global scale geology does this tectonic motions they do this human beings cannot do it fine so this state of stress exists in the system because of the geology because of the gravity and tectonic motions now this is what is known as a global plastic equilibrium in the soils and at this time what is going to happen your sigma h will be equal to Kp multiplied by sigma v hypothetical plane in the soil mass boundaries are flexible the state of stress develops in such a manner that the 2 planes come closer to each other what we call this as is the movement of the hypothetical boundary within the soil mass this is a soil mass and the boundary is moving within the soil mass clear opposite might happen so it so happens that sigma h might become negative we have talked about all the situations stretching of the material clear so when you are stretching this what is going to happen now this is what is known as delta sigma h might be negative or less let us not put less than 0 sorry is another state of stress let us say I mean okay I mean is increasing let us say and this is decreasing this is better way is this correct because negative positive I cannot use so in this case what is going to happen is is sigma h will be equal to K a into sigma v and this model is valid over here so this is the plane 1 1 and this is the plane 2 2 and this is how the state of stress is developing over here that means if I draw these lines in the entire soil mass they will correspond to the failure 1 1 how would I interpret all these are the slip surfaces along which the material will have a tendency to slide up one of the examples would be if I consider let us say this plane slip plane the moment this condition occurs this material has a tendency this becomes the parent soil mass and this becomes the block or the soil mass which is under passive or pressure alright has a tendency to slide up and you have done mechanics to solve this fine the only thing is that this angle is going to be 45-5 by 2 the moment I change the angle what is going to happen when I am stretching this block from the parent body will have a tendency to get detached and slide down alright angle is going to change because of what we discussed over there and that becomes your active earth pressure when this type of state of stress develops in the soil mass this is what is known as a state of plastic equilibrium in the soil is say means a normal so plasticity does not come in the picture at all the failure under active condition is going to occur when the slip surfaces are inclined at an angle of 45 plus 5 by 2 with respect to horizontal clear and the failure under passive earth pressure condition is going to occur when the slip surface is inclined at an angle of 45 minus 5 by 2 with respect to horizontal and what is horizontal sigma h plane here clear here this was sigma v which was cutting over here sigma v remains constant the first condition was two hypothetical surfaces are coming close to each other because of formation of this type of stress the slip surface which I have defined as this point along with the movement is going to take place as if this block gets detached from the parent body because of gravity only just because of gravity only clear so because of gravity the state of stress acts in the system which moves it up that means the system is coming close to each other fine and why gravity gravity plays the trick look at this all the tectonic motions that is what we had been discussing this is situation number one the second situation is if you stretch it out relaxation of the sigma h values this block has a tendency to move down the slip surface will change in compatibility with that this will become 45 plus 5 by 2 this is what is defined as the state of plastic equilibrium in soils.