 We had been discussing about analyzing the earth pressures and in the previous lecture I had talked about the theory which is proposed by Rankine. I will continue with this discussion and the discussion topic would be earth pressure analysis and by employing Rankine's theory which has already been completed. The second part of the analysis or second type of analysis which is normally done to solve most of the issues which occur in day-to-day life is the trial wedge. And this is what I will be discussing today and the third way or the tool for analyzing the earth pressures is the Coulomb's theory which I will be doing slightly later on. So as far as the trial wedge is concerned as the name suggests, suppose if I have a retaining wall which is sitting on the ground, this is the backfill and in Rankine's theory we have assumed that this is a frictional material and C is 0 and we have assumed that the state of stress in the soil mass exists because of the gravity. So gravity is nothing but the unit weight alright and of course the backfill happens to be dry and another assumption is that this wall is at 90 degree. We call this as a Rankine wall provided there is no friction between. So if I draw a shear stress component over here alright, so this is the shear stress component between the soil and the wall this is equal to 0 smooth condition no friction between the wall and the backfill material. Smooth wall vertical wall containing the backfill which is only frictional in nature no cohesion unit weight prevails dry condition and the backfill is horizontal. So inclination angle is equal to 0 alright. So this is a typical Rankine wall and we have analyzed this. Now in most of the cases what happens is I would like to analyze what is the inclination of the slip surface alright. So AB happens to be the slip surface and what we want to do is we want to optimize value of theta though we know this value. So I have written here theta critical alright. So I want to find out what is the value of theta critical for the height of the wall H there is no friction on the base also there is no friction between the wall and this this is the horizontal surface and how to analyze this. So this is where people consider trial wedge AB and C. Now this is a beautiful application of rigid body mechanics which you have already done where if I want to understand what are the pressures which are going to come on the block because of the wall is a action reaction. So the soil is pressurizing the wall and wall is trying to contain the pressures coming from the backfill alright. So the best way to deal this type of situation would be if I isolate this block which is in local plastic equilibrium you remember we have differentiated between global and local plastic equilibrium global is the phenomena which happens because of the natural activity tectonic motions. Now this is a local phenomena which is mostly prevailing in the manmade structures where the dimensions are finite. So if I isolate the block which is in either active or passive state of equilibrium plastic equilibrium. So what we do is we consider this as the block. So this is the AB and C and this is theta angle. So the statement of the problem is what is theta when there is a weight of the block acting is this determinate or indeterminate parameter determinate is it not. So if I know the value of theta this angle is going to be theta this is going to be h cot theta this is going to be h. So weight is equal to half into gamma intentionally I am writing gamma because I am not sure about the condition of the unit weight which is going to prevail in the backfill clear. I can always alter the situation by creating this is water up to this limit or the whole soil mass is dry or there is a partial submersion and whatever clear. So let it be gamma multiplied by h into h cot theta multiplied by 1 is this okay. So 1 happens to be the third dimension perpendicular to the board. So this is the total weight of the system that means this is half gamma h square cot theta as simple as this. What this indicates is that the weight of the block which is under active or passive conditions of failure would depend primarily on the height of the confinement or the walls multiplied by the critical theta angle which is slip surface fine. Now if I ask you to find out or to complete the free body diagram first of all. So what we will do is if I show p is this correct basically what is happening is if this is the wall and this is the backfill under active earth pressure conditions what is going to happen active earth pressure condition is the condition in which the wall is going to be pushed out by the backfill. If I see the interface C A and if I detach the wall with the backfill material this is how it will look like. This is the wall and this is the block a wedge. Now wedge is trying to push the wall outside clear. So that means the wall is pushing the wedge inside. So this is the pressure which is coming from the wall onto the wedge. Is this part clear? So what we have done is we have isolated two forces W and P. Now what we cannot remain in equilibrium because of the two forces there has to be a third force also. So imagine if this block has a tendency to attain active earth pressure condition this block is trying to slide down on the surface AB. So when it is sliding down the direction of the friction would be upwards. This is the shear strength which is getting mobilized in the system. Is this part clear? So if I show a resultant force which is going to act on this surface slip surface as R. Now this resultant force is going to have two components. One is the shear force and the normal stress clear. So this is the net resultant force fine. So I can exhibit the components of the R over here. So this is your T the force which is getting mobilized on the plane or the slip surface having the components of the shear strength and this is a normal stress. Is this part clear? Now the body is going to be in equilibrium because we have three forces and these three forces are P, W and R. Now this is a typical state of equilibrium which we draw for active earth pressure condition. What is going to happen in the case of the passive earth pressures? So in case of passive earth pressure nothing much is going to change. This is the wedge. This remains theta and you know the values of theta are going to be different. I will write this as now PA and this as PP. Is this correct? The weight is acting the way I have shown. What about the reaction which is acting on the slip surface? Imagine close your eyes and imagine how it is going to happen. Now that visualization is very important. So first of all let us go in steps. If I have to draw this type of diagram over here by definition the passive earth pressure is a situation where the wall has a tendency to get moved into the backfill, clear? When it is happening this block with respect to the parent block you remember this is the parent block or the parent backfill which is in elastic condition. Now when this block is moving up the friction is going to get mobilized in the downward direction. Is this correct? The shear strength somewhere here you have the normal stress. So this is where I was showing let us say R and this is what is going to be the normal stress and is this clear conceptually? So we consider the pressure which is going to come on the backfill or on the wedge. So in that case what you are observing is this is the direction of mobilization of the shear stress. This is going to be the normal component and the way the R is going to act is slightly trivial. Have you come across this problem anywhere in mechanics? You have suppose I lift a block it is kept like this alright where it is acting downwards. Now if I have to scoop it out this is a scooping out action. You know what wall is doing? Wall is trying to push the system in clear? So the whole block is going to get scooped out fine. So if I hold it like this you must have done these problems in mechanics where you are analyzing the weight of the block which can be held by a calipers or a forciple like this. So what I am doing is I am just trying to detach this block from the parent body. So look at the motion. The motion of the block is inside and upwards and hence the friction is getting mobilized downwards. So as per the thumb rules when the soil mass gets moved inside there will be a hump formation clear? And what is going to happen over here? Because the material is moving out there is going to be a depression. So wherever you find depression in the humps getting formed in the natural systems you can make out whether these systems are under active or passive state of equilibrium fine. Now this part is simple. Now what we have ignored very conveniently in Rankine's theory is that this wall is smooth. In most of situations not going to happen. Now we call this as the effect of wall friction on the stability of the block. That means there is a friction component which is acting between the interface of the wall and the backfill block or the wedge. So now can you draw the free body diagram of the block when the friction gets mobilized? So suppose if I assume that this was the case of let us say tau equal to 0 at the interface. But suppose if I say tau is not equal to 0 and this is a rough wall. So now what we are doing is you must have noticed that we have started manipulating the theories alright. It is a deviation from the Rankine's theory. So the moment you introduce the wall friction over here what is going to happen? The tendency of the block is to slide down. That means the wall is trying to lift it up clear. So there is a component of friction which is going to come over here. The PA is acting over. It will not act as it is the way it has been shown. And what is going to happen is if this is the normal from the surface now this will be the direction of the PA and what we define this as a value delta and this delta is known as roughness of the wall clear. So the two components of the PA will be one is the normal stress. Another one is going to be the shear component which is acting at the interface of the block and the wall. And I think now you can do the free body diagram very easily if the block is trying to slip down this wall tends to push it up. This is how the friction gets mobilized. Rest of the things remain same alright. Have you understood this part? So this is the effect of the roughness of the wall. Now the question is what roughness is going to do? Whether it is going to enhance the active earth pressure which is acting on the wall or it is going to decrease the active earth pressure on the wall. So in most of these situations ultimately the objective is to optimize theta obtain theta critical. And once you know the obtain once you know the theta critical value I can substitute and I can get the value of PA. So the principle objective is PA which is a function of theta and W and H and unit weight. Similarly what we will try to do is we will try to find out Pp as a function of all this. And because I have talked about the roughness in the wall also delta term. So delta term will also get included. Normally this delta is taken as 2 thirds of phi or the limiting value could be about 10 degrees alright. So these are the values of theta delta. So these are the concepts which are involved and ultimately what we will do from the next class onwards is we will try to utilize these simple mechanistic models and first of all optimize theta value and get the values of PA and Pp which is most critical. And incidentally this theta critical is going to be the slip surface along which the failure is going to take place and we will try to prove that this is theta equal to 45 plus phi by 2 or 45 minus phi by 2 for what cases. So this is going to be for active earth pressure condition and this is going to be for passive earth pressure condition. Now I can include as many as complications as possible. I will include a surcharge over here no issues. I may include a lateral pressure also which is coming because of let us say earthquake or which is coming because of some other activity which is going on the site. So what I can do is certain fraction of W I can include in this direction also which is because of let us say earthquake acceleration. So once you have done the free body diagram later on nothing much is required.