 So, I mean this is a good example of how would you start from the basics and analyze the factor of safety for the infinite slopes alright. Now suppose if I revert the direction of the seepage if it happens to be upward direction alright and that is what I cited sometimes back we have discussed this type of a situation there is a element of soil mass and then there is a seepage force which is acting at the base alright. So, the way we define seepage force is if I know the hydraulic gradient multiplied by yes gamma W per unit volume of the control volume or the soil mass per unit volume of the soil mass the total seepage force is I into gamma W. So, if I know the volume of the slice if I multiplied by I gamma W I know what is the order pressure in terms of the seepage force. So, can I compute here if the if the direction of the seepage is upwards n prime will be equal to what yes compute it quickly yes yes compute it. So, this will be gamma buoyant cos I minus of course B term is missing yes you are right gamma B was a buoyant what is the value of the port of pressure here if critical gradient I c if I depicted I c I c into gamma W into volume that is B into D. So, this is I c into gamma W into B into D that is it rest is same. So, what is the factor safety in this case? So, this is the case of upwards seepage n prime tan 5 prime and divided by this thing ok this whole thing is this divided by what W sin I. So, gamma B into B D sin I solve this expression what is that you are going to get 1 minus cos I also I will take out and this comes down. So, this becomes tan 5 prime over tan I multiplied by I c gamma W over what is the term which you are going to get gamma B cos I alright what is the significance of this gamma W upon gamma is taken as half alright. So, what is going to happen to this term 1 minus I c over cos I multiplied by here we are assuming that gamma W over gamma is equal to half. So, what is the significance of this if I c I c is the seepage gradient yes seepage force clear. So, the more the seepage force acting on the system the factor of safety is going to decrease that is right. Simple application of what we studied in the seepage theory how to find out the force acting per unit volume of the seepage and seepage pressure we have defined and seepage force and then we can find out the factor of safety. The last situation which I would like to discuss in case of infinite slopes would be let us extend this model to C phi material infinite slopes and then see what happens. So, principally what is going to happen you know wherever we have assumed C equal to 0 this component will come nothing more than that. So, if you are considering a C phi material when C is not equal to 0 your tau shear strength which is available would be C component plus sigma tan phi component includes C over here and that is it this becomes a C phi soil let us draw the free body diagram or the different type of pressure which are acting on the on the slice of the infinite slope. The only tricky thing here would be computing the pore of pressure acting at the base of the slice you can use the same concept still can mobilize C. So, for a C phi soil mass and infinite slope let us draw the free body diagram of the slice of the element which we have taken and show different forces which are going to come on that this is a ground surface sometimes people call this as ground level also does not matter there is a standing water table that means there is a seepage which is parallel to the slope and then I want to find out what is the stability of the system. So, this is a water table this becomes a standing water table this is the base of the slope infinite slope which is the critical one the way the failure is going to take place you have shear stress normal stress and this normal stress is because of the water table effective stress correct. Let us take the element how will you compute the pore order pressure at the base put the piezometer and let it cut the free attic surface this is the free attic surface alright what is the intuitive feeling if you place the piezometer over here what is the height up to which it will go up to here up to this line is cutting this is the free attic surface draw a nothing doing is not going to happen like that so this is your flow line free attic surface flow line draw the equipotential perpendicular to the flow line passing through the point which is sitting at the base of the slice is correct then only you can find out the pore order pressure so meaning thereby the equipotential line has to be touching this point crossing this point and perpendicular to the flow line so this is the point where I want to find out the pore order pressure if you put the piezometer here this is cutting this is the equipotential line equipotential line which is cutting the top flow line so this point has been obtained correct so this was made misleading why because you have to draw the equipotential line and the flow line and the intersection of the two so what is the height of the what is the value of hw here perpendicular from here to this that is it this is the concept so this remains b yes you are right this is d the depth of the flip surface this is the height of the water table normally we define as this is the slip surface there is a factor which is defined as n which is equal to z by d so can you compute now what is the value of pore order pressure at this point this is z yes so this thing will be z cos i and again projection of this on this plane so this is the value of hz so that means uw will be equal to what z cos square i and because we are finding out uw multiplied by gamma w so uw is as you know gamma w into hw that is it the moment uw is known what we have to do put the factor of safety term so factor of safety here will be equal to the shear strength available divided by the force which is acting destabilizing force correct what will be that value w sin i whatever steps I have followed over here you have to follow the same steps for computing everything the only thing is that this shear component will be equal to c multiplied by the length what is the length the base at which it is acting so what is the base length this will be b sec i correct so c prime multiplied by sec i and then effective stress upon w sin i so if you solve this expression what you will be getting is you will be getting I am just skipping the steps and writing the final step which will be equal to the factor of safety will be equal to c prime over gamma d sin i into cos i this term remains almost fixed sin i cos i into gamma d c prime by gamma d itself is a non dimensional number cohesion divided by gamma d alright now what is going to happen here we had this term 1 minus alright this term is becoming some parameter like n so this will become 1 minus what n times gamma w over gamma multiplied by tan phi prime over tan i so if you solve this expression by putting factor of safety equal to 1 because this is the most critical condition so for fs equal to 1 what is going to happen c prime by gamma into d can be written as cos square i tan i minus 1 minus n gamma w over gamma okay into tan phi prime this is the expression which will be getting this term is defined as r u the pore pressure parameter so in the simplest possible form this expression can be written as c prime by gamma into d will be equal to cos square i tan i minus r u 1 minus r u into tan phi prime so such a complicated situation we have brought down to a simple relationship and here the way you will read this is this is the critical depth so d may tend to become hc the critical depth of the surface so this d may attain criticality this is also a sort of a stability number c prime by gamma d if you remember when we are talking about the unsupported cut of the height of the vertical cuts I use this term c by gamma d is a sort of a stability number alright so this can be written as c prime by gamma hc equal to cos square i tan i yes c prime by gamma hc will be equal to cos square i multiplied by tan i minus yes the c this r r u term is n gamma w over gamma and suppose if I say n equal to 1 fully submerged case z becomes equal to d what a surface is on the ground surface what will happen then so this will become 1 minus gamma w over gamma gamma minus gamma w is gamma buoyant yes over gamma multiplied by tan phi prime so this is a situation for a totally submerged slope what is the value of gamma b by gamma this is equal to 1 fine so what is the significance of this c prime over gamma hc equal to cos square i multiplied by tan i minus tan phi prime how many parameters are contributing to the stability of the slope this is what actually you have to find out you should realize this by the fact that this term remains same alright c prime by this term actually gets added up so what cohesion is doing mobilization of cohesion is giving more factor of safety to the slope and that is right so when both cohesion and friction are getting mobilized both c and phi are coming to the picture factor of safety is going to get enhanced what cohesion does it induces power of pressures and the draining condition on the boundary conditions okay so this water table because of the presence of partial submergence of the slope the factor of safety has been computed and what we have done is we have extended this analysis to a situation where the entire slope has been assumed to be submerged and mathematically what we have done is we have put n equal to 1 so this water table goes and sits over here so as the water table rises in the slopes the factor of safety keeps on decreasing suppose if I ask you to draw the dependence of like this is the function I can always plot c prime by gamma hc as a function of i is this correct so more the inclination angle what is going to happen to c pi c prime by gamma h is going to reduce as i increases what is going to happen to the this function it is going to decrease okay as phi prime increases what is going to happen so more the value of phi prime what is going to happen the factor of safety is again going to decrease and we use this term n over here pore-water pressure parameter so I can plot this function with respect to pore-water pressure also what is the contribution of pore-water pressure so if n is lower this term is going to be higher so what is going to happen to the factor of safety optimize it you do realize the situation so we have discussed situations where infinite slope is made up of dry sense it is made up of dry sands and then there is a seepage pressure there is a upward seepage pressure which is acting on the system and we have also talked about a submerged situation and we have extended this situation to a complete submergence and what we are observing is that how the factor of safety can be obtained this is a complete submergence because this water table has reached up to this point what is critical here is just to obtain the pore-water pressure function and once you obtain this that is how the things are simple. So if I solve this equation further C prime over gamma h c this is equal to cos square i tan i minus tan phi prime one of the ways to interpret this would be that if I plot h c the critical height of the slope and as a function of i you know normally the solutions are valid for i greater than 10. So at 10 degree i if you plot the variation of h c with respect to i this is how you will be getting it so what it indicates is h c and i are inversely proportional in otherwise also you know you remember this is the slope and this is somewhere we have taken as d which is equal to h c. So more the inclination of this slope infinite slope which is having water table what is going to happen the h c value is going to drop. So otherwise we can also plot it as c prime by gamma h c as a function of i another way of plotting this could be you have c prime and c prime is plotted with respect to i the way we read this graph is if I have to design a slope or if I have to do some retrofitting for a given angle of the infinite slope given angle of the infinite slope what should be the value of c prime or vice versa if c prime is known what is the value of i. So this can be extended again as 10 degree and then we have a straight line and this straight line is c prime equal to gamma into h c multiplied by this factor one more parameter which can be plotted with respect to i would be c prime and the way we read this is for a given c prime value what will be the critical value of i which is going to be stable. So here again we will have this 10 degree as the threshold and beyond which we will have a nonlinear line okay and what is the function this function is equal to c prime into gamma h c multiplied by x this is the value of x. So these type of charts can be utilized quite well in the engineering practice whatever we have discussed so far there could be an alternate situation where the ground water table if I say that this is the ground surface this is the slope and at this point we have 5 d c d getting mobilized on the slip surface the water table is somewhere here alright and if I say that this unit weight is gamma 1 and this section is of h1 height the total height of the slope is h it is a reverse situation because what we did in the previous analysis we took the height of the water column above the critical surface as n times d that was the z value alright and suppose if I ask you to find out the stability number so you try to prove this yourself by using the same principles what we have discussed so far what we will be getting is as c d over gamma into h c where h c is equal to the critical height or the depth of the critical surface and this will be equal to cos square i now this term will get modified a bit 1 minus h1 over h gamma minus gamma 1 over gamma and this whole thing multiplied by tan of i minus gamma b over gamma plus h1 over h gamma 1 minus gamma b over gamma tan phi d and this is the stability number which we have got you should attempt proving this equation the concepts remain same the only thing is sign convention has got changed in the previous problem what we did is we considered this as z alright and the unit weight which was beneath the water table or below the water table was considered as gamma submerged thank you