 last class we have started the earth pressure in the retaining wall and different earth pressure theory this is the basic introduction as this earth pressure has been used design of reinforced earth soil mass so we have covered with this earth pressure at rest if there is a soil in a homogenous natural soil deposit for example here it is shown the ratio of sigma h prime to sigma b prime is called k that means that is your earth pressure called k that means k is your coefficient of earth pressure at rest coefficient of earth pressure at rest it is called coefficient of earth pressure at rest which termed as a this thing has been termed k zero so to arrive this k zero state that means coefficient of earth pressure at rest means there are no lateral strain during loading that means one d consolidation should be there now estimation estimating k zero for normally consolidated soils k zero is equal to generally we take one minus sign five for over consolidated soils k zero is equal to one minus sign five prime ocr over consolidation ratio to the power sign five to the power sign five for elastic analysis that means for linear or elastic analysis k zero is equal to mu by one minus mu mu is your poisons ratio we have started this also last class active and passive state or active and passive earth pressure if you look at here there is a smooth retaining wall now this wall is moving away from the field it has been shown moving away from the field because of your backfill material so backfill soil then this is called in active states active state means wall moves away from the soil if you look at here if you look at here because here is your soil mass it try to push this wall that means in this case particularly in this domain in this domain wall is moving away from the soil but in this domain wall is pushing the soil wall is wall is pushing the soil that means wall moves towards the soil in this region in this region walls moves away from the soil that means wall moves away from the soil it is called active state add walls moving towards jumps The soil it is called passive state If you look at here soil mass in the Snow in the active state Soil mass in the Sun You can see that it is a passive state or may be a wall movement because of the wall movement this happens Next part active earth pressure If you look at here erzähmy queen is минут fashion that is generally calculated as Gama Z So what included to a చోలాతోలాతోలషియైందిజిందిందానిందిదానితెదుం踏నఽిహాందాది నమారేపూతర క్సిలుపాస్లలా ప్తేక్రిందిందిందిత్దిసిదిందాిందిందిత� sigma h prime is k 0 times sigma v, k 0 is equal to sigma h prime by sigma v prime, so which is equal to k 0 times gamma z, as the wall moves away from the soil initially what happen wall is the constant there is no movement, now wall is moving away from the soil, so what will happen sigma v remains the same that means over button pressure remains the same at the same time sigma h decreases till failure occurs that means sigma h that is your lateral pressure decreases till failure occurs, so this is called your active state this is called your active state that means if I define in the active state a soil mass before the active state if there is a soil mass in the soil mass there are if I take it into consideration of plane strain conditions in two dimensional, so in this case there are two stress one is your vertical that is your sigma v and sigma h is your lateral, so once this soil is trying to push this wall that means wall is moving away from the soil, so in this case what will happen sigma v that means your over button pressure over button stress will remain constant it is not going to change however sigma h it decreases it is decreasing the sigma h is decreasing if this is the sigma h it is decreasing it is decreasing decreasing, so till the failure occurs, so if I take the for normally consolidated soil draw the Mohr circle for particularly active earth pressure conditions, so in this case this is your initial state of your stress if I draw the Mohr circle, so this is your sigma v prime this is sigma o prime and this comes out your Mohr circle, so initial this is your k zero conditions that means earth pressure at rest conditions, so once this soil movement starts what will happen sigma h decreases if you look at here this is your sigma v prime this is your sigma v prime this is your sigma h prime now what happen initially sigma h prime is your here now as the wall moves sigma h prime is decreases that means it is decreasing till failure occurs how do you know that till failure occurs that means the Mohr circle envelope Mohr circle will touch your failure envelope, so this case this case is called this Mohr circle is for your active earth pressure for your active earth pressure, now active earth pressures it has been given by rankine 1820 to 1872 WJM rankines, so if sigma h if I write it sigma h prime active because sigma h is decreasing sigma h is decreasing that means it is coming towards the that means it is your k a into sigma v prime k a into sigma v prime, so k a is equal to 1 minus sin phi by 1 plus sin phi which is equal to tan square 45 degree minus phi by 2 this is your rankine coefficient of active earth pressure k a, now failure plane once you once you draw the Mohr Mohr Mohr circle sigma v prime and sigma h prime active state the failure and failure plane is at 45 degree plus phi by 2 your horizontal and this this is your failure plane at 45 degree plus phi by 2 this horizontal to this horizontal then in granular soil if you draw the active earth pressure how it how how this earth pressure distribution become if there is a as wall moves away from the soil if you look at here this is a wall and this is your soil as wall moves away from the soil mass wall moves away from the soil mass, so what happen initially it is in k 0 state initially it is in k 0 state k 0 state means earth pressure are traced condition initially it is in k 0 state, so what will happen it will decrease decrease and it will remain at failure beyond thus there will not be any more further movement, so it will be a active state this is your wall movement beyond thus there is no further movement, now for calculating this active earth pressure in cohesive soils if I divide into two parts of the soil one is your cohesion less soil other is your cohesion soil follow the same step as for granular soil only the difference is that c is not equal to 0 in case of cohesion soil in case of cohesion soil c is not equal to 0 c is equal to your unit cohesion everything else has same as for your granular soil, so sigma h prime for active is equal to k a sigma b prime minus 2 c root over of k a, so sigma a prime sigma a prime is equal to sigma 0 prime sigma 0 prime into k a minus 2 c root over of k a this is the derivation for active earth pressure calculations in cohesive soils, now rankines has given because this active state has been given by your rankines based on that this theory whatever he has given based on his name this is called rankines theory, so rankines theory initially what we have discussed this is a plain ground surface earlier we have discussed for planar ground surface or may be plain ground surface there is no variation, now this ground surface has been inclined at an angle alpha that means with a slope backfill in that case in that case how this how this resultant of earth pressure acted if it is a straight if it is a plain then what will happen it will act like this and the distribution will become distribution will become like this, so it will act at a distance of h by 3 once this ground surface is slope that means slope backfill obviously this will make an angle this is your resultant pressure it will make an angle with your horizontal at an angle alpha which is equal to your slope angle of your alpha, so k a has been derived as cos alpha into cos alpha minus root over of cos square alpha minus cos square phi divided by cos alpha plus cos square alpha minus cos square phi root over where sigma a is equal to gamma z k a and p a this is more important p a is your resultant earth pressure which is equal to it is simple geometry or area of the triangle if I take it it is your half gamma h square into k a, so rankines earth force rankines active force rankines active force is parallel to your slope of the backfill that means this rankine active earth pressure or active earth force active force you can say active force is parallel to your parallel to your slope of the backfill this is your slope of the backfill parallel to your slope of this backfill now generalize case for rankines active pressure in granular soil the rankines active force is no no longer parallel to slope of the backfill in which cases one is your parallel to your slope of the backfill other cases it is some cases also parallel not to the backfill in which cases when this retaining wall if you look at here when this retaining wall is not purely vertical is not purely vertical it makes an angle this is called it makes an angle theta that means retaining wall itself it is battered there is an angle so this is your theta this angle is your theta and this angle is your alpha alpha is your slope angle this ground surface is sloping that means this is your this is your ground surface it makes an slope with an alpha now in this case this will be your beta the resultant earth pressure will make an angle beta and beta is equal to beta is equal to tan inverse sine pi sine jay one minus sine pi cos jay now jay has been defined as a sine inverse sine alpha by sine pi minus alpha plus two theta so then based on the jay and beta we can calculate it we can calculate it if you know jay and alpha k you can calculate it so pa is equal to half gamma h square into k a now there are certain examples we can discuss we can find it out also solved examples we can how this earth pressure calculation how there is a example suppose this is a retaining wall for example this is a retaining wall of height 6 meter 6 meter and gamma is equal to save this backfill has to be made is unit weight of the soil is equal to 16.5 kilo Newton per meter cube 5 prime is equal to 26 degree c prime is equal to 10 kilo Newton per meter square and q surcharge is equal to 15 kilo Newton per meter square now the question is find it out or determine the distribution of active earth pressure that means in this case in this case there is a surcharge q is equal to 15 kilo Newton per meter square q is equal to this is called your surcharge q is equal to 15 kilo Newton per meter square this is you can say that this is surcharge and if you look at this ground surface is plain it is not slope that means your alpha is equal to alpha is equal to zero it is a very straight forward very simple problem we are starting now if you come back to here sigma h active is equal to sigma h active is equal to because in this case if you look at here this is not purely cohesion less soil rather it is a c pi soil that means c is there also pi is there so in this case the soil is a c and pi both c and pi parameter is there so if you go back to this whatever they have derived for c pi soil so sigma h active is equal to k sigma v prime minus 2 c root over of k so k is equal to we can find it out k active earth pressure you can coefficient tan square 45 degree minus pi by 2 so pi is given 26 degree so it is your 26 degree by 2 it is coming about to be 0.39 now r z is equal to at initially how do you start this earth pressure distribution diagram if you put if you look at this equation what are the parameter one is your k a other is your sigma v prime sigma v prime is equal to gamma z that means you will start the variation gamma is fixed that means the variation of gamma is fixed that means the variation will be your z so you start with the z you start with the earth surface let us say z is equal to 0 that means sigma h prime is equal to k a sigma h prime is equal to k a k a is equal to 0.39 into sigma v prime this comes to be 0.39 so at z is equal to 0 so this is your sigma v prime then 2 c into root over this so this is coming about to be gamma into z this to be 0 gamma is equal to 16.5 this to be 0 it comes out to be minus 6.6.64 so if I take it if I take it this is this is wrong sorry this is your k a into sigma v prime if you look at here sigma v prime why it is not 0 sigma v prime in this case sigma v prime is equal to if you look at here in this case sigma v prime is equal to gamma z gamma z is your gamma z is your over burden plus your surcharge the surcharge is already there 15 kilo Newton per meter square the surcharge has to be surcharge effect has to be taken into consideration so that is why here z is equal to 0 gamma prime into z which is equal to 0 now sigma v prime is equal to q so q is equal to q is equal to 15 so it will be 0.39 into 15 minus minus which is your 2 into c and root over of 0.39 if you look at here minus 2 into c is equal to 10 c is equal to 10 and root over of your k a k a is equal to k a is equal to root over of this this is your 0.39 so sigma h prime is equal to minus 6.64 kilo Newton per meter square now once you have started at z is equal to this z is equal to 0 you can take also at intermediate point z is equal to 2 z is equal to 3 z is equal to 4 also you can find it out if you are a beginner then you can start with this z is equal to 6 that means at the base of the wall so z is equal to 6 sigma h prime is equal to this because 15 plus gamma into z gamma gamma into gamma prime into z so gamma is equal to your 16.5 into z is equal to 6 this comes out to be 31.97 kilo Newton per meter square what does it mean if you look at this example that means at the surface at the surface when z is equal to 0 when z is equal to 0 that means there is a negative force there is a negative pressure that means sigma h prime is equal to minus 6.64 it is a negative pressure so if you come back now this if I draw the pressure distribution diagram or earth pressure distribution diagram if you if you look at this earth pressure distribution diagram here it is coming about to be here it is coming about to be at the top it is your minus 6.60 64 kilo Newton per meter square at bottom we are getting or bottom we are getting 31.97 here 31.97 kilo Newton per meter square now if I draw this is the pressure distribution earth pressure distribution diagram this part will be a minus this is your positive or plus so now now next step you will have to find it out next step if you look at you will have to find it out at what distance because this negative will go up to what distance you take it you take it earth pressure as if in this case what will happen in this case your earth pressure will be 0 so you will have to find it out the point where is your earth pressure is equal to 0 if you take this 0 then you will have to find it out what is the z distance other thing you know c k and sigma b in terms of sigma b prime the z term is coming it comes out to be z is equal to 1.03 meter where from negative it will start it will become 0 then it will go towards your positive now second example second example determine the distribution of active earth pressure if this is a wall there is a retaining wall or this is a retaining wall you can say this gray color this is your retaining wall and there are two soils one is your one case one case the phi is equal to 0 that means it is purely cohesion less soil sorry it is c is equal to 0 so phi is equal to 30 degree that means this is a purely cohesion less soil and second case c prime is equal to c prime is equal to 0 phi prime is equal to 35 degrees second layer also it is also cohesion less soil that means both the cases cohesion less soil of gamma different and there is one case additional is there ground water table is lying at a distance of three meter below the below this top below the top so this is the conditions so find or determine the distribution of active earth pressure now as usual you find it out what is the value of k a 1 minus sin phi by 1 plus sin phi your phi for layer 1 your phi is equal to 30 degree for layer 2 for layer 2 your phi is equal to 35 degree for layer 1 what is the value of k 0.333 for layer 2 what is the value of k 0.271 so then you find it out at z is equal to 3 that means at z is equal to 3 this is the boundary that means two soil surface boundary and water table is there so sigma h prime is equal to gamma z into k a that means this is your k a k a gamma prime into z so for top layer it is your 16 into 3 into gamma a 16 kilo Newton per meter square then again z is equal to 6 at 6 the value is coming about to be 19.67 kilo Newton per meter square if you look at here if you look at here how this pressure distribution diagram comes into picture if you look at here if I am taking if I am taking let me let me taking into two parts for this z is equal to 3 meter where is your pressure will start it will start from 0 it will end here for z is equal to 3 meter this soil also comes into picture you can calculate also you can draw so this is your resultant this is your pressure distribution diagram now with this we have not taken into consideration of water table so additionally lateral hydrostatic stress also taken into consideration if you take it to u is equal to what is your hydrostatic pressure u is equal to 9.81 into 3 that is your 29.43 kilo Newton per meter square in this case this is your additional hydrostatic pressure will come into picture because water table is lying at the interface of both the soil layers so in this case if you go if you add your additional layer of hydrostatic stress then what will become with this this is your because of soil this is your earth pressure and this is because of your water if you add it how this earth pressure distribution diagram it looks this is your earth pressure distribution diagram now if i sum summarize it now if there are two soils of different properties in this case one soil gamma is equal to 16 gamma is equal to 16 kilo Newton per meters q other case gamma saturated is equal to 18 kilometer per meter q 5 prime is equal to 30 degree and here 5 prime is equal to 35 degree in this case in this if you look at this case in this case first you find it out at z is equal to 0 then find it out earth pressure at z is equal to 3 so once you come at the transition this is a phase where this soil as well as this soil is there you calculate for this soil z is equal to 3 how much is your pressure for this soil also you calculate how much is your pressure then you calculate at z is equal to 6 what is your pressure then you because of soil you find it you find what is your net means earth pressure distribution then after that once there is your water table is there additional pressure because of your hydrostatic stress that you add it then this becomes your sum of your stresses these are the two typical examples i have solved explained the next phase is your coulomb's active earth pressure distribution diagram this coulomb's active earth pressure distribution diagram maybe we can discuss in the next class thanks a lot