 We have finished this Rankine's earth pressure theories, now Coulomb's active earth pressure, so Coulomb's theory was developed in 1776, so in this case what will happen, the failure surface is assumed to be a planner and they have considered the friction between friction angle between soil and the wall, if you look at here this Rankine's case this wall is assumed to be a smooth wall, so the friction angle between this wall and the soil is not taken into consideration, in Coulomb's theory they have considered this friction angle between soil and wall and which is taken as delta, the active earth pressure is calculated based on the equilibrium they have taken a planner failure surface and based on the equilibrium means they consider the weight of the in that planner failure surface weight of the soil as well as PA active earth pressure resultant and they make the force polygon so that it has to satisfy this equilibrium conditions, so what will happen different failure surfaces are attempted, so this is your number one failure surface like this you can attempt number two, number three, number four until the largest PA is obtained until this largest active earth pressure is obtained from there coefficient of earth pressure active earth pressure kA is found out to be sin square beta plus phi, if you look at here beta, beta is your angle wall angle, wall angle it makes with this vertical that is your beta plus phi, phi is your angle of internal friction of the soil and sin square beta plus sin into sin beta minus delta into 1 plus sin phi plus delta into sin phi minus alpha, alpha is your, if you look at here alpha is coming out to be alpha is your searcher's angle this is your alpha, this is your alpha, this failure failure surface planner failure surface makes an angle with this, this is your theta 1 and theta 2 and from there you can find it out PA, PA is equal to half gamma h square into kA, now passive earth pressure in case of granular soil initially soils is in k0 condition, if this is your retaining wall, if this is your retaining wall, if you consider this is your retaining wall and this is your soil mass, initially the soil mass is in rest condition that means this is your sigma v prime and sigma h prime, then as the soil moves towards the soil as the wall sorry as the wall moves towards the soil that means this is your wall it moves towards the soil what will happen, the sigma v prime that means your searcher's it remains same, sigma v prime remains same, sigma h increases till the failure occurs this is the difference where in case of active state, active state sigma h decreases till the failure occurs, in case of passive state sigma h increases till the failure occurs, now this is called your passive state that means sigma h increases increases till the failure occurs, now if I come back to in terms of Mohr's circle diagram, if you look at here initially this is your k0 that means earth pressure at rest condition where is your sigma v prime and sigma h prime, what will happen as the wall moves towards the soil, so passive earth pressure in granular soil if you look at this in terms of Mohr coulomb, so Mohr coulomb failure and Mohr coulomb circles initially it is in a k0 state that means it is at rest then wall moves towards the soil, so what will happen it will go it will increasing the sigma h initially the sigma h value, initially sigma h value is here that means if you look at this value here sigma h is increasing from here to here, so till it touches your failure envelope touches the failure envelope that means at this point your failure occurs it touches this Mohr's circle failure envelope, so this is called a passive state or passive earth pressure this is your passive earth pressure, now sigma h prime of this passive is equal to k p into sigma v prime, so Rankine's coefficient of passive earth pressure, so k p is equal to 1 plus sin phi by 1 minus sin phi this is Rankine's earth pressure, so where k p is your 1 in terms of phi there is no delta as I said Rankine's theory assume that this wall is a smooth, so in case of coulomb's theory the friction angle between the soil and wall delta has been taken into consideration, so k p is equal to 1 plus sin phi by 1 minus sin phi where are in case of active case k a is equal to 1 minus sin phi by 1 plus sin phi, so this will be tan square 45 degree plus phi by 2, now passive earth pressure in granular soil, so failure plane is at 45 degree minus phi by 2 to the horizontal, so this is your sigma v prime this is sigma v prime, sigma v prime and sigma h prime passive state passive earth pressure, now if you take at this as wall moves towards the soil that means sigma h, this sigma h, sigma h increases, sigma h is increases this wall is moving towards that means wall is generating pressure towards this field till the failure that means this is your k 0 condition that means earth pressure are traced it will increase, increase, increase and it will reach here this is called your passive state in terms of graphical representation and as well as your wall movement for cohesive soil the steps are same for cohesive soil for calculation of your passive earth pressure the steps are same like your granular soil only difference is that c is not equal to 0, c is not equal to 0 that means cohesion force is not equal to 0, c is not equal to 0, so sigma x prime passive is equal to k p into sigma v prime plus 2 c root over of k p, now coulomb's passive pressure the k p this things I said earlier also p a is equal to half gamma h square into k a, suppose for example if you take the value of phi is equal to 40 degree, suppose this is a back field soil of interfictional angle of the soil phi is equal to 40 degree delta that means coefficient of friction between soil and wall is equal to 20 degree, so k p value comes out to be 11.77, so passive earth pressure that means what are the limitation the fundamental assumption of plane failure surface does not hold in practice, if you look at this assumption by coulomb's theory the fundamental assumption of plane failure surface does not hold in practice in practice it is not possible to be a plane failure surface the actual passive earth pressure is smaller than that coulomb's theory which is not safe whatever the actual passive earth pressure observe it is much smaller than your coulomb's theory, therefore passive earth pressure is estimated using either Rankine's theory or the result of ultimate equilibrium based on non-planar failure surface, generally it is recommended either you go for passive earth pressure calculations by using Rankine's theory or ultimate equilibrium based on non-planar failure surface, now if you look at case of non-planar failure surface that means logarithmic spiral if I take into considerations of a granular soil gamma and phi prime and c prime the chart has been given for different friction angle of phi, phi varying from 20 degree to 45 degree and k p value is given and delta by phi also varying from 0 to when delta by phi is equal to 0 it will be same as like Rankine's theory, so it is varying from 0 to delta by phi is equal to 1 delta by phi is equal to 1 that means the frictional angle between soil and wall and interfictional angle of soil is equal to same means is equal to 1 that means it is same that means soil and soil with wall act as a rigid body so with this chart once you know the once you know the value of phi then you can find it out what is the value of k p depending upon suppose this is the value of phi is equal to 40 degree and delta is suppose phi is equal to 40 degree for delta by phi suppose delta by phi is equal to 0.4 from this chart we can get it directly k p that means it is a non planner of failure surface, so k p is coming about to be 8.2 if I go by Coulomb's theory for phi is equal to 40 degree delta is equal to 20 degree k p is coming about to be 11.77 if you look at here by Coulomb's theory it is coming 11.77 and for by means of non planner theory it is coming about 8.2 and for Rankine's theory k p is coming about 4.6 that means actual value of k p is lying between Rankine as well as in between this Coulomb's theory so nature of active and passive earth pressure suppose you consider a 6 meter of wall 6 meter of wall so this is your this is your value of k 0 state at rest so you can say that active state it comes about to be 25 mm then passive state it comes about to be 250 mm how this nature of active and passive earth pressure is varying from active to passive state this is the variation then where it has been applied this active state and passive state we have discussed lot about design of retaining walls and design of reinforced earth walls these are the applications look at the few applications retaining wall applications now this is a highway there is a retaining wall already existing a building so first case is your if you look at this case one this is this one part is your retaining wall that means retaining wall suppose to retain the soil mass this soil mass has been retained soil mass has been retained so this soil mass will tend pressure on this wall it will be in the active so above the soil mass you can go for a construction also and nearby also here there is a highway this also exert pressure so this is one example where the active and passive state come into picture now second one is your gravity retaining wall cement motor and cobbles it is there one one way so it will act they rely on their self weight to support the backfill particularly gravity retaining wall they rely on the self weight they rely on the self weight self weight to support or to support the backfill because the backfill will exert some pressure by means of gravity retaining wall as I said also earlier in the beginning this will the stability analysis will be done based on its self weight so generally the self weight by means of self weight it will counter this pressure coming from the backfill to the wall now reinforced earth wall reinforced earth walls reinforced earth walls generally it is smaller than your gravity walls if you look at they act like a vertical cantilever this is just repetition this this I have already explained in the beginning of this beginning of this lecture what are the different types of retaining walls so in case of reinforced cantilever retaining walls what will happen cantilever retaining walls there they will support this earth pressure by means of cantilever actions this is this will be your by means of cantilever action they will support this earth pressure and it will be fixed to your ground surface now proportioning this retaining walls also I have discussed earlier so proper placement of concrete at the top and adequate with beam shear resistance adequate with beam shear resistance then adequate slide resistance then different failure failure of your retaining wall or stability analysis of retaining wall we have already also discussed overturning about toe sliding along its base this is overturning about the toe sliding along the base loss of bearing capacity deep seated shear failure so the design of retaining wall generally done in two phases that means check for slide stability as a whole and check each component for strength and steel reinforcement so this things has been covered also earlier this has been I have covered earlier so no need to discuss so bearing capacity failure it should be greater than one at this bearing capacity also we can calculate the stability analysis of bearing capacities so wall friction angle if coulomb's active earth pressure theory is used this is required generally this is required suppose if you are using by means of coulomb's theory active earth pressure theory then backfill material if you are taking gravel this range of delta is should be bearing between 27 to 30 degree for coarse sand it should be bearing 20 degree to 28 degree fine sand it is varying 15 degree to 25 degree steep clay it is varying 15 degree to 20 degree and silt clay it is varying 12 degree to 16 degree generally the delta value has been provided one half or two-third of phi half or two-third of this phi generally delta value has been taken this is all about this course of retaining wall means earth pressure theories and discussions