 retaining wall, because retaining wall is a very important structure in geotechnical engineering. So, to retain the soil, so you have to build the structure for this type of retaining wall. So, there are different types of retaining wall and we will discuss about various type of retaining wall, how to choose the dimension and what are the different factor of safety that we have to consider during the design of retaining wall. Then those things will be discussed in this class. So, first I will discuss about the different types of retaining wall. So, first the types of retaining wall, so first one is our gravity retaining wall, we can go for this type of arrangement, this is ground surface and this is, so this type is filling material. Now, depending upon its weight, we will get the stability for this type of gravity retaining wall and this is constructed for PCC plane cement concrete, we will not use any reinforcement for this type of retaining wall and this is not economical for large height. If the retaining wall height is very large, then this type of retaining wall is not very economical, but and then the next one is is semi gravity retaining wall, this type of retaining wall and this is ground surface, this is the retaining wall where the size of retaining wall is slightly reduced compared to the gravity retaining wall by providing small amount of reinforcement in the back face side. So, we can say that this gravity retaining wall, the most of the stability is given by the weight of this retaining wall and here reinforcement is not used. So, that is why we need, we have to use a very huge size of retaining wall, but in semi gravity retaining wall, the size of this retaining wall as compared to the gravity retaining wall is slightly reduced, is reduced by providing a small amount of reinforcement in the face of this retaining wall. So, this is reinforcement that is used and this is the filling material this side. Now, third type of retaining wall is cantilever retaining wall. Now, this type of retaining wall which is made of PCC plane cement concrete. So, this is the ground surface. Now, this type of retaining wall which is made by RCC reinforcement concrete and where these are the reinforcements and which is suitable for a height of 6 to 8 meter, suitable for a height of 6 to 8 meter. So, this and the next one another type of retaining wall is that is counter fort retaining wall. This is the retaining structure and this is the longitudinal side and where with some interval this counter forts are used. So, these are called counter forts and this is now this type of reinforced, this type of retaining wall the providing the use of this counter fort is to reduce the shear force and the bending moment in the vertical stream of the slab. So, this portion is called the stream of the retaining wall. So, this counter forts are reduced to this counter fort are used to reduce the shear force and bending moment which is induced in the vertical stream of the retaining wall or this and the base slab also. This is the base slab. So, to reduce the shear force and bending moment of this base slab and the stream this retaining counter forts are used. This counter forts are used with certain intervals. Now, economical for if the retaining wall height if this is the height of the retaining wall. Now, if height is more than 6 to 8 meter then this type of retaining wall is very suitable and this counter forts are used on the backfill side because this is our backfill side. So, this counter forts are used in the backfill side. So, these are the four different types of retaining walls. So, first one is gravity retaining wall where this most of the stability is getting from the weight of the retaining wall and where this we are using the plain cement concrete PCC. Then the next one is semi gravity retaining wall where the size of the gravity retaining wall is reduced by providing the small amount of the reinforcement. And the third one is counter fort retaining wall cantilever retaining wall where the RCC reinforced cement concrete is used for that type of retaining wall. Now, this cantilever retaining wall is suitable for the 6 to 8 meter height. Now, if a height of the retaining wall is more than 6 to 8 meter then you have to go for this counter fort retaining wall to provide the counter fort which is used to reduce the bending moment and shear force which is induced in the vertical stream and the base of the slab. And these counter forts are provided at the backfill side. Now, we have to design to these different types of retaining wall. So, first we will go for this gravity retaining wall and then before we go for the gravity retaining wall you should know what are the different types of factor safety you have to determine. What are the checks you have to do for this design of this retaining wall? First suppose if this is a typical section of a retaining wall say where this one is the backfill site and this is ground surface and this is foundation. So, this is our backfill site. So, this portion is filled with soil and this portion is void and here this is foundation soil. So, this retaining wall has to retain this soil pressure. So, that means the soil will give the lateral pressure on this retaining wall. So, now if I go through the free body diagram of this retaining wall structure. So these are the if this is the center of this retaining wall or if this is b by 2 this one also b by 2. So, total width is b and if this is the height of the retaining wall h then first this active earth pressure that will act at this is p a active earth pressure that will act at h by 3 distance from the base of this retaining wall because this and then similarly when we know that this retaining wall if it is move in this direction then this phase will be active earth pressure. Similarly, this soil this portion below the foundation soil this soil will also provide a lateral pressure in the opposite opposite to the p a. So, that is the opposite to the movement of this retaining wall. So, that means here act one passive earth pressure that will act. So, this is a passive earth pressure p p. So, this is and this one is the toe of the retaining wall. So, this passive earth pressure also act suppose this is the depth of the or the portion of the base of the retaining wall is at a depth of d from the foundation soil and where this soil will provide applied a passive earth pressure and this soil will apply active earth pressure. And now if we go for the reaction force for this base of the retaining wall. So, this because this weight of the retaining wall. So, that will act in this direction this is the weight self weight of the retaining wall that will act. Now, this is this reaction will act in this direction if I take the two components of this reaction R. Suppose this is this reaction is R and if we take the two components this is R H dash and this one R V dash. So, now this distance from the toe of this reaction force is say X bar and from the centre this one say E. So, now what are the forces that we will consider. So, that means we consider the active earth pressure we will consider the passive earth pressure we will consider the self weight of this retaining wall for the gravity retaining wall later on when you go for the cantilever retaining wall then for different theories you have to consider the weight of the soil also. So, this weight will act in the downward direction this active earth pressure will act from this direction and the passive earth pressure will act in the opposite direction. So, there is a friction that will act in between the base of the foundation or retaining wall and the soil. So, that means then this for this vertical and horizontal force. So, there is a two basically vertical and horizontal force then there is a reaction that will develop at the base of the foundation. So, if I take the two components of this reaction R one is R H dash one is R V dash. Now, say this R is acting as a distance of X bar from the toe of the retaining wall and so then that point of this reaction force is also at a distance of E from the center of the of the of the retaining wall. So, now if I go for this equilibrium condition say say R V dash is equal to W. So, that means this R V dash that is equal to the W this vertical components equal to this W another one this R H dash that is equal to this R H dash is equal to P A minus P P. Now, if I neglect this P P because this portion of force if I neglect this P P because this portion of force if I neglect because as this P P is very small compared to this P A and this P P will also provide additional safety for this retaining wall. So, if I neglect this P P and consider only P A by neglecting P P then R H dash will be equal to P A. Now, another condition we have to take moment from the with the respect to toe and that moment should be 0 for equilibrium condition if I take moment from the toe and this will be 0. So, we can write that R V dash into X bar that is equal to W and if W is acting at a distance of A or this acting at X distance of A from the toe of the retaining walls and this is W into A then minus P A into H by 3 as we are neglecting the P P. So, we can write that X bar is equal to W A minus P A into H by 3 divided by R V dash. So, we can write this all if we can write in the general form then W A that is the summation of all moment which is giving the resistance because this weight is actually giving the resistance and this P A is trying to overturn this retaining wall. So, it is pushing from this side and trying to overturn this retaining wall and this weight of the retaining wall basically providing the resistance. So, that moment is resistive moment minus summation of the overturning moment or M O and divided by all force of the vertical force. So, these are the vertical force. So, that means we can calculate the X bar in the summation of resistive moment minus summation of overturning moment divided by summation of vertical force. So, now we can write that our E eccentricity is equal to B by 2 minus X by. So, once we get we determine also we this X by by using this expression. So, we can determine the E value also by B by 2 minus X bar. Now, based on that what are the different stability checks that we will consider the first one is that there is a sliding between if I take this photographs here. So, there is a sliding between the soil and the retaining wall base. So, that means it may fail because of the sliding if I if the sliding is if the friction force is not enough in this base and of the retaining wall and the soil then there is a possibility of sliding. So, that sliding have to prevent. Now, sliding force that means for that means for no sliding condition we have to check the factor of safety. So, factor of safety F s for the no sliding condition the sliding force is a resisted by this vertical force into the coefficient of friction. So, this r v dash and mu and the sliding force is this is this is a possibility of slide because of this r h force. So, that is r h dash. So, where we can write that mu is the coefficient of friction between base of the wall and soil or you can write that is equal to tan delta. Now, in this case this factor safety should be equal to 1.5. So, that will give the no sliding condition. So, next one is overturning no overturning condition. That means this because of this force because of this p a force it will overturn this retaining wall and this weight will resist this overturning force. So, we have to no overturning condition that means we if we consider the first factor safety for no overturning condition that is the summation of m r divided by summation of m r o overturning. So, where summation of m r is sum of all resisting moment about 2 and summation of m o is sum of all overturning moment. So, for this previous case if I get the factor of safety for overturning case f a o or factor of safety for sliding f a s then f o that will give us for this previous case w a divided by p a into h by 3. And that should be also greater than 1.52. So, that is why we have to no overturning 2. So, next check that we have to do that for the bearing capacity check. So, that means next check is no bearing capacity failure. So, no bearing capacity failure means if the base soil or foundation soil is very poor then we have to check then there is a possibility that that soil will fail because of as the vertical load is coming on that soil. So, we have to check for the bearing capacity failure also whether this soil is capable to take the load of this retaining wall including that soil pressure that is coming on that foundation soil. Now, for that purpose we have to calculate the stress p max of the soil that will give us the r v dash all the summation of vertical force divided by width of the retaining wall into 1 plus 6 e divided by b. And or the p mean the minimum 1 is r v dash divided by b 1 minus 6 e divided by b. But here as we are considering the maximum stress so we work as taking this p max. And now for the factor of safety for bearing capacity failure that will be q n a divided by p max where q n a is equal to the allowable bearing pressure or this and this f s this should be greater than equal to 3. So, that means this soil pressure that it can take it can able to take and divided by p max so that we will get this factor of safety. Then first check fourth check that no tension failure no tension condition that totally depends on suppose if I get this is our base of the retaining wall. So, this stress distribution as we will go for this p max and p mean. So, you may get this type of distribution for the stress this stress distribution at the base of the retaining wall you may get this type of stress distribution this is p mean and this is p max. Now there is a possibility that we may get this type of distribution also where this will if this is positive then this will give us this p mean this is p max. So, this will give us the tension condition so, this will give you the negative stress was that will give us the tension condition. So, we have to avoid this no tension condition if this type of situation arises then you have to avoid this no tension condition and for the no tension condition it is occur when if a is greater than equal to b by 6. So, if e is less than b by 6 then this we have to redesign the dimension of the retaining wall to avoid the no tension to avoid the tension developed in the base of the retaining wall. So, for the no tension condition this e should be equal to greater than b by 6. So, these are the four checks we have to do during the design of retaining wall first we have to decide the rough step why if I go for the design of this retaining structure first we have to roughly decide the dimension of the retaining wall by based on the provided guidelines. Then we have to first see whether this retaining wall is safe against sliding and bearing or not if it is safe then you have to go for whether the this no tension condition is occurred or not then finally, you have to go for the bearing pressure calculation. So, all these four conditions if satisfy then we can decide. So, this dimension will provide for the retaining structure. Now, next one that for the other this checks addition to this checks what are the different other design criteria or guidelines. So, first we will go for the gravity retaining wall. Suppose, this is the retaining wall this is gravity retaining wall and this is the ground surface which is inclined at angle i with the horizontal surface this is our vertical stream. Now, provided guidelines suppose this is the existing soil and this is the back field. Now, this say is the depth of the retaining wall T and then for this if I consider the vertical line along the this face then you will give the this height of the retaining wall is H say. So, this is say height of the retaining wall H. Now, for the guideline which is provided that this top portion take around 0.3 meter or 300 millimeter. Now, this slope is 1 is 230. Now, this height of this base that is taken as H by 10. Now, with this guideline is we can consider 2 third of H. And this portion this extended portion is taken H by 6. Now, if I join this line with this extreme point and if this angle is alpha and this angle is eta then and what are the forces that will act. So, this is the weight of the concrete W c and this is weight of the soil you can consider. So, if I go that d value this d is at least 0.6 meter we have to provide this d value at least 0.6 meter. Now, as base width this base width of the retaining wall. So, base width we can consider 0.5 H to 0.7 H in between 0.5 H to 0.7. Now, this earth pressure that will act. So, now this earth pressure diagram if I consider this earth pressure. So, this earth pressure act this will act P a for this soil and here we also act the passive earth pressure this is active earth pressure. So, if we consider neglect this passive earth pressure. So, this earth pressure can be calculated either ranking formula or Coulomb's theory. Now, for this rankings theory if I consider this earth pressure can be computed either by rankings theory or by Coulomb's theory. Now, for rankings theory the shear zone should not pass through the stream. That means that this line is the extreme line of this condition if this line is passing through this stream then that is not acceptable if I consider the ranking theory. So, to satisfy that condition in this angle we can calculate this is 45 degree plus i by 2 minus phi dash by 2 phi is the friction angle minus sin inverse sin i divided by sin phi dash. Similarly alpha is equal to 45 degree plus i by 2 minus i by 2 plus sin inverse sin i divided by sin phi dash. So, now if i is equal to 0 then alpha is equal to 45 degree plus phi dash by 2 minus i by 2 minus i by 2. So, first you have to check whether this line is if I consider the ranking theory you have to consider check whether this line is passing through the stream or not. So, this for the shear zone this line should not be passed through this stream. So, this is the extreme point. Now, there is a few things that you have to clarify that if I consider suppose this is the retaining wall and we can use the rankings theory. Now, if I use the rankings theory suppose this is the two if I consider the same retaining wall. So, if I this is our foundation this is the ground circle this is backfill side. So, then we first we if I use the first case we use the rankings theory and next one if I use the Coulomb's theory. So, the same retaining wall if I consider for the Coulomb's theory for the different the forces that you have to consider that again if I join this point. So, this is alpha. So, here alpha as i is equal to 0 alpha will be 45 degree plus phi by 2 and then this angle. So now, this P a will act here. Now, this is our base now for the Coulomb's theory as we know this if I draw a vertical line or perpendicular line. So, this is 90 degree perpendicular line the face of this side of the retaining wall then P a will act with an angle delta of the vertical line. So, this perpendicular line P a will act with an angle delta whereas, in case of rankings theory it will act with the parallel to this backfill side size. If it is the angle is i then it will act with an angle of i whereas, it will act with parallel to this. So, here also this is the surface ground surface in parallel to ground surface. Now, what are the forces that we will consider this here the weight of concrete that we will consider again this is also the ground surface. Here the weight of this concrete we will consider in additional to that we will consider the weight of this soil also between this area. So, that means here we are not considering the weight of this soil here we are considering the weight of this soil and weight of this concrete also. But here this will act parallel to this ground surface here it will act perpendicular line with making an angle delta where delta is the friction angle between the soil and wall. So, delta is the interface angle between soil and the wall. So, these two things we have to remember when if we use the two different theories for this analysis part. Now, next one for the semi gravity retaining wall for the gravity we have discussed about different types of loading condition. Now, semi gravity retaining wall where this base width is slightly smaller as compared to the gravity one and rest of the design process are same. Now, next one you will go for the cantilever retaining wall design. So, for the cantilever retaining wall. So, for the cantilever retaining wall suppose this is the particular cantilever retaining wall. So, this is the ground surface in the backfill size which is making an angle i again. Now, this is the d depth of the retaining wall and this one is the h height of the retaining wall. Now, again if I consider the ranking expression then if we join this line this will give you the h and if I join the vertical line. Now this is the earth pressure distribution. So, this p a will act which parallel to the ground line and now the recommendation this as this top portion is 0.3 meter. Now, this angle is again 1 by 30. Now, this h this height of the base with the thickness of this base is 0.1 h this is 0.1 h. Now, this distance is also 0.3 meter. Now, this is 0.1 h this one this at the this t t junction this thickness is also recommended 0.1 h and this one is two-third of h roughly or this base width is two-third of h or 0.5 h to 0.7 h. Now, similarly thickness of this side is also is taken at it is either 0.1 h or h by 12 to h by 8. So, that means it is in between that is h by 10 this is considered here. So, either this is average value of this two is 0.1 h, but this range is h by. So, these are the guidelines to starting the to choose the initial dimension of the this retaining wall. So, now this one is this distance we can first choose then we will consider the weight. So, these are different weights first we will consider we will consider weight of concrete we will consider weight of soil if I take the ranking expressions. Now, for this condition what are the factor of safety we will choose for this type of retaining wall that first one that we will consider the factor of safety factor of safety against this is sliding. So, this factor of safety that will give us the total vertical force for the sliding into tan delta or delta 1. So, where this delta or delta dash where delta dash is the interface angle between the base of wall and soil. So, this is the interface of wall and soil. So, this is the interface friction angle between the base of wall and the soil. So, this is the summation of v into tan delta plus this if this total width is b. So, this base width is b b into c 2 c 2 is the cohesion as the base soil plus this p p if I consider the passive force p p that is also acting here. So, if I consider p p is the passive force. So, that will give you the base friction angle divided by p h or the p a that is horizontal force. So, that means this force this p p will act here this is the passive resistance that will act. So, p p b where b is the base width and c 2 is the cohesion of soil or foundation soil. So, there is two types of soil one is foundation soil where the retaining wall is resting another is backfill soil. So, c 2 is the cohesion of the foundation soil. So, that means the summation of all the forces into the friction angle divided by b into c 2 divided by p p this passive resistance plus this is plus b c 2 plus p p and divided by p a. So, this is for the sliding force that we can consider for this type of analysis. Now, if I consider the total free body diagram or the total forces of this type of retaining wall then this is the vertical portion. So, this is the retaining wall on this angle which is making an angle I this is ground surface. Now, for this edge so this is our active pressure that will act. So, with an angle I so p active with an angle I this is for the backfill soil. The next for this type of soil where this backfill soil generally this is a phi type of soil that means cohesion is not considered as a c phi type of soil and if it is a base soil if it is a c phi type of soil then you will get this type of distribution of the soil pressure. So, this is our c type of soil. Now, here we will consider this passive pressure and this is the reaction force that will act in the base in this form that is p min and this is p max. So, this reaction will act in this form. So, that will give you this is p min p max. Now, to calculate the weight suppose this for this top portion this will give us c 2 c root k p and for this bottom portion and this p p 1 that will act at a distance of d by 3 this is p p if this distance is d. So, this will give us c 2 c root k p and for this p p we can divide this part into two different portion one this is p p. So, one is p p 1 and another one for this one is p p 2. So, this is p p 2 that will act at a distance of d by 2. So, now finally, if I consider this stress again if I draw here. So, this is the passive force distribution for this portion so this is the d depth of the this foundation soil and then this stress value is 2 c 2 root k p and this value is k p gamma 2 d plus 2 c 2 root k p where k p is the coefficient of passive earth pressure and gamma 2 is the unit weight of the base soil. So, here this base soil. So, there is we can say the dimension this is the c 1 cohesion of the and phi 1 and this is gamma 1. Similarly, here also c 2 phi 2 gamma 2 most of the cases this c 1 is 0 for the black field. So, this c 1 is 0 for the black field and this c 1 is 0 for the black field case. So, this is the if I consider these forces. So, these are the pressure that will act. So, if I take the two parts one is this is p p 1 which is acting this is triangle at a distance of d by 3 from the base another is acting is p p 2 this is p p 2 which is acting as a distance of d by 2 from the base. So finally, so we can first we have to determine the stress of this triangle portion which is acting as a distance of d by 2 and then this stress of this rectangular portion which is acting as distance of d by 2 from the base. Then suddenly if we add these two stress then we will get the total passive force that is acting p p for this portion and then we can acting we can determine the point of application. So, suppose the if the h bar or d bar is the point of application of this passive force that we can determine that p p 1 into d by 3 plus p p 2 into d by 2 divided by p p 1 plus p p 2. So, total force so in this way first you have to calculate p p 1 from this triangular force this triangle that because we know the stresses at this edge and this edge and then p p 2 for this rectangular portion then we can determine d b bar. And this this stress p p p max and p min we can determine by using the previous expression then p a also calculate for the passive active force that is acting. So, now for the weight calculation we can divide this retaining wall into different section. Suppose so first we can consider this is the first section this is section 2 this is section 3 this is section 4 this is section 5. So, this 1 2 3 4 5. So, all these sections we can divide and you can take and then separately we can determine the weight of every section. And then finally, we can determine the factor of safety against the sliding overturning then bearing capacity and all those things. Now when we calculate this p a then we can calculate this p p p a value then p a this force will give us that half into gamma into h square into k a where h gamma is the unit weight of the soil h is the height of the retaining wall. Suppose this is the height of the retaining wall and k a is the active coefficient of active earth pressure. Now this coefficient of active earth pressure for the Rankine's and the Coulomb's there are different expressions are given. So, for the Rankine's theory if I use the Rankine's theory then this k a is equal to cos i into cos i minus root over h into cos i minus root cos square i minus cos square phi dash then divided by cos i plus root over cos square i minus cos square phi dash. And for the passive this passive earth pressure for the Rankine's case cos i that is cos i plus root over cos square i minus cos square phi dash divided by cos i minus root over cos square i minus cos square phi dash. Thus if i is equal to 0 then k a for the Rankine's case is 1 minus sin phi dash 1 plus sin phi dash and k p is just reversed 1 plus sin phi dash 1 minus sin phi dash. Now if I use the Coulomb's theory then you will get that k a is equal to for the case of Coulomb's theory k a is equal to suppose if this is our say retaining wall and here this force is acting here draw a perpendicular line is acting angle delta. So, and this is our horizontal line and this is our horizontal line and if this angle is beta say this angle is beta this is acting a delta is horizontal line and this angle will give us that 90 degree minus beta. Now here this is the value now for this type of k a will get by this expression that sin square beta plus phi dash. Now this is sin square beta sin beta minus delta then 1 plus root over sin phi dash plus delta into sin phi dash into sin phi dash into sin phi dash minus i divided by sin beta minus delta into sin beta plus i and total this is square for C 0 condition. Now similarly for the k p that will calculate by sin square beta minus phi dash divided by sin square beta sin beta plus delta 1 plus root over sin phi dash plus delta sin phi dash plus i sin beta plus delta sin beta plus i sin beta plus i sin beta plus then total square. So, by this way we can determine the passive and active coefficient of earth pressure for Rankine's theory and for Coulomb's theory if we use different condition. So, in the next class I will discuss about the different and solve a few problems to show that how to design design means to determine the dimension of the retaining wall by considering the different factor safety for sliding overturning bearing and no tension condition and to choose a proper dimension of the retaining wall. Thank you.