 In last class, I have discussed about the cantilever sheet pile. Now, in this class I will discuss about the anchor sheet pile. Now, the idea of this use of this anchor in a sheet pile is that as I have already discussed that the main difference between the normal retaining wall and the sheet pile is that in the retaining wall, the depth of foundation is less compared to the sheet pile, where the sheet pile most of the resistance is coming from this foundation depth. So, the required depth is very high as compared to the retaining wall, but sometimes to reduce this required depth is anchor is used. So, this type of sheet pile is called the anchor sheet pile. So, now first if I go for this anchor sheet pile condition, so similar to that this is the top ground surface, this one is the dredge level and this is the cantilever sheet pile. Now, here sometimes to reduce this required depth one anchor is used here. So, this is this that means this here apply tension is applied with the this is the sheet pile, where this tension is applied to this anchor, then it is anchored in the ground. So, now here this type of sheet pile that is called anchor sheet pile. Now, here this is basically the anchor, this is this one is anchored within the soil. Now, that means additional tension or force is applied. Now, here when we do the analysis that means ultimately here also similar to the cantilever sheet pile, we have to determine the required depth of this sheet pile below the dredge level, this is the dredge level. Now, here two methods that is applied, one is our free earth support method, this is the these are the analysis methods, one is free earth support method, another is fixed earth support method. So, two type of analysis that we can do, one is free earth support method and it is fixed earth support method, free earth support method, here we assume that surface this end is free. So, now if I consider the first free earth support method, then the deformation shape of this sheet pile is considered like this. So, that means it will deform in this form and then this will be the shape. Now, if I consider two different types of soil, one is your first will go for the granular soil and next we will do for this for the clay soil or cohesive soil. So, we will do this analysis for this free earth support method in granular soil and we will do the analysis in the cohesive soil also. Now, first that we will do this analysis for the free earth support method. So, first we will do for this is for the granular soil. So, if I consider one anchor sheet pile in the granular soil, so this is dredge level, this is the sheet pile, this is the top ground surface, this is dredge level, where this sheet pile, this applied force is applied here, that is the F force which is applied due to this anchor and this is the required depth of the foundation below the dredge level. Now, the water table condition, suppose this is the position of the water table, this is the ground water table, position of the ground water table. Now, the load distribution diagram that we can do that this would be the expected diagram of the, see this is the earth pressures diagram corresponding to the active passive combination. So, here F is applied here, then feed this force will act in this direction and this will act in this direction opposite direction. So, this is the earth pressure diagram of the sheet pile, so this is sheet pile. Now, these are the different position if this is A, this is base is the B, then this is dredge level, this is E we can consider. So, first these are different components here, we consider this one, this dredge means where earth pressure is 0 is at a distance of A from the dredge level. Now, this earth pressure at this level is P, A is the active earth pressure at E level. So, P A E is the active earth pressure at point E or at level E, E is this point. Now, now distance of this remaining part if I consider capital Y, so we can write that d is equal to A plus capital Y. So, the forces so that means here what are the forces we can consider this as the two parts, one is this portion is this is active zone and this is the passive pressure. Now, if I take this two different four that means what are the forces that is acting, one is F due to this anchor force and another is the P active and another force is P active. So, the force which is active that P passive, so the forces those are acting here. So, first one is F is the force due to the anchor anchor and then P A is the force or you can say active earth pressure, force due to active earth pressure and P P is the force due to this passive earth pressure. Actually this is the net pressure we can say. So, now these three forces that will act in this passive earth pressure. So, these three forces that will act in this total, these three forces means if I consider this is O, so one region is A E O this region that is P act A, another is O B in this region this is P P. So, now if I write this is this this anchor rod of force due to the anchor and now we can write that this F plus P P minus P A that is equal to 0. So, this among these three forces F plus P P minus P A equal to 0. Now as I have discussed already in previous class in during this cantilever sheet pile we can determine this A value with this expression that is P A E divided by P A E divided by gamma at this level below the dredge level if I consider gamma dash into K P minus K A, where P A E this is the pressure at this E level gamma dash weight of the soil below dredge level and K P is the passive earth pressure coefficient and K P is the passive earth pressure coefficient and P A is active earth pressure coefficient. So, these are the things. So, in this expression by the help of this expression we can determine the forces this distance A. So, this is our expression and this is our general force equilibrium expression. Now if in this other if I draw the same figure here. So, this is sheet pile it is a top level it is the dredge level now this is the ground water table position. So, forces that is acting this is the F anchor force this force is P active and this is P passive this distance is T or depth. So, this is the A E O and B and this point this is the P active at E level this distance is A remaining one capital Y. Now in this if I take the moment at anchor rod level or anchor level that is equal to 0. So, now this force P A say is acting at a distance of Y 1 bar from this rod level P A and P P is acting at a distance of Y 2 bar from the rod level. So, Y 1 is from the rod level this is P A and P P is acting Y 2 bar from this rod level. Now we can say that our P A into Y 1 bar that is equal to P P into Y 2 bar. So, this is the expression if I take the moment at the rod level. Now how will calculate this P A or P P? Now P P we can calculate because this P P value this value is given by this expression that gamma dash into K P minus K A into capital Y because this things already discussed in doing the cantilever retaining wall calculation. So, this one this is the net force or net pressure is K P minus K A into gamma dash into capital Y as this is the distance capital Y. So, P P will be half into gamma dash K P minus K A into capital Y into. So, this is capital Y into capital Y. So, this will be half into gamma dash K P minus K A into capital Y square and Y 2 bar will be Y 2 bar that will be that if total height is h say and the distance from the dredge level to the rod level is small h total height of the seat pile above dredge level is say capital Y 2 bar will be small h plus A plus 2 third of capital Y. So, this is Y 2 bar. Similarly we have to calculate the Y 1 bar by taking different components this small triangle then this rectangle then that again this triangle and then the remaining triangle. So, this how to calculate the Y 1 bar by taking different components this small triangle then this rectangle then that again this triangle and then the remaining triangle. So, this how to calculate this Y 1 bar that has been already explained in example in the last class. So, that thing has already been explained. So, in this way you have to determine the Y 1 bar and the P P also. Now, putting this Y 2 bar everything in this expression says about this expression 1 if I put everything in the expression 1 then we can write that P A Y 1 bar that is equal to half comma dash capital Y square K P minus K A into h plus A plus 2 third of Y. So, further if I simplify this thing. So, we will get capital Y cube gamma dash K P minus K A divided by 3 plus Y square plus capital Y minus K A divided by 2 gamma dash K P minus K A divided by 2 into h plus A minus P A into Y 1 bar that is equal to 0. So, in this expression this final expression if I look if we look this to this final expression then what are the unknowns here. The gamma dash is known the soil property K P K is also known then this A is known A we can calculate the expression which is shown h is h has to be known because we should know the position of this rod level from the dredge level. In P A we can calculate Y 1 bar also we can calculate. So, only unknown is Y this capital Y. So, first we have to determine the capital Y from this final expression capital Y once we get the capital Y next step we will get the d that is capital Y plus A and once we get this d this is increased by 20 percent to 40 percent by providing this factor of safety. So, once we get this d value or capital Y value then what we can do we can this P A we can calculate because this P A is the top portion where A we know we can determine P A we can calculate and if I know Y capital Y value then we can also calculate the P P. So, once we know the P A and P P then we can calculate this F force anchored force that is P A minus P P because that force is required. So, we have to design this anchor such that it can sustain under this F load. So, in and so this way we can determine how much depth required depth we can provide for this anchor sheet pile and what is the required anchor force. So, this analysis is valid for granular soil the similar one we can do for the cohesive soil also. So, next one is for the cohesive soil how to do this analysis for the cohesive soil. So, again this this next one is for the cohesive soil then. So, we will do this analysis for the cohesive soil. So, we will consider same sheet pile, but the soil is different this is the top portion this is the dredge level. So, required depth is D. So, here the same this is the position of the water table this is the anchor force that applying F. Now, here the difference is that here we consider in the previous one granular soil we consider that our foundation soil and the field soil because this is a field soil and this is the foundation both are granular soil. Here we consider that this is the field soil is granular soil where we consider C u is 0 and the soil below the dredge level of foundation soil that is cohesive soil that is phi u is 0. So, or you can consider the both in C phi soil also they are the pressure distribution diagram will be different. So, now here we can consider similar diagram for the, but here for the granular soil we will get this type of distribution instead of this triangular one. So, how we calculate this forces? So, this is also again this one p active at e level. So, this is a this is b this is e and here the forces that is acting again this F then this p active and again this p passive. So, again this F anchor force active force active force and passive force. So, these are the forces that will act again say distance between this anchor force from this rod level to this active force this y 1 bar and this is for the passive force this is y 2 bar this is y 1 bar this is y 2 bar. So, now when we try to calculate the what are the forces at different level then we can get this value this way that for this passive pressure. If I go so passive pressure diagram that is q bar k p plus 2 c u root k p. So, this is the common expression to calculate the passive pressure at any point similarly this is p active that is q bar k p minus 2 c u root k p where k p is the passive pressure coefficient and for that c u is the cohesion undrained cohesion. Now for phi u equal to 0 this k p equal to k a sorry here this one is k a active. So, this is the k p is the passive earth pressure coefficient and k a is the active earth pressure coefficient this is not the k p this is k a. So, k p minus k a that is equal to 1. So, at the dredge level now this q value we can calculate the q is the effective over burden pressure. So, q bar is effective burden pressure. Now, if I want to calculate what is the active earth pressure and passive earth pressure at this point e at dredge level. So, there is a two part one is the left part another is the right part. So, now if I consider that at dredge level the right point the right side of point e the q bar value is if this is h again h is the height of the sheet pile from the dredge level. Then that will be gamma into h this is effective this gamma into h or as this water table is here then we can calculate in other form also that is if this height is this is h 1 capital h 1 if it is capital h 2 then we can consider this density is if it is gamma and this is gamma dash then this will be gamma into h 1 plus gamma sub or gamma dash into h 2. So, this is the unit weight this is the submerged unit weight of the soil and this is the unit weight of the soil its natural condition. So, in this way we can determine, but it is gamma dash h for this calculation we are considering. So, now we can put that p a e this p active at e level that gamma dash is equal to q is equal to gamma dash into h as for this dredge level below this dredge this portion this phi u equal to 0 we are considering that phi u equal to 0 for this portion. So, at this level the stress that we are taking that phi u and if I consider this phi u equal to 0. So, k p will be 0 for this level. So, now we can write that k a is equal to 1 as e phi u equal to 0 then k p equal to k a equal to 1. So, this is gamma dash into h minus 2 c u similarly at the left hand side at the left hand side of point e q bar is 0 because here no surcharge is there. So, q bar is equal to 0. So, we can write that p p e because here the passive earth pressure p p e that is equal to 0 plus 2 c u. So, that is equal to 2 c u. So, at this point dredge level the right side active pressure is gamma dash h minus 2 u 2 c u and the passive pressure is 2 c u. So, net pressure we can write that net pressure that is equal to p p e minus p a e. So, that is equal to 4 c u minus gamma dash h. So, p p e at e and p a e. So, 4 c u gamma dash h. So, the net pressure that we are getting at this level that is 4 c u. So, this is the net pressure. So, that is 4 c u minus gamma dash h that is the net pressure. So, now if I draw this same figure. So, you can draw this is the sheet pile this is dredge level. So, this is water table position is here this is anchor force f. So, this is here net pressure we have calculated this one is 4 c u minus gamma dash h and this is the p a e that is e position this is a this is base b and this is the rod level. So, forces which is acting that is p active and this is p passive. So, this this is p passive distance from the rod level to p active is y 1 bar and this distance from the rod level this is p y 2 bar to this point. This is d is the required depth of the sheet pile h is the height of the sheet pile capital H. Now, again if I want to find the p p first because p a again we can calculate p a and how to calculate this y 1 bar that is also been explained. So, that part is fine, but we have to calculate this p p that p p is here 4 c u minus gamma dash h into d where d is the depth of the sheet pile d is equal to required depth of the sheet pile. So, now we can write this is this is expression say 1 and then summation r of the moment which is taken at the anchor level that is equal to 0. So, if I take the summation of this anchor level that is p a into y 1 bar that is equal to p p into y 2 bar. So, we can finally or we can write p a y 1 bar that is equal to p p is 4 c u minus gamma dash h into d and the lever arm is if this is the small h from rod level to d rage level the small h plus d by 2 because it will act at the center this is rectangle. So, this is d by 2. So, finally we can get this type of expression d square plus 2 d h minus 2 p a y 1 bar divided by 4 c u minus gamma dash h that is equal to 0. So, the final expression is this one where which are the unknown unknown p a we can calculate which is total force due to this active pressure y 1 bar also we can calculate c u and gamma as a soil property h is known. So, only unknown is d. So, what you have to do then you have to determine the d from this final expression then increase it by 20 to 40 percent due to the factor of safety. Then once you get this d we can calculate the p p then the finally f will be determined by p a minus p p like the previous case. So, these 2 are the different 2 cases where previous one is a granular soil and it is cohesive soil as it is in the cohesive soil then this phi u is 0. So, k a is equal to k p is equal to 1 that we are using. So, in this way we can analyze this anchor sheet pile for the granular soil and the cohesive soil, but here we have taken the free earth surface support method where the next one that I will discuss where this is analyzed we consider the fixed earth support method. So, for next we will discuss the fixed earth support method in that method that the next one is support method. So, that means here we consider this is our fit pile this is dredge level this is the top surface. So, that is the dredge level here this is the anchor this is the force anchor force and this is a this is e this is base b this is the rod level d. Here if I go for this deformation pattern then this will be the deformation pattern for the sheet pile. Here that means the difference is here one point of contour fracture will appear. So, here this point is say this is the point of point of contour fracture that means here the bending moment if I can draw the bending moment diagram that bending moment is changing its sign or here these two will go from positive to negative or vice versa. So, that means here bending moment diagram if I draw the bending moment diagram this type of case. So, this is the total sheet pile we can consider. So, these are the same points. So, this is suppose this is i point of contour fracture and then we can draw this type of bending moment diagram say here at a this is at d this is at e the next one is i point this point say i point and this is b point. So, we can get this we may get this type of diagram where. So, here this bending moment is changing its sign at this point of contour fracture and then we will get this type of diagram of the sheet pile. So, this is our fixed earth support method or this is the different position or we can say. So, this is i and this is the diagram of different points. So, we can say in here also we get different points where we will get say this another point g. So, there also we will get this g point and then we can get this type of diagram also. So, that is our the same this is the friction force that we are getting in opposite direction if I draw this diagram. So, this is our anchor position and this is the dredge level at the i this is the point of contour fracture then we will get at g level this value and also we will get some this type of pattern also. So, that means you will get this type of bending moment diagram for this for the soil for this fixed end anchor pile. So, now so this is the our b m d and this is also deflection pattern deflection pattern. So, now we will discuss how to analyze this system because here this is if this is the different type of this is the point of contour fracture where these forces are changing this is the g is another point. So, now if I get so this will be the g value is somewhere here and then you can get this type of diagram also. Now, if I go for the pressure distribution diagram for this type of sheet pile then we will get. So, suppose this is the sheet pile this is dredge level now the pressure distribution diagram we are considering this again this is a this is the anchor point d this is e and this is b then one position is i this is point of contrafection then below that this is say g. So, now if I get this diagram you will get the granular soil. So, this is the diagram where at the g suppose this is the maximum then you will get another change then you will get this type of diagram in the analysis. So, here this i is somewhere here say this is i point this is the where this is maximum this is g point. So, this is g point i is somewhere here. So, this is the diagram suppose this is h is the height of the sheet pile now here somewhere f is acting which is the force of the anchor force and then this is the required depth below the dredge level. So, we will get this type of distribution of the pressure distribution in this case. So, here also this is say a this distance is i say now from this total system we are considering only this up to g portion and if I draw this diagram for the g portion only up to g then you will get. So, this is the g value g point this is say i point this is the e point this is the d point this is the a point this is the i point. So, you will get this type of diagram. So, this is i point is here somewhere. So, and as we are taking this portion then you will get another reaction that will act is r g and this is the force will act at f. So, then what are the other forces then similar to this is the p active for this region that means this is a d e i in this region this is the total force a active and then the net forces then this force in the opposite direction this force p passive or p 2 that will act. Now, here suppose at i 0 point this force is p 0. So, these are the and this pressure is p a e the pressure at e level this is the pressure p 0 is the pressure at i level. So, once we get this force then you will get basically two if I consider two different beams of this total portion then this again if I take this if I consider the draw the same figure here up to a to g then this is the d d level e this is the d level where force will act f. So, this is g level this is i level. So, here this force is p 0 at e level p a e and g level one reaction r g will act. Now, for this one if I draw the bending moment diagram this is the pressure distribution then if I draw the bending moment diagram of this total portion. So, this is the at d level then it will follow at i point point of contrafecture it is 0 then it will go at g point is maximum. So, this is a point this is i point this is e point this is d point and this is g point. So, this is the bending moment diagram because this is this bending moment diagram is a similar diagram that I have previously drawn that means this is the forces then i at i it is 0 then g we are taking up to g portion. So, this is the point here also we are taking up to g portion. So, that means if I consider only up to g portion this will be the bending moment this stage in the just in the mirror image you can go opposite side. So, this is in the g portion. So, we will get this is the bending moment diagram. Now, we consider this total portion is taken to divide into two parts and two different beams. So, we can consider one beam is from a to i another beam is from i to g. So, we consider these two beams one beams is from a to i. So, this is our beam number one and this is our beam number two because this sheet pile you are considering this is two beam this is one is a to i another is i to g. Now, what are the forces in the first beam? So, that force in the first beam. So, that force f at d level that force will act this anchor force then this reaction r 1 that will act at the i level in the beam two opposite reaction r i r 1 will act at beam level and this reaction r g that will also act at g level. So, these are the forces then additional to that in beam one and beam two the force will also act due to this earth pressure. So, that means this is f r 1 r 2 and r g. Now, due to the if I consider the forces this is the forces and then this beam two then in addition to or other force that will act if I consider earth pressure also because of this additional earth pressure some forces will act. So, what are the these forces that will draw here? So, that means the forces that we are talking about. So, first we will draw the earth pressure diagram of this a to g portion. So, the forces this is the dredge level e. So, we will take this is the fourth earth pressure diagram here. So, then it is your maximum then it will follow this path. So, this is the diagram we have drawn. So, this point is say i where moment is 0 and this is g point and this is d point where this f is acting. So, now we have taken two beams one is from a to i a to i another is from i to g. So, now what are the forces that is acting here? So, from the a to i. So, the first force from this beam one one is acting f anchor another force is acting r 1 reaction the similar opposite reaction is acting here r 1 and here r g acting. So, now the forces that is acting for this for a to i beam. So, this force from this acting. So, this will act as p 1. So, p 1 is the forces due to this areas pressure similarly this beam. So, one pressure will act here because of this small triangular pressure this is in the second beam. So, this is a beam one this is beam two and so that means p 2 will act here and this side for the this triangular opposite direction one pressure p 3 will act. So, if I take separately this beam one what are the pressure? So, this is the beam one the pressure is acting this is beam one. So, one pressure is acting f at a d level then this reaction r 1 that will act and this force p 1 that will act force for this earth pressure. Now, for the beam two. So, this is a to i now for the beam two what are the forces that will act this is i 2 g this is beam two. So, this reaction r 1 also will act here one reaction r g is also act because this as the reaction because we are taking the some segment not the total beam. So, this is some segment. So, this is r 1 is this r 1 because this is beam one beam two then additional force this p 2 will act this p 2 is due to the small triangular force which is acting this side this is also acting this side and this is also acting on the opposite side. So, another force p 3 that will act here. So, these are the forces that will act now we have to calculate all these reactions all these forces then we can determine how determine this T value. Now, these are the forces diagram for this in the force this diagram on the different beams what are the forces acting what are the reaction forces what are the external forces acting here. Now, with the help of these forces we have to determine the required depth of the sheet pile and anchor force. So, how to determine this depth of the sheet pile and the anchor force that I will discuss in the next class. Thank you.