 Now, today I will discuss about the design of sheet pile or basically in this lecture I will explain how to determine the depth of the sheet pile, because this sheet pile is very important structure in geotechnical engineering, this is a another type of retaining structure. Now, this structure is very flexible and the difference between the retaining wall and the sheet pile is that that the depth of the retaining wall or the foundation depth of retaining wall is very less compared to the depth of the sheet pile, because the resistance of this sheet pile is getting from the soil pressure itself and that is why the required depth of the sheet pile is more as compared to the retaining wall. Now, for example, if I first explain that what is sheet pile, now the basic difference if this is our retaining structure. So, this is one cantilever retaining wall and this is say the dredge level or this is the existing ground level and this is the field material. Now, here if we consider this is the depth of this retaining wall and similar type, so here this portion is void and this is the soil exist field soil and similar type of retaining wall for the sheet pile if I consider. So, this is the sheet pile and here the ground surface here we can say this is the dredge level and here if I consider this is the d is the depth of the retaining sheet pile. So, this is the retaining wall and this is sheet pile. Here we can see that this type of structure is very flexible that is why it is used for this excavation purpose it is used this type of sheet pile can be used in excavation purpose. So, then after the excavation one place is over then this can be used or this can be taken out from the soil and that this type of sheet pile can be used for any other location for the excavation purpose even in for the water in offshore structure or in for retaining the soil near the sea course that sometimes use this sheet pile. Now, here this is also the soil and this portion can be the this can be the void or this can be the water. So, now here we can see that here also the lateral pressure is acting on the retaining wall here also lateral pressure is also acting on the retaining wall and here. So, that means, when we are designing the retaining wall traditional retaining wall then we consider that this side is active pressure is acting and this side this is the passive resistance is active. So, this side we can say this is the passive resistance P P and here this is the active resistance P A. Now, this active pressure is acting from this side and passive pressure is acting from this side. So, as during the design also as the depth of this retaining structure foundation is very small compared to the height of the retaining wall then sometimes we neglect this passive resistance and we design by considering this active resistance only. But here that means, here the this resistance is this passive force which is coming from this side this active force that is resisted by this retaining wall itself. That means, this retaining wall is wall is if it is a gravity retaining wall then the size of retaining wall is huge. If it is cantilever retaining wall then to resist this moment this reinforcement is used. So, that means that that is why the required depth of the retaining wall is less compared to the sheet pile. Whereas, in sheet pile here also this lateral force is acting this side and this side also soil pressure is acting. But here the the force which is coming from this side suppose this is the active pressure is acting this side in here also the passive pressure is acting this side. That means, the load which is acting from this side that is resisted by the load which is or the stress which is act pressure acting from the bottom of the wall. So, that means the so the most of the resistance this sheet piles is getting from the soil itself. So, that is why the required depth of the sheet pile is more as compared to the retaining wall as the resistance from the soil because below this dredge level the soil resistance is giving the stability of this retaining stops of this sheet pile wall. So, that is why here we cannot neglect this passive resistance because that resistance is not small as the depth of the sheet pile is more as compared to the retaining wall. So, that passive resistance we have to consider in the design. So, now so first if I consider that how this sheet piles are joined suppose this is our one sheet pile segment join with another sheet pile segment similar sheet pile segment by this ball and socket joint. So, this is first pile second pile which is joined in this portion by ball and socket joint then next one is placed here. So, that means the shape of the sheet pile structure may be in this form. So, this is the plan view of the sheet pile structure. So, so that is why and then this side is the elevation or cross sectional view and this is the plan view of the sheet pile structure and then if I take the cross section. So, in the design purpose we consider this vertical and which is very flexible. So, next one we can consider the different types of sheet pile. So, that the sheet pile can be one is cantilever sheet pile another one is anchor sheet pile. So, this may be the one type of cantilever sheet pile is suppose this is the sheet pile structure this is dredge level and this is the foundation or this is the existing ground level. So, we can call this is our cantilever sheet pile then another one this is ground level where one anchored is attached with the sheet pile this is anchored sheet pile. Now, use of this anchor as for the cantilever sheet pile the required depth of this is this is the depth of the sheet pile required depth is more. So, that is why to sometimes reduce this required depth this anchors are used in the sheet pile. So, this type of sheet pile is called anchored sheet pile. Sometimes this retaining wall there is a free cantilever sheet pile also. So, this is the cantilever sheet piles and sometimes this is free cantilever sheet piles, where this portion top portion is free. So, that basically these are the two types of sheet pile, one is cantilever sheet piles, one is anchor sheet pile. Then the first I will discuss about the cantilever sheet pile, then how to determine the depth of the sheet pile all these things. So, first problem or first discussion with cantilever sheet pile, piles wall it is in granular soil. So, that means this sheet pile wall can be in the granular soil, it can be in the cohesive soil. So, first we will discuss about the sheet pile is in the granular soil. So, suppose if this is the sheet pile wall and this is ground surface and this line is the dredge line. So, this portion is may be in the void or this is water this side and soil this side. So, this one is the height of the sheet pile above the dredge line and this one is the depth of the sheet pile which is up to the base of the sheet pile. Now, as I mentioned that here this depth of the sheet pile is not negligible one. So, when this due to application of this lateral force it will deflect. So, that means the deflection pattern of this sheet pile will be something like that. So, that means it is rotated with a point below the ground level. So, that means this is the deflection pattern of the sheet pile. So, now from this deflection pattern. So, if we can say this is the A this distance is A from the dredge level. So, you can see you can say that up to this point from the top of the sheet pile there will be a passive and there will be active pressure that will act and in this side. So, if I consider the two sides of the retaining wall that is one is this side or this is the right side this is left side. So, that means the right side from the top to this point O say. So, this portion active resistance will act and from this O to the base of this retaining wall can say if this is A this is B. So, from A to O in the right hand side active resistance will act because the active resistance this deflection is towards the direction of the force and from O to B in the right hand side this passive resistance will act. So, the from A to O from the right hand side active resistance and from O to B the passive resistance will act. Similarly, from this left hand side this side from D to O this passive resistance will act and from O to B active resistance will act. So, now if I draw the active pressure diagram this is the cantilever sheet pile and this is if I draw the active pressure diagram. So, this is the retaining wall. So, as I have mentioned this is the sheet pile wall as I have mentioned that from A to O. So, this is O this point is D O A this is base B. So, from A to O the active resistance active pressure will act. So, this is the active pressure that will act from A to O this is A 2. So, this value we can calculate this form that at this level the active pressure will be K A into gamma into H if gamma is the unit weight of the soil. Similarly, at this point because this distance is A this will be K gamma H plus A. Similarly, in this portion the right hand side here if I extend this pressure diagram. So, here also the active pressure will act in the left hand side from O to B. So, this will give us the active pressure. So, the value of this active pressure if I write this value here this is K A gamma into H plus. So, this value, but at this point this is gamma K A gamma H plus A. Similarly, here this side this value is K A gamma into K A gamma into K A gamma into H. Now, if I draw the passive resistance of this same of passive earth pressure this is A point this is same as D point this is O point this is B point. So, if at this side this is active resistance then similarly this side this side this side this side this side area that will be passive resistance. So, this is the passive pressure diagram when this side is active definitely the opposite side will be passive. So, this is the passive resistance. So, this value is K P into gamma into A because this value is A where K A is the coefficient of active earth pressure K P is the coefficient of passive earth pressure. And similarly if we extend this value I will extend this line. So, will be in this form similarly when this side is active that means definitely this portion will be passive. So, now if we extend so, this portion will be passive. So, this value is K P gamma into H plus D. So, now this is the net diagram is that this is active from A to O in the right hand side similarly in the opposite direction because this portion is void also that is soil is not present here. So, the passive resistance will act from this side point to D to O. So, I mean D to O is the passive opposite to this portion here this is active from O to B and definitely passive will be in the other hand side this is passive. Now, this is the passive and active pressure diagram for this cantilever retaining wall based on from this deflection line also you can see. Now, if we draw the net pressure diagram considering the active and the passive. So, this is retaining wall. So, this is the dredge line O this is A sorry this is D A and B. So, we can draw this will be the net and this will be the pressure diagram because you can see from this figure that here this is passive and this side is active as the net pressure that mean the passive minus active as passive pressure is more compared to the active pressure. So, this side the pressure net pressure diagram this side the pressure will exist and similarly here that means up to this point here active pressure is increasing and then up to this point there will be active pressure. So, this is the line straight forward we can draw after that the net pressure diagrams means that this active and the passive both will act. So, that means when this will start going from right to left and that means when this pressure active pressure and then the passive pressure if I take the net. So, at the point where this line will cross this sheet pile that means from here this will start and then it will go this side and cross this diagram and then again it will cross. So, two portion it will cross. So, that means the final diagram if I draw this is up to this point is active then the net pressure diagram this passive will start acting then it will go this side then again the passive force is more in this portion then it will go this side and then it will follow this pattern. So, this will be the net pressure diagram. So, here we can see this is active pressure zone this portion is passive pressure zone again this portion is passive pressure zone. So, from this figure we can say so this is active pressure zone then again this portion is more this is passive force passive pressure zone and then it will go in this side. So, this passive force will act. So, this is also passive pressure zone. So, there is a three parts one is active pressure zone then passive pressure zone then again passive pressure zone. So, that means this force is that will act P A P P 1 and then another force that will act P P 2. Now, though during the analysis so this is the actual net pressure diagram in cantilever sheet pile over. Now, during the analysis if we consider this type of pressure diagram then it is slightly complicated. Now, to simplify these things we can consider one passive pressure one net pressure diagram or this net pressure diagram where same sheet pile if we consider A D and this is B then this portion as it is active pressure zone then instead of taking this circle or non-linear this non-linear variation or the then we consider we take one linear type of variation. So, the same net pressure diagram is representing in this form where which is in same two points is crossing but it is in linear form. So, that means here also this is active pressure zone this is passive pressure zone and this is also passive pressure zone. So, these are the this is the net pressure diagram. So, this is the this is the converted net pressure diagram. Now, based on this net pressure diagram we will do the analysis. Now, if we start this though the net pressure diagram again this is dredge line. So, this is the net pressure diagram. So, we can see that this force is acting this side. So, this P A this active force will act in this side. Now, here so further simplify what we will do we will extend this line and we consider that this is P P 1 and this is P P 1 this is 2 or you can say suffix B because this is acting at B. Now, this distance is A from the dredge line where this is passing first crossing this sheet pile dredge diagram. So, this is A. Now, you consider this distance is acting at B this is Y capital Y and this one is this is Y 1 Y and this is say Z. So, this one is Y minus Z. So, this distance where this is 0.1 point. So, this is say this point is this is B this is C 1 and this is C 2. So, D 2 C 1 is A B 2 B 2 this point where the edge of this triangle is Z and then this one is Y minus Z. So, total from C 1 to B is so you can write that D 2 C 1 is equal to A then B 2 C 1 is equal to Y and this edge point from the B point is Z. So, now, if I write this expression this is P A P A will be K A because this distance say height is H and total is the depth is the D is the depth of the sheet pile. So, below the dredge line. So, this P A is K A into gamma into H. So, if this gamma is the unit weight of the soil where gamma is the soil and K A is the coefficient of active earth pressure. Now, this point at this C 1 point point that net pressure is 0 that means the active pressure is equal to the passive pressure. Now, if I write in this form that at C 1 point what will be the active pressure? At C 1 point the active pressure will be K A gamma into H plus A because at C 1 point is active pressure is equal to passive pressure because at C 1 point is because the net pressure diagram. So, that means at this point active pressure will be H plus A into gamma into K A that is equal to the passive pressure which is K P gamma into small a because this is the passive and active pressure. So, now, if I draw the previous figure that means at this point that this point any point the active pressure is equal to passive pressure. So, passive pressure at this point is K P gamma into A and active pressure will be K P gamma K A gamma into H plus A. So, if we equate these two because at this point is passive pressure and active pressure this is equal to equal to this is equal that means the net pressure is 0. So, if I simplify this thing A value will be K A gamma H divided by K P minus K A into gamma. So, K P K A gamma H that is equal to P A divided by K P minus K A into gamma. So, we will get first we will get the A value. Now, next one will determine this P B 1 P P 1 B. Now, this P P 1 B this value that will be the net pressure at this P P 1 B that is K P gamma H plus D minus K A gamma D. So, you can see from this figure that at this point the passive force is K P gamma H plus D and here the active force is K P gamma into D. So, if we take the net then this will be K P gamma into K P gamma into H plus D minus K A gamma D. So, this is K P gamma H plus D minus K A gamma D. So, this is the net pressure diagram for this P P 1 B. Similarly, P P 2 B that is equal to because at A is P A into K P minus K A into gamma. So, we can write this value that P P B that will be K P minus K A into gamma into Y. So, this is Y the net pressure is K P minus K into gamma into Y. So, this is the net pressure for this P P B 2. Now, we will start to calculate the forces F H. What is F H? Because what are the forces they are acting here. So, now if I consider these things in three parts that this is the P A is the net pressure this is the total pressure for this portion. That means, from A D C 1 this total pressure is acting as a P A and then if I consider this triangle this triangle this is say 1 2 3. So, if I consider 1 2 3 triangle and subtract from this triangle C 1 2 and B then ultimately that means, we are first considering this total triangle then we are subtracting this triangle C 1 2 B. Then what will happen? This portion will be cancel out. So, this thing minus this thing that means, these are acting opposite direction because this force acting this side and this force is acting this side. So, these are the three forces for one for this portion lower portion one for this triangle middle triangle one for lower triangle which is acting this side one for the middle triangle which is acting opposite triangle one for the upper portion. Now, what we are doing that we are taking this triangle 1 2 3 and then subtracting from this 1 2 3 triangle forces with C 1 2 and B. So, ultimately this portion is cancel out and this is acting opposite direction. So, now if I do in this form then F H will be R A then this triangle plus half into P P 1 B plus P P 2 B into this z z is the height of this triangle 1 2 3 into z then the subtracting this other triangle. So, that is height is y half into base is P P 2 B into y. So, that forces will be 0 for the equilibrium these forces will be 0. So, now from these forces see if I write that again that is R A or we can write this force this is P A P A because here this is P A. So, we can write this is P A plus P P 2 B plus this triangle this P A is the total force for this upper portion. Now, we can write if I write this expression again P A plus half P P 1 B plus P P 2 B into z minus half P P 2 B into y this is equal to 0. So, now from this expression we can write z will be P P 2 B into y P 2 B into y minus 2 P A divided by P P 1 B plus P P 2. So, from this expression we can also calculate the z. Now, we will take the moment at base capital P A minus capital B that is also equal to 0. Now, if I take the moment and we can say that from this center point from this point C 1 the distance of this forces that is acting at the top portion is y bar. So, that distance is from the C 1 where the net pressure is 0. So, that from the C 1 point this force P A is acting at the distance y bar. So, that we can easily determine and this is the capital Y. So, from the base B the distance of P A will be capital Y plus small y bar. So, and so we can write this P A if we take the moment into capital Y plus small y bar plus for the first angle that is half into P P 1 B plus P P 2 B into z that is the area into that will act as a distance z by 3 from the base. Then minus the area for the next triangle is half into P P 2 B into y and that will act y by 3 from the base. So, and that is equal to 0. So, ultimately after simplifying so after application we can write 6 P A into capital Y plus y bar plus z square P P 2 B plus P P 1 B minus P P 2 B y square that is equal to 0. So, in this expression what are the unknowns that unknowns is z that we will get from this expression and then P P 2 B P P 1 B these we will calculate and P will calculate and then y that we will also calculate and we will ultimately we will determine the value of y. So, from this expression we will determine capital Y then if we add A then D will be capital Y plus. So, you have to calculate capital Y from this final expression last expression then if we add the A the A A calculation that we will do from this expression number 1. So, now if I give the name this is number 1 from here we can determine A then this is equation number 2 from here we will determine z and this is equation number 3 from there we will determine the capital Y. So, that means and this z we will put here. So, capital D you will get capital Y plus A. Now, once we get these things then you have to increase this day D by 20 to 40 percent for given additional factor of safety. For these factor of safety purpose you have to increase this day by D by 20 to 40 percent that is the factor of safety. Now, this expression will get the depth of sheet pile based on this calculation from this 3 expression and this is for the sandy soil. Now, for the clay soil what we will do that if the same thing we calculate for the cohesive soil that where phi u equal to 0 and same sheet pile this is ground level this is dredge level. So, now for the cohesive soil. So, this will be the pressure diagram for the cohesive soil so this is minus this is plus and from here we will get this type of pressure diagram. So, now why we will get this type of pressure diagram first we will explain this is H this one is D this is the depth of the sheet pile. Now, here first expression of P A will be that is equal to Q bar into K A minus 2 C u root K A. So, Q into K A minus 2 C u into root K A is the coefficient of active earth pressure. Now, here Q bar is equal to effective vertical stress at any depth and here phi u is 0. So, your K A minus 2 C u into root K A is that is equal to K p that will be equal to 1 as phi u equal to 0 for this expression. So, K A equal to K p that will be 1. Now, you can write that P A that is equal to Q bar minus 2 C u where C u is the cohesion and then cohesion of the soil and P p that will be simple Q bar plus 2 C u. So, P A will be Q bar minus 2 C u P p will be Q bar plus 2 C u. So, now at E point or at say D point this is A D and this is B. So, at D your passive the net force net pressure the net pressure will be that P p minus P A at D that will be P p because at this here the P p that passive resistance that will act in this side and active will act in this side. So, but at this point at this D point if I consider the passive that Q is 0 because in this side this is there is no soil. So, Q we can consider this is 0. So, at this side this is 0 plus 2 C u minus Q bar this is the Q bar will be this Q bar is gamma H gamma is the unit weight of the soil Q bar minus 2 C u if there is a same soil C u is here also and C u is also here. So, that net force that is 4 C u minus Q bar. So, at this point this value is 4 C u minus Q bar. Similarly, at B point the net pressure is P p minus P A at B that will be equal to. So, at this point B here this side at right hand side there will be passive pressure and in the left hand side there will be active pressure because at this point this right hand side is active pressure and left hand side is passive pressure, but at the base right hand side is passive pressure and left hand side is active pressure. So, passive pressure if you write that is Q bar plus this load that is gamma into D plus 2 C u is minus then this Q bar will be gamma into D into minus 2 C u. So, now net pressure will be 4 C u plus Q bar. So, this value is 4 C u plus Q bar. So, again we have to take the. So, again we have to take the forces. So, that means if I take this F H summation of the force and we can write that this force is P A at total active force and which is acting at a distance y bar from the D point. Similarly, we can write that this distance is Z that means where this is taking a this distance is Z. So, what we are doing we are taking this force this force is acting. So, there is another force is acting inside and another force is acting this side. So, we are taking adding this force and then adding the force of this triangle then we are subtracting this force which is acting opposite direction. So, ultimately this portion will be cancelled out. So, in this way. So, F H we can write this is F H is 0 this is P A total force of the upper portion plus half into Z into half into Z that means this triangle the half into Z into 4 C u minus Q plus 4 C u plus Q bar. So, we can write this is 4 C u minus Q bar plus 4 C u plus Q bar then minus D is the depth into 4 C u minus Q bar is equal to 0. Then 4 C u means this D then for this rectangle this portion is 4 C u minus Q bar into D that is the area. So, from this upper portion force plus this triangular force then minus this rectangular force. Then ultimately we can write this is P A plus half into 8 C u minus D into 4 C u minus Q bar that is equal to 0. So, the Z will be D into 4 C u minus Q bar minus P A that is divided by 4 C u. So, this is one expression one from which we cannot determine the Z value. So, now next one we will take the moment with respect to this B that is also 0. So, moment that will be P A into capital Y plus Y bar then minus that D is half the this triangular one plus that is Z half into Z and this one will be here also this is Z and is that Z 1 this is half 8 C u into Z then we can determine the Z from this expression. So, half into Z into again 4 C u minus Q bar plus 4 C u plus Q bar and that will act at a distance Z by 3 from the base B. Then minus this D into 4 C u minus Q bar and that will act distance D by 2 from the base because this is a rectangle that is equal to 0. So, after simplifying this we will get D square 4 C u minus Q bar minus 2 D P A that is minus P A 12 C u Y bar plus P A into 2 C u plus Q bar that is equal to 0. So, after putting the value of Z here expression 2. So, this is the final expression. So, where this Z value is replaced by this expression 1. So, you put the Z value in this expression and then we are simplifying this we will get this final expression. So, from again this Q bar is equal to gamma into H. Now, again from this final expression expression 2 we will determine the value of D and then increase the distance of the base B. So, this is the final expression and then we will get this expression. So, from again this Q bar is equal to gamma into H. Now, again from this final expression expression 2 we will determine the value of D and then increase we determine the value of D and then increase it by 20 percent to 40 percent for the factor of safety. So, these are the two this is the determination of the depth of the sheet pile for two different cases for cantilever sheet pile. One is for the granular soil, one is for the clay soil or cohesive soil. So, next class I will solve few examples. And then we will find how to determine the depth of this sheet pile wall for different soil condition if for the cantilever sheet pile. Then later on I will discuss about the how to determine the depth of the anchored sheet pile also. Thank you.