 Now we are doing the derivation for cantilever sheet pile, first case is your for granular soil. Now as I said there will be a point of rotation in case of granular soil for cantilever sheet piles. If this is my point of rotation O, the next stage how this derivation has been derived if I, so this is acted upon by horizontal load H k p minus k line, this is your horizontal force, this is your horizontal force. If you look at this diagram complete diagram and now if I take it the original what I explained earlier, this is my cantilever wall and there will be like it is your dredge line. So it says it will rotate some point at point O, this is called pivot point acted upon by horizontal force. So then as I said earlier, as I said earlier if it is rotate, if this cantilever wall is rotating at point O, so what will happen? The top part if I divide into two parts, one is your above the point of rotation, other is your below the point of rotation. If I say above the point of rotation this wall is moving away from soil, suppose this is my soil mass, so that means this part will be active, this is your passive state and if you look at the bottom it will be reverse and this is your passive state and this is your active state. So generally passive state we write k p line, active state we write k a line or k a, we write it k a. Now the same principle if I apply here, let us consider this is a cantilever sheet pile wall and this is my dredge line, dredge line and below this there is soil mass. Now this is a point point O, now above this dredge line there is nothing, so what will happen if this wall cantilever sheet pile wall, now it will rotate at point O, it will rotate at point O that means if it rotate at point O what will happen above this, this is the passive state and this is the active state. Now I can draw this is my k p line, if I draw the passive earth pressure it will be kind of like this and active earth pressure it will be kind of like this. So that means I have taken consideration of passive state and active state up to at point O, up to at point O. Now this is the passive line or passive state of the soil and this is the active state of soil that means always remember k p is greater than k a, passive state is greater than your active state or passive earth pressure is greater than your active earth pressure. So that is why this triangle I have taken this your k p that means passive state or passive earth pressure it is bigger than this line is your, this triangle is your smaller, this is your active state. Now come back to if this is your first diagram, second diagram, third and fourth diagram, now come back to second diagram, in second diagram what is supposed to be happen, this is my point O, this is the point O look at the point O below point O what will happen below point O this wall is moving towards and this part is it is this wall is moving away that means if I draw a line here somewhere else below this, this is your active state and this is your passive state, passive state. Now same concept has been applied here if I draw it completely below point O below point O that means this is my active state or active earth pressure will act this is passive earth pressure will act and I will remove the pressure from O to dredge line because the concept below this is different than the concept above your point O below this point O it is different passive state active state above the point O this is passive state and active state this is in a reverse. So above point O this has been this has been neglected and below point O it is considered so this will be your k a line and this is your k p line. Now for complete derivation it has to be superimposed both the cases has to be superimposed. Now this has been superimposed what happened this is k p and this is k a so this will be k p minus k a line and this is your k p and this is your k a in this case this is your also k p minus k a line. So now the diagram has been drawn so this point is your point O this point is your point O now next question you can ask yourself will it be the failure line will it be the failure line no why because from passive state to active state soil cannot transit in the all sudden transition cannot be occur sudden transition means if you look at point O if you look at point O all of sudden it is in passive state other side then it is moving moving and it is changing then all of sudden it is changing. So this transition is not acceptable this transition is not acceptable that means there will be a smooth transition from one state to other state one state of failure to other state of failure there should be a smooth transition. So that is why it has been drawn so that it is slowly slowly decreasing and it is achieving this other failure transition. So now this is your final failure mode of cantilever retaining wall below means in granular soil in granular soil having a point of rotation O above the base now how it looks it looks like this now these are the assumptions for particularly considering not non-linear actual case of this how it it should be looks like it should be a kind of curvilinear it is a curvilinear it has been approximated to simple straight line or may be linear failure you can say it. So if I take it this is my final this is my final failure surface final failure surface of this cantilever retaining wall cantilever retaining wall having embedded in granular soil that means it is starting from one state to other state there is a smooth transition this concept has to be used for derivation of cantilever sheet pile wall for derivation of equation that means what is the capacity of cantilever sheet pile wall in granular soil if I remove all one two three I am keeping only this part final form of this equations. So we will start this derivation how it how the derivation has been made now instead of suppose I say there is there might be a soil mass above the cantilever there might not be a soil mass of the cantilever as I said cantilever sheet pile walls are two types one is your free cantilever other is your free cantilever other is your simple cantilever pile cantilever sheet pile in case of free cantilever there is nothing in case of this is a typical case of free cantilever so both the sides there is nothing this will this will this stability will be achieved by means of your embedded depth but the moment I say that cantilever sheet pile that means it has been embedded at one end it will retain it will retain your soil mass it will retain your soil mass now if I modify this as only this is case of free cantilever now I am saying that only cantilever sheet pile wall how it looks this is the cantilever sheet pile now this is your dredge lane line dredge line now it rotate like this now here at other end here at this end it is retaining soil it is retaining soil because this is not a case of free cantilever sheet pile so it is retaining the soil now how the failure surface will be there if I take this principle of this look at this this derivation of this failure surface below the dredge line for free cantilever sheet wall has been taken it has been taken it has been put it here now if you consider the top part as it is retaining the soil mass it will generate active air pressure up to this now this is the final form of your this is the final form of your soil mass this is the final form of this failure surface of cantilever sheet pile wall one side it retain your soil mass one side it retain your soil mass now you can say that with these conditions what are the most adverse condition may occur while the derivation the water table may be at the ground surface may be somewhere else at the ground surface so there are conditions are there but this failure surface whatever we derived from free cantilever sheet pile wall that has been applied for your cantilever sheet pile wall now coming back to your complete derivation if I draw it this is the dredge line now below this let us say this is my point of rotation now let us say at any distance below the surface ground surface at the top there is a water table this is a symbol of water table now I am drawing the pressure distribution diagram a b now this is your p a this is the complete failure surface how the derivation has to be done now if I take it this is my dredge line this is the water table so let us say this property is gamma phi k p and k a and below the water table the properties are let us say equinoid bunker so this property I can write it because water table is there this will be gamma prime phi prime k p prime and k prime now let us say this distance is h 1 and this distance h 2 and a b let us say name d this is point o this is your c now it will be acted upon by let us say a resultant force this will be your p a active earth pressure prime let us say all these forces this is your resultant force r a particular from this to this this is your resultant force r a it acted upon by it acted upon by say suppose say at a distance y bar y bar from point o now this distance say a because why I am naming this derivation has to be made for this to find it out what is the capacity what load this wall can take this is your distance a let us say from here to here this is your distance y now this to this let us say y minus z and this is your distance say z this your c so now if I put it like this this to this let us say it is p p prime and this is your say p p 2 prime and in between if I take it this pressure distribution will go like this so in between from o to this I say p p 1 prime now this failure surface concept has been taken from once again I am saying from free cantilever sheet pile to cantilever sheet pile wall and this below this whatever this free cantilever sheet pile the failure surface it has been adopted and this failure surface is drawn now this I put it most adverse conditions what what may possible in most adverse condition may be possible during rainy season may be this water table fluctuate from ground surface from the below from the dredge line this water surface water table may be raised to your ground surface so now with this complete of this what we are supposed to get we are supposed to get what is the capacity and what distance d what distance d this is your complete distance d means I should know depending upon the capacity how much depth how much depth this cantilever sheet pile can be embedded below your dredge line or can be put it below your dredge line this is your output this this parameter to be required to find it out as well as the capacity as well as the capacity h what capacity it can take these two parameters to be find it out so k and k p is your rankine earth pressure coefficient so let us start with this p a for this diagram complete if you look at this for this diagram this is your p a prime so what is your p a prime p a prime is your active earth pressure from a to up to your dredge line active earth pressure from a to up to your dredge line so p a prime is equal to k a gamma h 1 plus k a prime gamma prime h 2 now what is your p p prime p p prime is your effective passive earth pressure at the base of pile effective passive earth pressure at the base of the pile this is your p p prime which is equal to gamma prime k p prime minus k a prime into y basically earth pressure is your gamma either k a if you look at how the derivation your earth pressure will be a k a k gamma h h is your distance gamma is your unit weight of soil and k should be either k a or k p from this concept it has come here because water table is lying here below this this will be a some earth unit weight that is why it is gamma prime k p prime is your k p prime minus k a prime as i derived here this is a k p minus k a and the distance will be your y now it will be gamma prime k prime y so k p prime is nothing but your k p prime minus k a prime and k is equal to k p minus k a now what is your p p 1 passive earth pressure at point o what is the value of p p 1 now p p 1 prime is equal to at point o it will be gamma h 1 k p minus k a and k is equal to gamma plus h 2 plus a gamma prime k p prime minus gamma prime a k a prime so it will be your gamma h 1 k p gamma h 1 k p this diagram plus gamma prime k p prime h 2 a this is your h 2 plus a up to this how much it is coming minus gamma a prime a k a if you look at a this part will be your active state k a k p and minus this is your active state because this part is also soil is there so this part will be active so gamma prime a k a prime now similarly you can find it out p p 1 sorry p p 2 prime p p 2 prime which is equal to nothing but p p 1 prime plus gamma prime k prime y gamma prime k prime y which is your gamma prime k prime y gamma prime k prime y which is your gamma prime k prime and y is your this distance it should be added if the value of phi remember if the value of phi for both the person is same for both the person is same value of phi phi value then it will be your k p is equal to k p is equal to k p prime and k a is equal to k a prime now these are the values you get it p a p a p a prime p p prime p p 1 prime p p 2 prime based on your earth pressure distribution diagram now you have to find it out what is the result and earth pressure what is the resultant earth pressure so let us say r a is equal to resultant of all forces above point o r a is equal to resultant of all forces above point o above point o the point o is located at a distance a below your dredge line point o is located at a distance a below your dredge line so now the pressure at point o is 0 now the pressure at point o is 0 with this concept pressure at point o is equal to zero with this concept a gamma prime k p prime minus k a prime is equal to p a prime is equal to p a prime p a prime minus K a p a prime now a you can find it out the distance is equal to p a prime by gamma prime and k prime k prime now this distance z now what are the other other parameter this distance z has to be find it out before before finding out r a this distance to be find it out in terms of k a and k p now distance z you can find it out so distance z can be find it out z you can get it by taking summation of force in horizontal direction is equal to 0 now r a this force this is acting in this direction r a plus your p p prime p p 2 prime into z by 2 into z by 2 if you look at here p p 2 prime and p p prime p p prime into z by 2 this is a triangular distribution this will act with a z by 2 plus minus p p dash minus p p dash minus p p dash this is your p p dash minus p p dash y by 2 is equal to 0 from this you can find it out z is equal to p p prime y minus 2 r a by p p prime plus p p 2 prime now how many unknowns there are 2 unknowns you do not know z you do not know r a so that means additional equation is required that means you take you take movement at the bottom should be 0 so taking movement at the bottom is equal to 0 now it will be r a y plus y prime plus z by 3 p p prime plus p p 2 prime into z by 2 minus p a prime y by 2 into y by 3 is equal to 0 that means I have consider there are 2 unknowns 2 equations is required 1 is your force in horizontal direction is equal to 0 other is your movement taking movement at point but the base is equal to 0 then you can find it out what is the value of this by simplifying this you can find it out 6 r a y plus y bar plus z square p p prime plus p p 2 prime minus p p prime y square is equal to 0 now with this putting this value of z putting this value of z here to here now this will be a fourth order differential equation will be derived now this fourth order differential equation if I write in a form it will be y fourth plus y cube p p 1 prime by gamma prime k prime minus y square 8 r a by gamma prime k prime now it will be minus gamma 6 r a by gamma prime k prime whole square into 2 y prime gamma prime k prime plus p p 1 prime minus 6 r a y prime p p prime plus 4 r a square by gamma prime k prime whole square is equal to 0 this is your final form of equation this is your final form of equation in a fourth order differential equation sorry fourth order equation it is not differential equation it is a fourth order equation so how you can derive it can be derived by means of trial and error method by means of trial and error method one end you put this values we put this values then whether it is becoming 0 or not satisfying or not you see first what happen you take a value you see that whether it is negative or positive depending upon that you take the second set value then it will be it has to be approximated by means of trial and error method you have to solve it and with this solution with this solution factor of safety can be provided by two methods once you get it once you get it d and r a value you can provide your factor of safety in by two methods one is your the value of d obtained should be increase by 20 to 40 percent or or second factor of safety you can take it 1.5 1.5 factor of safety 1.5 factor of safety you can take in k p end or or k a that means while designing at the end what you are suppose to get at end you are suppose to get the value of d you are get at the end suppose to get your value of d how much your depth d so how to take the factor of safety either whatever you are getting the point d means distance d this can be increase 20 to 40 percent for your factor of safety or you can take 1.5 factor of safety in k p and k a in k p and k we will solve a problem may be next class.