 Next part after cantilever seat pile wall, now let us start with anchor seat pile, so there are in anchor seat pile, let me draw the anchor seat pile, as I discussed earlier anchors has been provided in the seat pile, so that this deflection and as well as bending movement will be reduced, if I draw a seat pile wall and one at this is my anchor or anchor rod and this is the water table, now draw the first you would this draw pressure distribution diagram, so this is a complete kind of seat pile with your anchor dredge line, now where this anchor has been used, it is particularly dock and harbour structures, it may it will assist ship may tie of ship may tie of alongside, it will reduce as well as also it will reduce lateral deflection reduce lateral deflection bending movement and depth of penetration of pile more than one anchor more than one anchor may be provided, if I take a section like this, so this is one side of the wall, if you look at here this is my seat pile wall, this is the seat pile wall and it is going in this direction, in this direction and this side is your water sea level water sea level, so it has to be embedded below the dredge line, so what will happen once ship comes it will tie of here, so once it tie of will then it will pull the pile wall for more stable, if you provide the anchor rod what will happen if without anchor rod, if this is your deflected ship without anchor rod, once you provide anchor rod the deflection will be drastically changes, that means once deflection will be drastically changes what does it mean, that means this failure risk of failure you can avoid it also, second part is that bending movement also reduced by providing the anchor rod, now if this is your wall generally the wall design has been done in width wise per meter width, one meter one meter in this directions, so anchors may be provided not one anchor may be multiple number of anchors you can provide along the depth direction, if this is my depth direction suppose say in one meter depth one meter one meter depth means one meter long width direction, if this is the anchor you can provide two anchor three anchor then it will continue same way, so what will happen it will continue means in that depth direction sorry length direction if it continue it will reduce its bending movement also to more factor of safety or may be more you can add in depth direction also anchor can be provided in both way length as well as depth direction anchors can be provided, so this is the advantage of anchor sheet pile earlier there was a sheet pile as I said, so this is a free and fixed sheet pile, now this is anchored sheet pile in anchored sheet pile also it has been classified into two parts, one is your free earth method one is your free earth method other is your fixed earth method, in free earth method what is the assumption pile is rigid pile is rigid and rotate at anchor rod passive earth pressure develops soil in front of piling active back of wall, now if you see free earth method again this is the condition there are two cases free earth method one is your for granular soil and other is your for cohesive soil other is your cohesive soil in fixed earth method the anchor is fixed there is no rotation, if you look at this free earth method it has been assumed that this sheet pile is rigid once it is rigid that means it will not fail by means of bending, it will not fail by means of bending it will fail by means of rotation and another assumption is that whatever the rotation is going to happen it is going to happen along the anchor rod along the anchor rod, so that means failure occurred by means of rotation and rotation will occur at the anchor rod, so passive earth pressure develops soil in front of the piling and active earth pressure develops the soil back of the wall back of the wall, we will start with this granular soil we will first start with the granular soil of free earth method this derivation of particularly granular soil in case of granular soil what is the derivation and how do you get for anchored bulk head embedment depth how do you get it let us say this is my wall this is the dredge line this is f a r now this is your water table water table let us put this will be h 3 and this is your distance will be completely h now by drawing as it is said there will be a rotation draw the pressure distribution diagram now if I draw it this is my a and this is the distance say x this is your d and let us say this resultant passive earth pressure is your r p and this part will be gamma prime k p dash minus k dash into x, so which is equal to let us say c x and this is my x, so c is equal to gamma prime k prime and r p is equal to c x square by 2, now let us say force earth here will be p a prime and let us say resultant force above the base this is your r a then distance from the anchor rod because will take the rotation at the anchor rod say y bar and force in the anchor rod let us say f a r a r means anchor f subscript a r means subscript a r means force at the anchor red anchor rod then this distance from here to here this distance from here to here it will be y prime which is equal to h 3 plus a plus 0.67 x and this is the anchor rod now if you look at this this is the complete pressure distribution diagram of anchor sheet pile wall in cohesion less soil or granular soil we are deriving now case one for particularly granular soil we are now doing now we are taking all adverse conditions now there is a water table somewhere else this is the anchor rod now as it is assumption said it is a rigid and rotation has to be assumed at the anchor rod and failure anchored by means of rotation of rotation at the anchor rod. So let us say this from here to here this resultant pressure is r a and from here to here resultant pressure is your r p. So r a acted upon below y bar from the anchor rod and r p acted upon acted at a distance below y prime from anchor rod which is equal to h 3 plus a plus 0.67 x h 3 this is your distance how it has come plus a this is your distance plus 0.67 x because this is a triangle. So this will be two third two third of x. So that is why this is your 0.67 x now this gamma prime and k prime has been assumed as a c and this is your c into x now what is this r p this force you can find it out c x square by 2. So f a r is your tensile force if you see this kind of things this is your tensile force at the anchor rod this direction going away the direction going away means outward direction of r o it says that this is tension if I put in this this way that means this is a compression member. So that is why this is anchor rod it will take complete tension. So that it will not rotate or deflect more as expected. So it will restrict the deflection and rotation. So that is why this is f a r now f a r force in the tensile direction that means in tension f a r now if you take this point o at point o where is your point o this is your point o at point o the wall pressure is 0 point o wall pressure is 0 that means if I take point o a gamma prime k p dash minus k a dash is equal to p a prime a this is your distance gamma prime k p minus k a this should be equal to whatever force coming here p a prime because at point o it should be 0 now you can get it a is equal to p a prime by gamma prime k p minus k a which is equal to p a prime by gamma prime k prime or p a prime by c now take movement take movement is equal to 0 at anchor rod this is our assumption it will fail by means of rotation and this rotation will occur at anchor rod. So taking movement at anchor rod and equating to 0 so it will be what is this r p into y prime into y prime is equal to this force anticlockwise and r a this is your r a into y prime this is your also anticlockwise so it will be equated to 0 now if I put it in this manner y prime r p which is equal to y bar r a now from there you can get it y prime r a y bar r a is equal to c x square by 2 h 3 plus a plus 2 third x now if I combine both this term if I combine both the terms that means taking the value of a from here to putting it here now we can get it 2 x q plus 3 x square h 3 plus a minus 6 r a y bar by c is equal to 0. So what is the second condition for this taking forces at h is equal to 0 horizontal forces acting is equal to 0 that means f a r is equal to r a minus r p the bottom line of this yours bottom line of this why you are doing there are 2 parameters you want to find it out 1 parameter you want to find it out what is this embedment depth d that means a as well as x you have to determine x plus a is your d then once you get the d then you can apply the factor safety later on 20 to 40 percent another one with this case what is the maximum force this anchor low anchor rod can take what maximum force it can take it if I make this equilibrium conditions horizontal forces is equal to 0 then force at anchor rod will be r a minus r p r a is your this direction r p is your this direction and this from there you can find it out your f a r now we will solve a problem we will see now with this derivations are there let me put in such a way that let us say this is example problem and here surcharge is equal to 0.5 keeps per square feet and given data is your phi is equal to 5 prime which is equal to 30 degree delta is equal to 20 degree c is equal to 0 gamma is equal to 0.105 keeps per cubic feet and gamma prime is equal to 0.066 keeps per cubic feet this is the value of the soil profile phi is equal to 5 prime which is equal to 30 degree delta what is the value of delta delta is your friction angle between wall this is the wall and the soil delta is your friction angle between the wall and the soil delta is equal to 20 degree r c is equal to 0 it is purely cohesion less soil and gamma is equal to 0.105 keeps per cubic feet gamma prime is equal to 0.066 with this conditions with this condition what is your pressure diagram what is the other part has been given other part is given this is this to this is equal to 4 in 4 4 4 feet and this to this is your 4 feet and below the water table this is given 22 feet and now and total height is your from here to here it is given 30 feet now if I draw if I draw the pressure distribution diagram if I draw the pressure distribution diagram this pressure distribution diagram is not like this now the pressure distribution diagram will change because there is a surcharge there is a surcharge at the top so pressure distribution diagram will come this part will be your surcharge if I draw the entire part if I draw the pressure distribution diagram here completely how it looks this height is your 8 feet now water table here is your 4 plus 4 it is your 8 feet then so this will be your p let us say this is your p 1 let us say this is your p 2 then this part is your k a q q is nothing but is your surcharge 0.5 your k f now this will be your k a gamma h now then this is your 22 feet earth pressure diagram from here to here this is your 22 feet now this if I draw it this will come as a p 3 this is your p 4 and this is your p a bar this is your k a gamma prime h and now I draw the pressure distribution diagram up to 0.0 up to 0.0 then here it will be will have to find it out what is the distance of this point a this is unknown now will have to find it out from there now once you get this from there you can find it out value of delta and phi value of delta and phi as I explained earlier from value of delta and phi you can find it out k a is equal to k a dash which is equal to 0.297 and k p dash is equal to k p which is equal to 6.10 and c is equal to gamma prime k prime which is equal to 0.38 now will find it out r a now what is your r a resultant forces acted above your dredge line from here to here because if you look at this is my dredge line everything is given your surcharge you know your phi now you can find it out what is your resultant earth pressure so you can find to not to make mistakes you can find it out individual force p 1 p 2 p 3 p 4 then you can add it so p 1 is equal to h into k a q which is equal to 8 into 0.5 into 0.297 which is equal to 1.2 k p then similarly p 2 is equal to 1.0 k p 3 is equal to let me calculate your p 3 p 3 is equal to 22 into k a q plus k a gamma h which is equal to 8.8 kips p 4 is equal to half k a gamma prime h so which is equal to 4.5 k a gamma 7 kips now p a prime p a prime is equal to k a q this is your k a q plus k a gamma h plus k a gamma prime h which is equal to 0.83 kips per square feet so all forces are equal to 0.5 kips per square feet so what is your resultant earth pressure in this case we have calculated why I do it I can calculate directly complete taking this diagram trapezoidal I can find it out force but not to do mistakes you make it small small and individual area we can find it out so that chances of doing mistake will be less now from there you are a is equal to p c bar by c c is nothing but your gamma prime and k prime now from there I can get it a is equal to p a bar by gamma prime k prime which is equal to 0.83 by 0.066 into 6.10 minus 0.297 and which is equal to 2.4 kips per square feet so 2.2 feet now this distance is we are getting 2.2 feet now once you get it so from there you can find it out what is the value of p 5 so once you get a is equal to 2.2 feet p 5 you can find it out very easily 0.9 kips now summation of all forces summation of all forces that means p 1 p 2 p 3 p 4 p 5 this is nothing but your resultant force resultant force r a so summation of forces is equal to r a which is equal to 16.6 kips now once you get r a what is your next step your next step to find it out next step to find it out what is the resultant distance from here y bar that you can find it out very easily as the pressure distribution diagram is there so from there y bar we can get it 13.4 feet from o 13.4 feet from o that means this is not y bar that means from anchor rod y bar is the distance from anchor rod and from here to here this is your y prime so we are intent to find it out the movement at anchor rod that is why we are going to find it out y bar so y bar is equal to that means from anchor which is equal to 14.8 feet now which is equal to 14.8 feet now this distance is your 14.8 feet now everything is there now we can find it out all these things this way we can find it out x and others all the calculation we have done now if i calculate this it is a third order equation first find it out this coefficients then put it solve the equation so now what is the value 3 h 3 plus a which is your h 3 this is your h 3 that means 22 plus 4 that means it is 26 so it will be coming around 326 plus 2.2 which is equal to 84.6 now next one is your 6 r a y bar by c which is equal to 3879.2 now if you find it out you can find it out it has to be solved by trial and error so x and d now if i write x q x square sorry this is your x q if i write this equation again if i rewrite this equation again this will be your x q plus 3 x square h 3 plus a x square this will be your x q plus 3 x square a by 2 minus 6 r a bar r a y bar by c 2 is equal to 0 so it will become x q plus if i put this value from here so it will be 42.3 x square which is equal to 1939.6 now you write x q plus 3 x square which is equal to 1939.6 by trial and error if i start with 6 this value your 216 and this is your 1522.8 which is equal to 1738.8 if you buy trial and error this is not achieving we are not achieving 1939.8 it is less now second part is your 6.5 you are achieving your 2062.2 now it is slightly better slightly better means slightly greater than this value not as low as like x is equal to 6 now take 6.4 value x now you are getting the value of it is approximately 6.41995 almost close to that so that means once you are getting x is equal to 6.4 that means you got x is equal to your 6.4 feet now with this value with this value you can find it out one parameter is over that means this distance a is 2.2 x you are getting 6.4 6 plus 2.2 8.6 you are getting total embedded depth now you have to find it out how much tension it can take now doing this f a r plus r p minus r a is equal to 0 from there f a r is equal to 9.1 keep 9.1 keep now after getting all these things x and your anchor force anchor force once it is you are getting then take movement about this all the values this movement should because this is assumption this movement should be 0 it should come around 0 that means 0.001 or 0.01 if it is not 0 that means there is something wrong then recalculation has to be done once you get the value of f a r and x plus a with this depth d are factor safety of 20 to 40 percent that means increase this depth means value of 20 to 40 percent this is complete solve the example of case one that is your anchor bulk head in Poisson