 Already, we have finished this anchor bulkhead free earth method. Now, we will start next one anchor bulkhead. This is your fixed earth method. Let me draw these three diagrams. This is your elastic line diagram. Then, if I draw the pressure distribution diagram, if I draw the pressure distribution diagram, this is anchor rod. How the pressure distribution diagram looks as we have drawn earlier. Now, if I name it A T, this is J D C E F. So, this distance will be somewhere else around say X and this distance will be say A and this is your completely D. Now, movement distribution diagram also in the same profile, if I extend the movement distribution diagram, if this is my anchor rod and this is your X point of contra flexure where it changes point G. This is your J. This is J at a distance X at a distance X. Now, look at these three diagrams. Look at the three diagram. One is your elastic line diagram and this is your pressure distribution diagram for fixed earth method and this is your movement distribution diagram. Means, fixed earth method means this bulkhead particularly if you see this anchor wall below the dredge line. Below the dredge line, it has been pushed to a certain depth. So, that it will be fixed. It will be fixed. It will not rotate. As earlier case, it will not rotate below the dredge line. It will be fixed. Once it is fixed, it will be considered as a propped cantilever. If you look at this, this is my cantilever wall and this is the load is coming. This part will be we can simulate as if a cantilever wall with a propped cantilever wall. So, if I take this three diagram elastic line pressure distribution diagram and movement distribution diagram, then let me write what are the point G tends to G tends to point of fixed d t point of fixed d t below which below which the sheet pile below which the sheet piles remains remains vertical. That means if I take this is my point G, this is the point G. That means if this is the elastic line, the elastic line will change up to point G, then it will remains vertical. As it is a fixed earth method, then point J, J is nothing but is your point of contra flexure, point of contra flexure. That means earth point J, earth point J, contra flexure means the bending movement or movement distribution bending movement will be 0. So, that point will be a point of contra flexure J. For simplicity pressure distribution diagram, if you look at their E F E F and C G, if you see that E F and C G, E F and C G, if I mark it E F C G, this part generally below the G as it is said below the G it is a fixed one. That means the wall will be vertical, there is no further change in deflection or anything else. So, below point G as it is a fixed, so instead this part may be replaced, may be replaced by means of a simple force instead of pressure distribution by means of a simple force. So, E F C G, E F C G can be replaced by a force R B acting at G, R B acting at G. So, how do you solve? It can be solved by means of elastic line method. It can be solved by means of elastic line method. What is that elastic line method? We will discuss later. There are two methods, one is by means of elastic line method, other is your by means of equivalent beam method, other is your by means of equivalent beam method. In particularly elastic line method, the discussion will be done later on. Now we will start with this, there are two ways to solve it. If I write it, there are two ways to solve it, one is your by elastic line method, other is your by equivalent, equivalent beam method. No, just I am starting this both these methods, elastic line as well as equivalent beam. So, there are two methods to solve this, one is your elastic line method, other is your equivalent beam method. We will start the basic, how we will solve it by elastic line method. What are the assumptions? In elastic line method, what are the assumptions? The basic assumptions, deflection of sheet piling form an elastic line, deflection of sheet piling form an elastic line. Then elastic line changes its curvature, elastic line changes its curvature, changes its curvature at point J, at point J. So, this area, the force below G, pressure distribution below point G, may be replaced by R B, may be replaced by reaction force R B. If you look at the assumptions, the deflection of sheet piling form an elastic line. That means, there will be deflection, it will form an elastic line. This is called elastic line and elastic line changes its curvature at point J. Elastic line changes its curvature at point J. That means, the moment I say that changes curvature at point J, means bending moment is equal to 0 at point J. And pressure distribution below the G, because as you have assumed, this elastic line below G, there will not be any deflection. The pressure distribution below G will be replaced by a concentrated force R B, concentrated force R B. Now, the concept of, concept of elastic line theory, if you see it, particularly analysis. Now, you consider this as a, you consider this as a cantilever beam, consider this as a cantilever beam, as if this is a beam, this is a beam. Once I say it is cantilever, that means one end is fixed. One end is fixed means, if you come back here, this end as I said, this is a fixed support, this is completely fixed. This end is fixed and it acted upon by this concentrated load as well as U D L. So, then you can apply whatever may be this, there are the load distribution will be above the cantilever. It can be, it can be analyzed as a cantilever beam fixed at one end, then draw the bending moment and shear force. Then, where is your maximum moment comes into picture? As it says, point J is your point of contrafecture, then you find it out, where is your maximum moment. Once you find it out, your maximum moment taking into consideration of maximum moment, then you can find it out force of the anchor rod F A as well as distance D, as well as distance D. This is called by means of elastic line method. Second one is your equivalent beam method. Equivalent beam method, in equivalent beam method what will happen? If you look at here, if you look at here, let us take the, let us redraw the pressure distribution diagram, pressure distribution diagram again. Pressure distribution diagram again, if I draw the pressure distribution diagram, one is up to distance x. Look at here, this distance x because the point J is your point of point of contrafecture. That means, here there is no moment, there is no moment. So, if I consider up to point J here, then below the point J, below the point J, this is my point J and also below the point J, if below the point J, if you look at is, this is x, this is your A minus x. If I take it total, this is the distance A and this is your x, this distance will be A minus x. Now, if you take it completely this force and put it equivalent, it is as it is say equivalent beam method R B prime and here also R B prime. Can you see what is the assumption here? Two opposite force, that means it has been assumed a hinge has been taken at point J. A hinge has been considered at point J, so that the bending moment will be zero at the point of contrafecture. That means the moment you take the hinge at this point is equal to zero, that means force it will be nullified, there will not be force are equal to each other. So, that the bending moment will be zero. So, this is consider equivalent beam, that means one beam above the point of contrafecture, other beam below the point of contrafecture. If I summarize it, if I summarize it, what are the assumption a hinge at the point of contrafecture, at the point of contrafecture, where bending moment is zero, then the part of bulkhead above the hinge can be treated as separate freely supported beam, can be treated as separate freely supported overhang freely supported beam or you can say that can be treated as freely supported overhang beam or freely supported beam. The beam reaction R B dash acting at the hinge can be evaluated and this R B dash what is the procedure if you look at here, once again I am saying that it is a equivalent beam method. First one is your by elastic line method, that means it has been assumed as a cantilever, then consider the bending moment and see your force diagram, find it out maximum bending moment and from there you can find it out what is the anchor force, then you can find it out the distance that you got your first method. Second method is your equivalent beam method in this condition, in this case particularly what is the assumption, as there is a point of contrafecture in the beginning for fixed arc method, that means point j is your point of contrafecture, if point j is the point of contrafecture, that means a hinge at the point of contrafecture has been taken, that means where bending moment is zero, as a hinge has been assumed at the point of j, so what will happen the part above this, the part above this has been consider as separate freely supported beam, part above this has been consider as separate freely supported beam, so that means you know the gamma, you know the phi, you know the all the property you know, so you calculate your arc pressure diagram, you calculate your pressure diagram and from this pressure diagram you find it out what is your reaction force r b dash, because this has been consider as a separate beam, as if it is a overhang separate freely supported beam, so overhang part has been assumed as a freely supported beam and you will get your reaction force r b dash, once you get r b dash as there is a hinge at point j, that means this reaction for lower part of the beam, lower part of the beam it should be opposite, so once you consider your r b dash opposite, once you know the force you can calculate you can find it out your distance, so this is the case how the equivalent beam method has to be solved, both the methods can be used, so we will solve a problem based on the equivalent beam method also, draw pressure distribution diagram as I said earlier, draw pressure distribution diagram, that means based on your soil property based on your given soil property and water table, draw the pressure distribution diagram, so this is my pressure distribution diagram, find a this is my distance a, find a the distance of point zero pressure, the distance of point zero pressure below the dredge line, if you look at here this is my dredge line, if this is the dredge line this distance is a, this distance has to be evaluated first, because at this distance this pressure has been changed from positive negative to negative positive or negative positive to positive negative, that means pressure has been changed and point d the pressure will be zero, so find the distance a, so as we have discussed earlier for free earth method from there a is equal to p a prime by gamma prime k p dash minus k dash, which is equal to p a prime by gamma prime k dash, then find x, how do you get x, the point of contra flexure how do you get x, there are charts has been provided to find the x, if you look at this chart I will redraw the chart again also, find x, x is your distance below the dredge line, where there is a point of contra flexure, that means bending moment is equal to zero, find x from chart and compute pressure intensity p prime, now where is your p prime comes into picture, if you look at here this is my p a prime, so this will be your p prime, this will be your p prime, so find x find the distance x and compute pressure p prime, so p prime is equal to a minus x into p a prime by a, then your fourth point find r b dash by equating moment is equal to zero, moment is equal to zero at tie rod, this is the tie rod or anchor rod, find r b dash equating moment is equal to zero at tie rod, at tie rod, then after you find it out your r b prime what else, then you have to find it out your distance y minus x, where is your distance y has come into picture, this is the r b and from this r b to this r b to here, this is the distance y, if this is my distance y, so definitely this distance will be y minus x and this distance will be a minus x, so your fifth point is your find y, y minus x, y minus x, then find d is equal to one point two of y, find d is equal to one point two of one point two of one point two of y, now if I summarize, if you look at here, if I summarize first based on the pressure distribution diagram, based on the based on the soil profile, draw a anchor, draw a anchor, sheet pile wall along with the anchor and locate where is your water table and write this, means what are the value is given, value of soil parameter has been given, so calculate first step is your calculate active earth pressure coefficient, passive earth pressure coefficient, all parameters once you calculate, then next step draw the pressure distribution diagram, so this is your pressure distribution diagram, so draw the pressure distribution diagram up to your, up to your hinge line, this point x, so I draw the pressure distribution diagram up to the hinge line, so this is my pressure distribution diagram, once you get the pressure distribution diagram, can you find it out distance a, if you look at here, this is the distance a below the dredge line, where pressure has been changed that means, where it is 0. So, you know the value, then once you draw the pressure distribution diagram for not doing mistake, you can divide into small parts, because you know the value, you know the height, you can calculate what is the total pressure. Then once you get the total pressure, find distance a, find distance a by means of p a prime, gamma prime, k p prime minus k a prime, if you look at here p a prime, the pressure up to the dredge line, this is your p a prime, the pressure up to the distance x, this is your p prime. Now once you find it out a, then find distance x, x will be find it out from the your chart, I will give the chart right now. Then once you find your distance x, then you can easily find it out pressure p 1, p 1 that means, at the base p 1 is equal to a minus x by a p a prime. Once you get your pressure p, pressure p prime, then find your r b, because bottom line you have to find it out by means of equivalent beam method, what is your force, what is your force or reaction force r b prime. Once you get the r b prime for this case, as it is a hinge support, it should be equivalent opposite, so now it can put it opposite. So you will find it out r b prime by considering movement about your tie rod is equal to 0, this is also another assumption at the beginning of your bulk head or anchor rod, that means bulk head with your anchor bulk head. In that case, the assumption is that movement about anchor rod is to be 0, then once you draw this and movement about anchor rod is equal to 0, you can find it out r b prime. Once you get r b prime, then find distance y minus x, so this is the distance y minus x. So you know r b prime, you know y minus x, you can find it out from there, what is the distance d. So once you know y, once you know y minus x, as you know the x, then once you know y minus x, you can find it out y. Once you get y, you will find it out d is equal to 1.2 times of y. So this is the computational procedure, we will discuss also, we will solve a problem may be sometimes later may be next class. I will give you the chart from where you can find it out this x, distance x, these are the standard charts has been given. For value of phi, this is the value of phi, so it is varying from 0 to 40 degree, so this will be your 10, this is your 40, this is your 30, 10, 20, 30 and 40 degree. Now 0.1 h, 0.2 h, so it will up to go up to 0.5 h, so this graph will be coming in this way, this is the distance x. That means from the chart, you know what is the value of this x is equal to what part of this x. Then once you know the value of phi, you can find it out from the chart, if you know the phi, then you can find it out what is the value of x. Suppose phi is at this case, suppose phi is equal to, phi is equal to 25 degree. So from 25 degree draw a line where it intersect from there, you draw the vertical intercept, so it is approximately 0.15 h, so this is your total height h, so you know it will be given. Once you know total height, from there you can find it out x is equal to 0.15 into total height. That is it, I am stopping here, next class we will solve a complete problem, example problem of this equivalent beam method.