 imperfect ditch conduits. These are the special conduits to be used, the reason being it has to be reduce the load, reduce the load on a conduit under a high embankment, high embankment. So, imperfect ditch conduits are special type of conduits and it has been used particularly to reduce load on the conduits. What we have seen generally the load coming on the conduits generally it is very higher. So, it is reducing this load on the conduits under a high embankment. If the embankment is very high, the construction generally it has been done by two ways. One is first conduit is install, first is your conduit installed as a positive projecting conduit, as a positive projecting conduits, then it is covered with earth field, then it is covered with covered with earth field up to a height, up to a height p b c, p is varying from 1 to 2.0 and of course, the field will be well compacted. This is your construction there are two ways, then second way the construction is that trench is excavated directly above the conduit, means trench is excavated directly above the conduit from the initial level of embankment to top of the conduits, then trench will be back field, afterwards trench will be back field. So, if I start with the A 1, if I draw the figure A and B, how it looks? So, this is your initial level of embankment, you can say that initial level of embankment, then second one is, second one is if you look at here, second one is this is your top of final embankment and this part is your as I said plane of equal settlement and this is where it lies this, if I draw the diagramically the two cases, this is your compressible material and this part is your p b c. If you look at this two parts as I said earlier, this is a special type of ditch conduit, imperfect ditch conduit would say and it has been used particularly to reduce the load on the conduits under a high embankment. If this is a high embankment, what are the load coming to the conduits? It has been reduced by this method. So, there are two ways of construction first one is your let the, let the conduits placed, let the conduit placed and it should be installed like a positive projecting conduit. What is positive projecting conduits? That means, the entire conduit will be above the surface that is why this is called positive projecting conduits. Then after that covered with earth field, then you can say covered with earth field, then p b c p is equal to 1 to 2.0. Second one is, second one is after placing this, after placing this a trench will be directly excavated, a trench will be directly excavated above the conduits. That means, first you excavated the trench. If you look at the two cases, two ways of construction, first way of construction is first place the conduit, first place the conduit, then fill this material, fill your soil material. In this case, instead of excavating, placing this first to excavate the trench, first to excavate the trench that means, from this embankment, this excavation has been made. This is of your plane of plane of equal settlement, plane of equal settlement that means, first to make the trench, first to make the trench in this case, then put your conduit down, then this trench has been filled up. Look at the two difference between these two construction method. In this case, first conduit has been filled, first conduit has been filled, then soil has been, soil has been, first conduit has been placed, then soil has been filled, then embankment construction has been done. In this case, whole entire embankment is there, first make a trench, first make a trench, then conduit has been placed below this, then you can fill the material. So, these are the, there are the two ways of doing these conduits. So, this is a special case of conduit. This is called imperfect ditch conduits. Then another special case is coming that is your tunnel conduits. Conduits passing this beneath of below the existing embankment or hilly area are known as the tunnel conduits, particularly conduits, you can write it conduits passing below or beneath hilly areas, hilly areas or existing embankment or existing embankment or existing embankment or existing embankment are known as tunnel conduits, are known as tunnel conduits. If you look at the tunnel conduits, definition that means conduits passing beneath the hilly areas or existing embankment that means this embankment in this case imperfect ditch conduit, what will happen? The embankment has to be constructed above the conduits. In this case, already the embankment is there. So, beneath the embankment you have to pass the conduit or beneath this hilly area, you have to pass this conduit. In this case, it is called tunnel conduits. So, particularly these are all your sewer pipes, example in a highway or railway lines, sewer pipelines in a highway or railway lines. If you look at this, how it figure wise, how it looks, this is your top of existing conduits, this is your total height h. Now, if you come back, this is called tunnel or jacked sleeve and this is your tunnel conduit, this part is your called bt. bt in this case load on the conduit can be found out gamma bt square ct minus 2c bt ct. bt is your width or diameter of tunnel or diameter tunnel, c is your cohesive strength of soil, c is your cohesive strength of the soil, c is equal to cohesion, ct is your as I said earlier, this is your coefficient coefficient of tunnel conduit. And which is equal to cd or means equivalent to ditch conduit, this is your coefficient of tunnel conduit and which generally ct is equal to cd. If you look at here, particularly in case of tunnel conduits, now there is already this embankment or hilly region is there. Look at this kind of, it may be an embankment or this kind of hilly region is there. You want to make a sewage pipe inside, make a sewage pipe inside, so that without disturbing your existing embankment. That means making a hole inside, then pass sewage pipe inside that. So, this is called particularly tunnel conduit. So, if you look at here, this is your existing existing, top of the existing embankment, existing embankment, then below this existing embankment at an height h, this conduit has been inserted. So, best example is particularly sewer pipelines. Sewer pipelines, if you look at there, without doing anything else, without much excavating, we push this sewer pipelines below the embankment, below the embankment, these are called tunnel type of conduits. So, you can get it like as earlier, only this 2 c bt ct has to be deducted. So, ct is equal to coefficient of tunnel conduit, which is equivalent to your c d. Earlier, we have derived for these conduits, coefficient for these conduits. So, you can find it out the pressure on your conduit that is your WC. These are the 2 special type of conduits. So, one is your imperfect ditch conduit, second one is your tunnel conduits. So, imperfect ditch conduits has been used particularly to reduce the pressure above this conduit. Tunnel conduits has been used particularly below this hilly area or an existing embankment. Example is your best example is your sewer pipelines below this rail road, below road or below any hilly area. So, this is your best example particularly this sewer pipelines. Then, we will come back to next part that is your about shaft. We will start new one that is stresses in the soil in the vicinity of vertical shaft. These are all about conduits what we have finished this is about your conduits. Now, we will start about this shaft stresses in soil in the vicinity of a vertical shaft. Next part is your stresses in the soil in the vicinity of a vertical shaft. If a vertical shaft is going inside the soil. So, what is the stress around the periphery of this vertical shaft? Let us draw this diagram. This is r 0 and this depth is your vertical shaft of gamma z. Now, stresses in the soil vicinity around the periphery of a vertical shaft. If this is the vertical shaft, if I draw it sigma theta, this is your sigma z and this is your sigma r and this will be k 0 gamma z, k 0 gamma z. Let us consider a point around this vicinity. So, let us say p into r z. So, this will be total will be let us say 2 k 0 gamma z, 2 k 0 gamma z, 2 k 0 gamma z. Let us consider for example, let us consider a vertical shaft of internal means this radius is equal to r 0, vertical shaft of radius is equal to r 0 and with this vertical shaft at a depth of gamma z, this is your at a depth of up to gamma z. Now, we are interested to find it out what are the stresses along the soil means if it this is my vertical shaft what are the stresses variation along the vicinity of the vertical shaft. Let us say stress at any point, let us consider a point p at any distance here. Let us consider a point p, p is equal to gamma z. Let us consider stress at any point, any point say p gamma z before the exhibition of start are given by. Let us consider p gamma z before we have done any exhibition a point at this point at a distance of gamma z at this point. So, let us these before this exhibition, before this exhibition will stress at any point p gamma z before exhibition. So, sigma z i is equal to gamma z sigma r i is equal to gamma z, sigma r i is equal to k 0 gamma z, sorry this is not r this is gamma z sigma theta i is equal to k 0 gamma z. So, sigma z is equal to vertical stress sigma z is equal to vertical stress sigma r is equal to vertical stress sigma r is equal to horizontal radial stress, sigma theta is equal to hoop stress sigma theta is equal to hoop stress or it may be a horizontal, circumferential stress horizontal, circumferential stress gamma is equal to unit weight as you know gamma as you know gamma is equal to unit weight k 0 is equal to coefficient of earth pressure. The shear stresses are 0, you see the shear stresses are 0 because before any excavation look if you look at here suppose there is no vertical shaft for example, there is no vertical shaft the ground is full means this is a soil this is a ground then in this case what are the stresses. So, stresses at point vertical stresses obvious vertical stresses is equal to unit weight gamma is equal to unit weight times of z then horizontal or radial stress horizontal or radial stress that also horizontal or radial stress if you look at here a ground is there at this point I am asking what is the stress. So, vertical stress is equal to gamma and z whether radial or hoop stress these are all k 0 times gamma z k 0 times gamma z means k 0 is equal to earth pressure at rest coefficient of earth pressure at rest into gamma z this is your lateral stresses or lateral pressures. Now once the excavation has been done in this case once the excavation has been done suppose the excavation has been done then what will happen. Now let us say let us say the material let us say earth shear stresses on this cylindrical surface within the soil mass are 0 the material located within the boundaries of the proposed shaft can be replaced by material located within boundaries of shaft boundaries of shaft means this is your shaft material located at the boundary of the shaft can be replaced can be replaced by a liquid by a liquid of unit weight of gamma l equal to equal to equal to k 0 gamma. What I have taken what we have taken in this material if this is a shaft around the periphery around the periphery what are the materials material is soil and what is the stress coming into lateral stress k 0 into gamma. So, this can be replaced as a equivalent fluid that is your unit weight k 0 into gamma unit weight is equal to gamma l. Now the horizontal pressure p due to equivalent liquid load. So, this horizontal pressure now it will become because of a liquid p is equal to gamma l z which is equal to k 0 gamma z at the pressure is equal to gamma r i gamma theta i there is no effect on stress in the soil in the vicinity of the shaft this pressure horizontal pressure if you look at this horizontal pressure is nothing but sigma r and sigma theta this is nothing but is your sigma r and sigma theta sigma theta. So, now this stress due to equivalent fluid pressure can be found by lambe's formula lambe has given lambe has be given stress due to equivalent fluid pressure lambe what we have taken how it has been modeled because here what will happen there is a shaft the shaft has been penetrated. So, there are the soil. So, lambe earlier what he has given the equation suppose there is a shaft inside the fluid instead of soil there may be fluid water may be fluid inside the fluid what is the stress what are the stresses coming into picture. So, that formula has been used here in this case suppose there is a fluid around the periphery of the shaft then the sigma z p p is your point p is your point this point where we are discussing about the stresses sigma z p is equal to 0 sigma r p is equal to p r 0 whole square by r square which is equal to k 0. So, r 0 gamma z r 0 whole square by r square sigma theta p sigma theta p is equal to minus p r 0 whole square by r square which is equal to minus k 0 gamma z r 0 whole square by r square. So, p is equal to small p these are all small p these are all your small p small p is your stress are due to pressure p of the equivalent fluid the small p is your stress due to equivalent fluid around the periphery of this shaft what is the equivalent fluid. Now, after the shaft has been excavated once the shaft has been excavated along the depth the shear stresses and radial stresses on the interior surface are 0 the shear stresses after the shaft has been excavated after shaft has been excavated the shear stress and radial stress are 0 radial stress are 0 on the interior surface on the interior surface. So, stress at any point after the excavation of the shaft can be obtained by super position of initial stress and those due to pressure you see if you look at here if you look at here what we have done what we have basically done initially we consider there is no shaft there is no shaft as if there is a ground then at point p what is the what is the stress what is the pressure coming these are all your pressure coming. Then second part what we have done q 1 shaft has been put inside. So, both these sides of the soil has been taken into equivalent fluid equivalent fluid. So, with this what are the stress has been come as if a shaft is inside the fluid. So, what are the stresses coming into picture if I want to bring into this picture. So, what will happen I will take out this shaft that means I will super impose case one without shaft with what is the stresses and second is your with shaft what is your stresses then shaft has to be taken out then by super position method of super position we can find it out stress due to the pressure of x vertical shaft by method of super positions. So, stress at any point after excavation of shaft can be obtained by super position of by super position of super position of initial stress and those due to pressure and those due to pressure. So, now this becomes sigma z is equal to sigma z i i is equal to initial I forgot i is equal to initial case before your excavation i is equal to initial case. So, this minus sigma z due to pressure due to pressure equivalent to fluid or due to may be earth which comes out to be gamma z in this case is equal to 0 that sigma r which is equal to k 0 gamma z minus sigma r p which comes out to be k 0 gamma z into 1 minus r 0 whole square r 0 whole square r 0 whole square r by r square. So, similarly sigma theta also sigma theta comes out to be k 0 gamma z 1 plus r 0 whole square by r square. Now with this if you look at the variation if you look at the variation the variation has been drawn the variation has been made for sigma r how it is varying and sigma z theta how it is varying wherever what is the variation of sigma z if you look at the variation of sigma z around the vicinity this is my shaft has been excavated in the vertical direction. So, what will happen my sigma z at this point it is gamma z at this point it is gamma z at this point also it is gamma z at this point also gamma z at this point also gamma z there is no change. So, that is why this is a straight line this is a straight line now come back to sigma r sigma r is equal to sigma r is equal to if this is my sigma r k 0 gamma z 1 minus r 0 square by r square r square is nothing but this distance from here to here this point this point is distance from here to here this if you look at the coordinate is your r and z z is your coordinate from here to here this is your z vertical distance r is your radial distance as I increase the radial distance as I increase the radial distance what will happen the variation will start the variation will start like this. Similarly for sigma theta it is also 1 plus r 0 square by r square the variation has been given. So, these are all your stress in the soil in the vicinity of a vertical shaft if I exhibit a vertical shaft vertical shaft then what will happen what are the stress variation in the soil as I go radial distance as I go radial distance what will happen where it is in practical importance suppose a vertical exhibition or vertical shaft has been inserted or may be vertical exhibition has been made. So, in that case you can find it out you can find it out you know what is its influence what suppose there is another structure is there nearby what is it influence how the variation of sigma r and sigma theta. So, this r distance is your design parameter distance based on the r distance you can go for another construction may be embankment may be any other man made construction you can do it this as a this sigma theta and sigma r as a significant roll in stress variations particularly in the soil in the vicinity of the shaft. Thanks a lot.