 Yesterday, we have derived this equation for stresses for these conduits and it is not possible for these conduits, vertically cut generally the side slope has been provided in case of these conduits. So, this is your distance b d and we have derived this, if this is my b c, b c is your diameter of the conduit, b d is your width of your ditch. We have derived this w c is equal to yesterday c d gamma b d square, where c d is equal to c d is called load coefficient, c d is called load coefficient for these conduits, load coefficient for these conduits. Now, the value of c d is equal to, we derived it 1 minus e to the power minus 2 k mu prime capital H by b d and divided by 2 k mu prime. So, it has been this coefficient of load coefficient for these conduits has been given graphically for these conduits, if I plot between this H by b d versus is your load coefficient c d, start with this 0, 1, 2, 3, 4 and this is 5, now it starts with 1 up to 14. So, the range is coming about between this, if you look at here this has been given graphically. Now, this value of if I take it k mu prime value of 0.12 and this value is equal to k mu prime is equal to 0.11 and this range is for clay for clay and this is if I am taking k mu is equal to k mu prime is equal to 1.11 to 194, 1.11 to this is your k mu prime is equal to 1.94. So, this is this range is particularly for sand or cohesion less soil, this is your dimension less parameter if I take it H by b d, this is a dimension less parameter and c d is equal to load coefficient, if you note the load coefficient because this load coefficient has been derived for different value of H by b d, different value of k and mu prime. k is your lateral earth pressure coefficient mu prime is your friction between your d's and your side fill soil and if you know this k mu H and b d you can find it out this c d value. So, it has been plotted for different soils it has been taken for two kinds of soil, one is your cohesion less soil that is sand other is for clay the range is well defined for clay it is varying between k mu prime 0.1, 1, 2 k mu prime is equal to 0.1, 2 within this band of this range this is for clay soils and this for sand k mu prime is equal to 1.11 and k mu prime is equal to 1.0, 0.1, 1, 1, 2, 0.194 I did a mistake 0.1, 1, 2, 0.194. So, the range is well defined you can say that for particularly 0.1, 1, 2, 0.194 you can say that this is for sand 0.1, 1, 2, 0.1, 2, 0, we can say that this is for range for clay. In case of this this derivation is basically for rigid ditch. So, in case of flexible pipes conduits. So, what will happen in case of flexible piles some degree of stiffness should be applied for flexible flexible pipes some degree of stiffness at the pipe itself value we can say that degree of stiffness w c which is equal to b c by b d the load on flexible pile would then be then it will be this is your degree of stiffness. So, w c is equal to c d gamma b d square into b c by b d. So, this comes out to be c d gamma b d and b c gamma b d and b c if you look at this if this pipe sometimes what happened actually in practice not a single pipe or maybe all pipes are not perfectly rigid there will be some flexibility. So, this flexibility degree of stiffness has been added with this the modified value of load is coming how much to the vertical means particularly above the ditch this is your c d that means load coefficient into gamma b d into b c b d is your width of your ditch b c is your diameter of width of your ditch and b c is your diameter of your conduit. So, in this case this is for rigid conduits and this is for your flexible pipes or flexible conduits. Now, we will move forward for the other classification one part is over as I said earlier there are three types of basically we use this conduits one is your ditch conduits other is your positive projecting conduits third is your negative projecting conduits. Let us start with the second one this is your positive projecting conduits positive projecting conduits means it will be rest if you look at here it will rest for example, why it is called positive projecting conduits if you look at here it just slightly embedded below the ground surface and it will this pipe or the conduits it will above the ground surface above the ground surface that is why it is projecting this some part is projecting above the ground surface that is why it is called positive projecting conduits. Now, where it has been used as I said earlier it has been used for culverts for highway culverts for highway then railway or air field if you look at here if there is a ground surface here if this is your positive projecting conduits what will happen generally this has been allowed passage of water. So, what happened the road has been constructed above this, but actually this conduit is lying above the ground surface your ground surface is here. So, it has been practically used particularly culverts or railways culverts for highways culverts for railways it has been practically applied. So, now if I am I am considering this positive projecting conduits. So, there is a differential settlement between the central zone directly over lying the conduit and the side zone let me draw two figures. So, this is your top of embankment this is your top of embankment and this is your ground surface this is your ground surface or ground level. Now, what will happen if I let me draw it completely. So, in this case this is your shearing stress shearing stress this is your central zone this called central zone more settlement and this I can term as a flexible conduits. Now, same thing I draw for rigid conduits now in case of rigid conduits if you look out the graph this is your shearing stress this is your shearing stress this part is your central zone less settlement. Now, I can say that this is your rigid conduits there are two difference if I am taking this positive projecting conduits I can separate out this what is your shearing stress and central zone these are the two conduits positive conduits here it is your ground level or ground surface. If it is a flexible what will happen the central zone there are two zone one is your central zone at the middle part other part is your side or shearing stress zone what will happen the central zone will settle more than your side shearing stress develop. So, what will happen the shearing stresses will be vertically upward in case of flexible conduit what is the reason that in flexible conduit central zone will settle more because it is lying above your conduits because it is a flexible conduit it will also settle. So, the central zone will settle more as compared to your side. So, shearing stresses will be acting upward in case of rigid conduits rigid conduits if you look at here it is perfectly rigid those is settlement of the rigid conduit as compared to flexible conduits will be less. So, the shearing stresses will be vertically downward and the side friction means side settlement will be more as compared to your central zone central zone will be your less settlement this is the difference between flexible conduits and your rigid conduits particularly positive projecting conduits. Now, if I start with this detail how this settlement occur as compared to top and bottom let me draw another two figures critical plane has to be shown this critical plane where it is lying the direction of shearing stress can be determined from the settlement of a plane means particularly the direction of the shearing stress can be determined from the settlement of a plane that is called critical plane. So, let me draw it now this is my say you can say that this is your ground surface as compared to this. So, this distance this is your completely let me write it this is p b c b c is your b c is your diameter of your conduits and this is your this part will be delta g and now how the conduits will behave this conduits will go now this part to this part is your delta a this part is your delta f and this complete part will be your delta f plus d c d c is your width of your means d c is your width of your means vertical deflection of your conduits d c then this is called this is called critical plane after settlement. Now, if I take this side shearing stress as compared to your vertical means central zone now one more figure this is particularly in case of what is the mechanics what is the mechanism I am showing this is in case of flexible conduits this is in case of your rigid conduits. Now, this is my critical plane after settlement and this will be delta m plus delta g and this becomes your delta f plus d c now this is your delta f now this is your original ground surface now this is your delta g now I can write it out ratio known as settlement ratio ratio known as settlement ratio with respect to this figure I am coming back let me finish it ratio called settlement ratio this is called r p r p this is not gamma this is a r p called settlement ratio which is equal to delta m plus delta g minus delta f plus d c by delta m. So, delta m is your let me define it delta m is equal to compression of the field compression of the field on the sides of the conduits in the distance p b c. So, delta m is equal to compression of the field on the sides of the conduits in the distance in the distance p b c delta g is equal to settlement of natural ground surface settlement of natural ground surface. Adjacent to the conduits adjacent to the conduits delta f is equal to settlement of conduit into the foundation settlement of conduit into foundation and d c d c is equal to vertical deflection of conduits d c is equal to vertical deflection of conduits. So, if you look at this p is equal to projection ratio p is equal to projection ratio b c is equal to diameter or width of the conduits. If you look at here now there are in case of positive projecting conduits there are two phenomenon one is your flexible conduits other is your rigid conduits. Let us start with one by one this is your flexible conduits in case of flexible conduits what will happen as the conduit is flexible what will happen it will settle. That means the soil above the conduit that we called as a central zone and soil side by this conduit this is called side zone the central zone will be shear more as compared to your side zone that is why what will happen the shearing stress will be vertically upward. Now if I look at this if I look at this look at there is called critical plane it cannot settle up to a certain depth it will settle up to the critical plane after the settlement. So what will happen if you look at here delta f plus d c what is d c d c is your vertical deflection of conduits d c is your how much vertical deflection of your conduits and delta f is your settlement of conduit into the foundation settlement of the conduit into foundation vertical deflection vertical deflection of conduits if you look at here d c is your vertical deflection of your conduit delta f is your settlement of conduit into the foundation. That means once there is a flexible conduit what will happen the conduit itself deflect this is called d c also the conduit also goes inside your foundation. That means it will also penetrate inside your foundation. So this is your critical plane p is your projection ratio what do you mean by projection ratio how much it is projected above your ground level or ground surface how much it is projected if this is my conduits from this how much it is projected above your ground surface if this is my ground level how much it is projected above your ground surface as compare if this is my d or d c or b c whatever you say if it is the diameter with this respect to diameter how much it is projected above. So with respect to p b c the deflection at the top will be total deflection at the top will be delta f plus d c because it is a flexible conduit now if I write it in terms of settlement ratio in terms of settlement ratio it will be delta m plus delta g minus delta f plus d c by delta m what is your delta m compression of field on the sides of the conduits in the distance p b c compression of the field delta m compression of your field on the sides if I look at this is your this value is your this is your delta g and this value is your this complete value is your delta m plus delta g delta m plus delta g delta m is your compression of field on the sides of the conduits in the distance p b c on the sides of the conduits how much soil sample both sides both the sides of the conduits it will compress it will settle and delta g is your settlement of natural ground surface adjacent to the conduit settlement of natural ground surface if this is my natural ground surface how much it settle delta g adjacent to your conduits how much it settle now this with respect to the settlement ratio we can find it out what is your settlement of your central zone of your flexible as rigid conduits now you come back to rigid conduits in case of rigid conduits what will happen you see this is your delta f plus delta d c the delta f what is your delta settlement of conduit into the foundation vertical deflection of conduits this part will be less as compared to your flexible as compared to your flexible in case of flexible delta f plus d c will be more than your delta f plus d c in case of rigid that is why what will happen central part of the soil above this conduit will settle more once it will settle more whole mass will be settling as compared to your side soil that is why this shearing stress will be vertically up in this case what will happen the central part will be settle less as compared to your side soil what will happen your shear stress will be vertically downward this is the physics behind it so these are all these p is equal to projection ratio b c is your diameter of this conduits now with respect to the settlement ratio with respect to the settlement ratio then if I say settlement ratio is negative the moment I say that settlement ratio is negative that means conduits settle more than your critical plane settlement ratio when it will be possible to be negative delta m plus delta g delta m plus delta g is your settlement of adjacent soil adjacent soil compared to your compared to your conduits if you come back to delta f plus d c this is your settlement of your this is your settlement of your conduits now if I the moment I say settlement ratio is negative that means this value is more than this value so what will happen settlement ratio if it is a negative conduit will settle more than your more than your critical plane this conduits particularly this conduit will settle more than your critical plane this exactly happen in case of delta f plus d c will be more than delta m plus delta g in case of flexible conduit but in case of rigid conduit it will be more than this case so accordingly that this physics will change so settlement ratio of various settlement ratio for different conduit the value has been given some values I am just writing this some few of this values based on your settlement ratio we can decide whether it is a flexible conduit or rigid conduit so now the values of the settlement ratio has been given conduit condition and this is your settlement ratio now rigid conduit on a foundation rock just few cases you can get it chart from there are different charts means different values also there I am just enumerating few cases first one is your rigid conduit rigid conduit on foundation of rock this value will be exactly plus one this value will be exactly settlement ratio is equal to plus one because it is rigid conduit again it is on the surface of the rock rock mass means the settlement of ground surface will be less as far as possible and the settlement of rigid means conduit also it will settle less means it is very negligible that is why the settlement ratio will be your plus one now second part is your rigid conduit foundation of ordinary soil ordinary soil means any soil it is varying from plus 0.5 to plus 0.8 why it is plus why it is not negative the why it is plus because this is a rigid conduit the settlement ratio if I write it the settlement ratio is equal to once again for your reference r p is equal to delta m plus delta g minus delta f plus d c by delta m if you look at it is positive because this term settlement of the side of the soil will be more than compared to your settlement of your conduits that is why it is plus and this range is plus 5 to 0.5 to 0.8 second part is your flexible conduits flexible conduits with poorly compacted side soil with poorly compacted side soil this comes out to be minus 0.40 to 1.0 then fourth part is your just this flexible it is a flexible conduit plus well compacted plus well compacted well compacted soil this comes out to be minus 0.20 to sometimes plus 0.80 if it is sorry 0.4 to it will be 0.0 if you look at this particularly flexible conduit if the soil has not been compacted properly it will be always in the minus side it will be always in the minus side because deflection because of your conduits deflection because of your soil below the foundation conduits below the foundation will be more as compared to this now for if it is a well compacted soil generally it is varying from minus 0.2 to plus 0.8 these are all your typical range I will stop it here next class I will start about your negative projecting conduits thanks a lot.