 finished last class positive projecting conduits in this case one thing will continue that is your plane of equal settlement plane of equal settlement when a high field exist over a conduits that means what do you mean by plane of equal settlement if I say that means when a high field exist over the conduits then the settlements are more in the top it is more on top and progressively decreases you can say that progressively decreases top of here top of conduit you can say that progressively decreases towards top of the embankment at certain height he above the conduit you can say that at certain height he above the conduits a horizontal plane exist a horizontal plane exist at which differential settlement is 0 at which differential settlement is 0 differential settlement is 0 this plane called plane of equal settelment this plane is called equal settlement plane of equal settlement above the plane of equal settelment central zone this high zone settles equality if I draw it, if I draw it diagramically in figure a positive conduits with this, this plane is called plane of plane of equal settlement, this is your top of embankment, top of embankment and this height is known as h e and total height is your capital H and this is called this is called your critical plane. If you look at the definition plane of equal settlement that means when a high fill exist over the conduits that means the fill of height of the fill is very high, so that over burden pressure will be more this is called a when a high fill exist over the conduits and in this case what will happen exactly the settlement is more on top of the conduit look at here this settlement is more at the top of the conduits and what will happen the settlement will be progressively decrease the settlement will progressively decrease as you proceed from top of the conduits this is your conduit top of the conduit to your top of the embankment condition one there is a high fill there is a high fill, so once there is a high fill what will happen the settlement above the top of the conduits will be very high and this settlement will be slowly slowly it will decrease slowly slowly it will decrease it will decrease what happen at certain point at certain point there is an horizontal plane exist there is a horizontal plane exist at which the differential settlement is zero differential settlement is zero this plane is called plane of equal settlement that means plane of equal settlement means the settlement here here here everywhere else around and above the top of the conduits will be equal whatever it settle will be equal this is called plane of equal settlement so that means h is your height below your equal plane of equal settlement to your conduit top surface h is your total height below which from the top of the embankment where the where the conduits is there so why you are interested for this plane of this plane of the equal settlement or plane of equal settlement line if you see type of positive projecting conduits if i write it type of positive projecting conduits for a complete these conduits complete these conduits h is less than equal to h c h is less than equal to h e h is less than equal to h e so r p is equal to negative then complete projection conduit complete projection conduit in this case in this case sorry for complete this conduit h is less than equal to h c for complete projecting conduits h is less than h e in this case r p is equal to positive then incomplete these conduits incomplete these conduits incomplete this conduits in this case h is greater than h e where r p is equal to negative then incomplete projection conduits incomplete projection conduits in this case h is again greater than h e where r p is equal to r p is equal to positive so these are the all classification based on your plane of equal settlement from this because once the moment you identify your plane of equal settlement then you know your h e height above the conduit that is your h e that is called plane of equal settlement and depending upon h e is greater than h or h is greater than h e it should be classified as it should be classified as complete this conduit h is less than h e complete projection conduit h is less than h e r p is negative or r p is positive then incomplete this d i t c h incomplete this conduit r p is negative then incomplete projecting projection conduit r p is positive it depends upon that how h and h e are there so some relationship has been given by some relationship have been given by for this positive conduits plane of equal settlements by this author this is m a r s t o n mars turn as given the following expression for positive projecting conduits in this case gamma b c square c p where c p is your c p is equal to coefficient coefficient this is equal to e plus minus two k mu prime h by b c minus one minus one divided by plus minus two k mu prime this is for your positive conduit this is for your expression for positive projecting conduits similarly for incomplete conduits similarly for incomplete conduits similarly for incomplete conduits your c p comes out to be c p is your e plus minus two k mu prime h e by b c minus one divided by plus minus two k mu prime plus h by b c minus h e by b c into e to the power plus minus k mu prime h e by b c so these are the two relations based on your plane of equal settlement mars turn has given for positive projecting conduits and incomplete conduits now next part of this we are going to start that is your negative projecting conduits of your complete derivation as well as a theory behind it so negative as i said earlier this positive conduits negative conduit as well conduits as well as imperfect conduits the definition has been explained by diagrammatically. So will start this negative projecting conduits case now first case is over for positive projecting conduits now second one is your negative projecting conduits if I draw a diagrammatically how it looks it has been used for culverts example for culverts so this is your top of embankment this is your plane of equal settlement plane of equal settlement then if I start with this negative projecting conduits below this top of the embankment there is a culvert and drawing this how it works slightly increasing diagrammatically let me put it in this way this is your delta g p b d and this part will be rho d plus sorry delta d plus delta f plus delta g and this is your critical plane critical plane and this is your critical plane after settlement and if I take it this two this is your kind of dc bc bd and this part will be your coming delta f so as I said in case of negative projecting conduits r p comes out to be delta g minus delta d plus delta f plus delta c by delta d so where delta g is equal to settlement of ground surface delta d is equal to settlement of backfill in trench delta f is equal to settlement of conduit foundation dc is equal to vertical deflection dc is equal to vertical deflection vertical deflection of conduit so as usual it is weight w c is equal to gamma for negative case bd square into c n c n is your c n is your coefficient of negative projecting conduit c n is equal to coefficient of negative projecting conduit and c n comes out to be e to the power minus 2 k mu prime h by bd minus 1 divided by minus 2 k mu prime when h is equal to less than he and this will be with the same term same term plus h by bd minus he by bd e to the power minus 2 k mu when h is greater than he if you look at this negative projecting conduits where it has been used negative projecting conduits example is a culvert so there is a culvert here i have drawn the culvert there is a culvert here in this culvert this conduits is there above this is your top of the embankment top of the embankment that means filling part is there so if i draw this if you look at here one is your plane of equal settlement one is called plane of equal settlement it will be below your top of embankment so that means up to from plane of equal settlement to top of the embankment whatever the settlement is going to happen it should be equal it should be equal now the settlement below this if you look at here there are things like critical plane this is your critical plane then this is your critical plane after settlement there are particularly four type of settlement observed delta g if you look at here delta g settlement of ground surface the ground surface in this particularly conduit at this level so this has been settled with this your dotted line because what will happen once there is a conduit try to understand once there is a conduit above this there is a filling material what will happen the conduit also settle then the material above the conduit also it will also settle along with side by soil also it will settle so in this case delta g is your settlement of ground surface that means it is nothing to with this it is only the ground surface how much amount it settle it is your delta g then delta d is your settlement of back filling trench what is your delta d if you come back here delta d delta d is your settlement of back fill in the trench that means whatever back fill you have done particularly above the trench because this is a trench this is your trench above the trench so that is your delta d is settlement of back fill in the trench and delta f is your settlement of conduit foundation if you look at here conduit is resting over the pressure of the soil what will happen the conduit also settle the conduit settle where below this there is also soil so conduit settlement of conduit foundation this is called below this called conduit foundation so this part is from here to here it is called delta f that means this is your conduit settlement dc is your vertical deflection of conduit if you come back to your dc at the top part how much it will because once the conduit will settle at the top conduit will settle at the bottom at the top what will happen this is we called a vertical deflection of conduit the conduit will be deflected some part of vertical deflection then at the bottom conduit will settle at the bottom conduit will settle because here there is a soil here below the conduit there is no soil this is a water pipes or maybe something it is a concrete so this is called deflection here we are calling it as a settlement so there are four type of settlements one is your settlement due to your ground surface because of your ground surface then second is your settlement of backfill in the trench third is your settlement of conduit foundation this is your conduit foundation fourth is your vertical deflection of conduit fourth is your vertical deflection of this conduit this is vertical deflection of the conduit and now with this this is this is complete how the settlement of conduit how the settlement of foundation soil how the settlement of particularly your kind of ground surface all if you look at here conduit settles above the conduit soil settle below the conduit what what will happen ground surface settle and below the conduit also foundation also settle so everything has been taken into consideration based on that as I said r p r p is your delta g minus delta d plus delta f plus delta c by delta d so based on that based on that the w c is coming about gamma b d square into c n c n is your coefficient of negative projecting conduits c n is your coefficient of negative projecting conduits in this case c n comes out to be this top once h is less than equal to h e where is your h e as I said h e is your distance of the conduit distance of the conduit above the conduit to your plane of equal settlement this is my plane of equal settlement this part is your this is your h e now what is your h h is your from here to top of the embankment this is your h now as I said when this plane of equal settlement you are going to observe if this height above the m conduit is more high that means filling material is more high so it will be in that case you are going to get plane of equal settlement that is the condition otherwise you will not you may not get your plane of equal settlement so these two terms depending upon that you can say that if h is less than h e what is that c n coefficient of negative projecting conduits if h is greater than h e what is the coefficient of negative projecting conduits so these are all about this how it settle how the behavior will settle what will happen what will happen in this case what will happen this settlement will be kind of this this kind of settlement it will go all of a sudden it become a plane of equal settlement plane of equal settlement means the settlement at the ground surface settlement above the conduit settlement off conduit settlement of foundation surface settlement of ground surface all will be equal all will be all will be suppose it is suppose somebody say if it is a 10 mm of settlement are the plane of equal settlement that means delta g is equal to ten m m delta d is equal to ten m m delta f is equal to ten m m delta c is equal to ten m m. So this is called plane of equal settlement. So how this in this case negative projecting conduits what are the what are the effect how the effect has come it has been completely drawn in a figure where b c is equal to b c is equal to diameter of conduit b d is equal to width of your trench width of a trench because this is your culvert this is your culvert width of trench where trench has been made then culvert has to be erected. So this is your b d is equal to this. So this is all about your negative projecting conduits may be next class we will start about some special conduit cases of the complete derivations. Thanks a lot.