 Welcome to lecture series on advanced geotechnical engineering course, we are into module number 6 which is on buried structures or buried conduits this is lecture 1. So the contents of buried structures or buried pipes in this advanced geotechnical engineering course are load on pipes, martens load theory for rigid and flexible pipes and trench and projection conditions and minimum cover requirement and pipe flotation and liquefaction issues. So we will try to look into the trench and projection conditions first and thereafter we will introduce ourselves to martens load theory. So these buried conduits or the buried pipes are very important from the infrastructure point of view like many underground utilities which are required nowadays to be embedded below the ground level and these are on offshore as well as on onshore. So these buried pipelines are divided into two main categories and these are called as ditch conduits that is conduits which are embedded in a ditch or a trench and as not all conduits can be put below the ground level so that some of the conduits will be projecting out or above the ground surface they are called embankment conduits or projecting conduits. So based on the method of installation according to Spangler and Handy 1973 they have been divided principally into two categories first ditch conduits and projecting conduits. So in this particular slide what we are seeing is a trench and with a pipe of certain diameter D and H is the height above the pipe from the central line height above the pipe. So you can see that this is the natural ground the side walls and this is the backfill so the trench is excavated the pipe is placed and it is again backfilled and then brought to the natural ground surface. So this type of conduits are called ditch conduit and also called trench conduit condition. So the pipe is installed in an arrow trench generally the trench width will be less than or equal to 2 times the diameter in undisturbed soil then backfill to the natural ground surface. So pipe is installed in a narrow trench and the width of this trench which is BD or BE is nominally is less than or equal to 2 times the diameter of the pipe then the backfill to the natural ground surface. So the examples of this type of conduits are sewers, drains, water mines, gas mains and buried oil pipelines. So various classes of conduits in installation we are actually trying to look into it and first category we have discussed is the ditch conduit or trench conduit where the pipe is installed in a narrow trench generally the trench width is less than or equal to 2 times the diameter and is installed in undisturbed soil then it is backfilled to the natural ground surface. So examples of this type of conduits are sewers, drains, water mines, gas mains and buried oil pipelines. There is also another type which is called projecting conduits. The projecting conduits are further divided into 2 classes positive projecting conduit and negative projecting conduit. So in the case of a positive projecting conduit it is a conduit or pipe installed in shallow bedding with the top of the pipe cross section projecting above the natural ground surface. So if this is the top of the embankment surface that is the fill above the ground then the pipe is actually placed on the ground surface and h is the height above the center of the pipeline. So a positive projecting conduit is a conduit or a pipe installed in a shallow bedding with the top of the pipe cross, top of the pipe cross section projecting above the natural ground surface. So this is actually projecting above the natural ground. So basically highway and railroad culverts are often installed in this way. Highway and railroad culverts are often installed in this way. So highway and railroad culverts they are basically called as you know the positive projecting conduits. So a positive projecting conduit is a conduit or a pipe installed in a shallow bedding with the top of the pipe cross section projecting above the natural ground surface. There is also another class of projecting conduit is called negative projecting conduit. A negative projecting conduit is a conduit installed in a relatively narrow and a shallow ditch with the top of the conduit below the natural ground surface and the ditch is then back filled with the loose soil and embankment is constructed. So this is you know negative projecting conduit is a relatively is installed in a narratively shallow trench and the trench is and which is below the top of the pipe is below the natural ground and the ditch is then back filled with loose soil and embankment is constructed. So this is basically effective in reducing the load on the conduit especially if the back filled above the conduit is, back filled above the conduit is loose soil. So this is effective in reducing the load on the conduit especially if the backfill above the conduit is a loose soil. So the negative projective conduit is basically installed in a relatively narrow and a shallow ditch with the top of the conduit below the natural ground surface and the ditch is then back filled with the loose soil and an embankment is constructed. Then there is also another class of you know in category of conduits which is called imperfect ditch conduit. This is a special case similar to negative embankment condition but more favorable from standpoint of load reduction on pipe used in very deep installations and difficult to achieve for large diameter pipes and this type of construction is called imperfect ditch conduit or induced trench conduits. So this is a special case similar to negative embankment condition but more favorable from standpoint of load reduction on pipe used in very deep installations and difficult to achieve for large diameter pipes and this type of construction is called imperfect ditch conduit or induced trench conduit. So here these you know here the natural ground is there and the pipe is you know above the natural ground surface the top of the pipe is above the natural ground surface and it is this portion of the soil above the right above the pipe is excavated and refilled with loose soil and then embankment is constructed. So although effective in reducing the load on the conduits this type of construction with loose backfill encourages channels of seepage flow through the embankment and not recommended for wet areas. So for example this particularly use of this loose fill encourages basically in reducing the load on the conduit but this type of construction with loose backfill encourages the channeling of seepage flow through the embankment hence it is not recommended for wet areas. So this type of you know the projecting condition of the installation condition of the conduit is called as a imperfect ditch condition or induced trench condition, induced trench conduit. This is induced trench conduit is that the top of the pipe is above the natural ground and it is you know used in very deep installations and here this portion is excavated and refilled with a loose soil but because of this loose soil there can be encouragement of the formation of channel of seepage flow through the embankment hence it is not recommended for wet areas. So in this particular slide typically types of buried pipes or conduits are shown once again and this is the ditch conduit where this is the you know the side walls and where this backfill with you know certain soil and this is the ground surface. So this is BD is the breadth of the trench and D0 is the external diameter of the pipe and this is the backfill. In case of a positive projecting conduit where you have the diameter D0 and the embankment top is here. So the top of the pipe projects above the ground surface. So here in the positive projecting conduit top of the pipe is projects above the ground surface and above that the embankment construction will happen. And negative projecting conduit pipe is placed in a shallow trench and top lies below the ground surface the top of the pipe lies below the ground surface. So this is the negative projecting conduit and this imperfect ditch conduit is again shown here schematically where you have got top of the pipe above the natural ground surface and a portion which is actually you know filled with loose soil basically to reduce the load on the conduit and then embankment is constructed on the top. So this type of construction of conduit is called installation of conduit is called imperfect ditch conduit. So we have seen different types of you know the conduit installations and accordingly the Marston has actually proposed in a load theory in 1913. So in 1913 Anson Marston developed a theory to explain the characteristics of soil columns above a buried conduit. So because of the shear resistance provided by the walls of the trench known as the soil arching. So we need to understand what is this soil arching? Soil arching is a phenomenon in which yielding mass transfers the forces or stresses to the non-yielding zones. So a significant fraction of the weight of the soil above the conduit is transferred to the walls of the ditch thus reducing the load on the conduit. So because of the shear resistance provided by the walls of the trench known as the soil arching action a significant fraction of the weight of the soil above the conduit is transferred to walls of the ditch and the reducing the load on the conduit. So we have this soil arching phenomenon is actually is visible in number of you know applications in geotechnical engineering namely in retaining walls or in tunnels and in buried conduits. So with reference to from buried conduits point of view we will try to see what is the soil arching and there are two types of arching one is called active arching and passive arching. So the Marston's load theory we will introduce ourselves and then see that how the you know load transferred on to the pipe is a can be computed for a given condition. So the let us now try to understand about the soil arching, soil arching can be you know the best described as a transfer of forces between a yielding mass of geomaterial and adjoining stationary members. So arching can be best described as a transfer of forces between a yielding mass of geomaterial and adjoining stationary members. So stationary members are nothing but non yielding zones in case of buried conduits it is you know the side walls. So a redistribution of stresses in the soil body takes place. So the shear resistance tends to keep the yielding mass in its original position resulting in change of pressure and both the yielding parts you know support and then adjoining part of the soil. So a redistribution of stresses in soil body takes place because of this participation of this arching and the shear resistance tends to keep the yielding mass in its original position resulting in a change of pressure on both the yielding parts support and the adjoining part of the soil. So this topic of soil arching we have actually also discussed in you know when we are discussing about slope stabilization using piles that is piles slope stabilization and we said that when piles are placed very close to each other with centre to centre distance within the slope at an optimum location within the slope then we said that the participation of arching is visible and when the piles are spaced apart like s is equal to 8d then d is the diameter of the you know pile then we said that the piles will behave like individual piles and then visibility of or participation of arching is marginal or negligible. So similarly we have the phenomenon here that when the trench is narrow and when the fill is actually is trying to settle and the redistribution of the stresses takes place and the shear resistance tends to keep the yielding mass in its original portion resulting in a change of pressure on both the yielding parts support and the adjoining part of the soil. If the yielding part moves downward that means that the soil above the conduit the shear resistance will act upward and reduce the stress at the base of the yielding mass. So what will happen is that if the yielding part moves downward the shear resistance will act upward and reduce the stress at the base of the yielding mass. If the yielding part moves upward suppose if the yielding part moves upward the shear resistance will act downward to impute this movement and cause of increase of stress at the support of the yielding part. So if the yielding part moves upward the shear resistance will act downward to impute this movement and the cause of increase of stress at the support of the yielding. So this causes to the stress increases this causes to the stress increase at the support of the yielding part. So if the yielding part moves downward the shear resistance will act upward and reduce the stress at the base of the yielding mass. So this will actually happen in the narrow trench conduit or ditch conduit condition. If the yielding part moves upward the shear resistance will act downward and to input this movement and also causes increase of the stress at the support of the yielding part. So consider let as we said that there are two types of archings one is called active arching the other one is called passive arching. So here if the structure is compressible so the displacement under pressure PS when the structure is more compressible than the surrounding soil. So this is a typical structure which is considered and this is the surrounding soil. So the pattern of displacement at plane AA and BB will be like this and displacement under pressure PS when the structure is more compressible than the surrounding soil. You can see that the soil in this portion will undergo in a settlement and so this is the displacement patterns under the pressure when structure is more compressible than the surrounding soil. So in this case what will happen is that active arching occurs when the structure is more compressible than the surrounding soil. So in active arching occurs when the structure is more compressible than the surrounding soil. So if the structure deforms uniformly on plane AA and plane BB the stresses on it tend to be lower toward the edges due to mobilized shear stresses in the soil. So if the structure deforms like this and the structure is more compressible than the surrounding soil then at plane AA or BB the stresses on it tend to be on the lower toward the edges due to the mobilized shear stresses in the soil. So because of the mobilized shear stresses the stresses will be lower and then you can see that the surrounding areas that is the non-yielding portions are actually receiving the higher stresses. So this is a terminology of what we are defining is the active arching. Active arching occurs when the structure is more compressible than the surrounding soil that we need to note down active arching occurs when the structure is more compressible than the surrounding soil. Now we as we said the other type of arching is passive arching and the pattern of displacements will be like this in case of passive arching where the structure is less compressible than the surrounding soil. When the structure is less compressible than the surrounding soil so that means that here these settlements in this zone will be less and the surrounding soil actually yields more. This is that there will be differential settlements here, there will be differential settlements here and there will be you know the there will not be any settlements here. So this is the displacement pattern under pressure PS when the structure is less compressible than the surrounding soil. The structure is less compressible than the surrounding soil. So in case of passive arching the soil is more compressible than the structure. So passive arching occurs when the soil is actually more compressible than the structure. So as a result the soil undergoes large displacements and mobilization of shear stresses which increase the total pressure on the structure while decreasing the pressure on the adjacent ground. So as a result what you can see is that the soil undergoes large displacements and because of the reduction in the occurrence of settlements you can see that mobilizing the shear stresses which increase the total pressure on the structure while decreasing the pressure on the adjacent ground. So this is the typical stress distribution across plane A or B where you can see that and at the edges you will see that the stresses are highest and the lowest in the centerline. So this is because as the assuming that the structural deformations are uniform being in the sense that rigid compared to the surrounding soil the stresses are highest at the edges and lowest at the centerline. So you can see that this distribution is because the structural deformations are uniform being rigid compared to, inherently rigid compared to surrounding soil the stresses are highest at the edges and lowest at the center. So in the passive arching the soil is actually more compressible than the structure and assuming that the structural deformations are uniform the stresses are highest at the edges and lowest at the centerline. So as a result what you can see that the soil undergoes large displacements mobilizing shear stresses which increase the total pressure on the structure while decreasing the pressure on the adjacent ground. So after having seen active arching and passive arching we said that active arching occurs when structure is more flexible, more structure is in case of passive arching the soil is more compressible than the structure. In case of active arching soil is, structure is more compressible than the surrounding soil. So when the rigid in the case of rigid pipe now very you can see that there are 2 types we will come across that is called rigid pipes and flexible pipes. And rigid pipes when the side columns of soil are the external pressures are more compressible than the pipe due to inherent rigidity and this caused the pipe to assume the load generated across the width of trench. So when the side columns of soil areuis external pressures are more compressible than the pipe due to its inherent rigidity and this cause the pipe to assume the load generated across the width of the trench. So just now we have seen in passive arching you know the soil is more compressible than the structure. So similarly here when the side columns of the soil or the external presence are more compressible than the pipe due to inner end rigidity this causes the pipe to assume the load generated across the width of the trench. And the shearing stresses or the friction forces that develop due to differential settlement of the external presence and the central presence of soil are additive to the load of the central presence alone. So this is defined as rigid pipe. So rigid pipes show the signs of distress before being vertically deflected at 2% of the diameter. That is rigid pipes show the signs of distress before being vertically deflected by 2% of the diameter. So in the rigid pipe when the sides of columns of soil or the external presence are more compressible than the pipe due to its inner end rigidity and this causes the pipe to assume the load generated across the width of the trench. And the shearing stresses are the frictional forces that develop due to differential settlement of the external presence and the central presence are additive to the load on the central presence alone. So in this condition this type of pipe is called rigid pipes and rigid pipes are show the signs of distress before being vertically deflected at 2% of the diameter. The typical rigid pipes include reinforced concrete cylinder, pre-stress concrete pipes and vetrified clay and polymer concrete and cast iron, asbestos cement and cast concrete. So what we can see is that in effect of the soil settlement will be being the structure is rigid here. So this is a type of passive arching condition, this is a type of passive arching condition what we discussed. The surrounding soil undergoes the differential settlements and where here because of the inherent rigidity the structure will not undergo any different movements but because of this as the settlements are uniform the stresses will be higher at this point and at this point and the stress at the center will be lower. So this type of situation is very close to rigid pipe embedded in a certain depth in a soil is very similar to this passive arching phenomenon. So typical rigid pipes include reinforced concrete ones are made of pre-stress concrete or vetrified clay or polymer concrete, cast iron, asbestos cement and castings with two pipes these are all the rigid pipes but effect of the soil settlement on rigid pipes is actually shown here. The surrounding soil actually settles undergoes settlement here and because of the inherent rigidity this portion is actually not and does not undergo any movement. So you can see that both sides it undergoes surrounding soil prism undergoes movements and this particular portion will not actually undergo any settlement because of the rigidity of the pipe. Now consider the flexible pipe if the pipe is more compressible than external soil columns as a result of vertical deflection allowing the central prism to settle more in relation to the external prisms then the actual load on the pipe is less than the load at load of the central prism due to the direction in which the shearing stresses are acting. So this type of pipes are called flexible pipes. So you can see that the settlement D at this portion the soil undergoes in this portion undergoes differential settlement and the surrounding soil does not undergo any movements are relatively less compared to in the zone which is actually above the pipe. So this condition is very close to active arching condition what we have seen active arching condition where the surrounding soil is actually more compressible the structure is actually more compressible than the surrounding soil. So if the pipe is more compressible than external soils as a result of vertical deflection allowing its vertical direction allowing the central prism to settle more in relation to the external prisms. The actual load on the pipe is less than the load of the central prism due to the direction in which the shearing stresses are acting. So hence this is a condition is called flexible pipes. So a flexible pipe has been defined as the one that will deflect at least 2% without structural distress. So generally there is also another definition for the flexible pipes is flexible pipe has been defined as one that will deflect at least 2% without structural distress. And the deflection is limited for less than 2% with rigid lining and coating and if you are having rigid lining and flexible coating then it is 3% the deflection is to be limited less than 3% of the diameter with rigid lining and flexible coating and if you are having rigid lining and coating then it is should be affected only due to the deflection is limited only to less than 2%. So the flexible pipe condition is active arching condition and rigid pipe condition is the passive arching condition. In case of flexible pipe the active arching condition this pipe the soil portion above the pipe undergoes differential settlements compared to the adjacent soil prisms and because of this what will happen is that load acting on the pile load acting on the buried pipe will be less depending upon the because of the you know the shear resistance offered by the side walls on the both sides of the both sides of the pile that is this portion because of this what will happen the shear resistance will act in this direction and the load is actually a portion that is the load which is actually imposed on the conduit will be less. So if the pipe is more compressible than the external soil columns as a result of vertical deflection allowing the central prism to settle more in relation to the external soil that soil in the external prisms the actual load on the pipe is less than the load of the central prism due to the direction in which the shearing stresses are acting. So the flexible pipe for examples include steel pipes and ductile iron pipes and thermoplastic such as polyvinyl chloride and HDPE pipes, high density polyethylene pipes and thermo setting plastics such as fiberglass reinforced polymer FRP pipes and bar-wrapped concrete cylinder pipes. So the for examples for flexible pipes the materials which are actually used for making these flexible pipes include steel, ductile iron, thermoplastic such as PVC and high density polyethylene pipes and thermo setting plastics such as fiberglass and reinforced fiberglass reinforced polymer that is FRP pipes and bar-wrapped concrete cylinder pipes. Now here the typical favorable arch conditions are actually shown arching effect in underground conduits can see that here this is a flexible pipe so where the direction of redo settlement will be here in this direction the resistance from the side walls actually act upwards because of this load acting in the central portion will be less. So in this case inverted arch action and this is predominant in rigid pipes where the surrounding soil actually settles more compared to the soil above the pipe. So this is inverted arch action which also there in rigid pipes and favorable arch action that will be there in the flexible pipes. So we have seen that flexible pipe and rigid pipe in the flexible pipe wherein the soil above the pipe will undergo settlement and so because of this the shear resistance participation of active arching and or we can say favorable arching there is reduction of the load coming onto the pipe. So the buried conduit will experience the less load. So there is also rigid pipe and then we also said that there is one more category it is called semi-rigid pipe. Some pipe materials exhibit characteristics of both rigid and flexible pipes primarily controlled by their diameters and they are basically referred as semi-rigid pipes. So semi-rigid pipes deflect between 0.1% to 3% without causing armful or potentially armful cracks. For example bar-wrapped concrete cylinder pipes is actually an example for the semi-rigid pipes. So some pipe materials exhibit the characteristics of both rigid and flexible pipes and primarily controlled by the diameters and basically they are referred these type of pipes are referred as semi-rigid pipes. So in going to understand about Marston load theory for narrow trenches and let us look into the assumptions which are put forward. Here the cohesion if any between the trench fill and the soil in the trench sides is ignored. So primarily the cohesion if any between the trench fill and the soil in the trench sides is ignored because of its variable and uncertain value depending upon the moisture condition. So the ignorance of this cohesion also would have lead to the conservative side basically so that we will actually calculate more load on the pipe. So as one is that the considerable time would have to be elapsed before cohesion could develop and second thing is that assumption of no cohesion would yield the maximum load on the pipe. That assumption of no cohesion will yield the maximum load on the pipe. So the cohesion if any between the trench fill and the soil in the trench sides is ignored because of its variable and uncertain value depending upon the moisture condition. And the soil density and the frictional properties are assumed to remain constant over depth and the soil friction is assumed to vary in the direction, direct proportional to the active horizontal pressure of the fill against the trench bases. So the soil density and the frictional properties are assumed to remain constant over the depth and the soil friction is assumed to vary in direct proportional to the active horizontal pressure of the fill against the trench bases. So we have when a pipe is embedded in a narrow trench we have the two side faces when you are actually taking per unit length of the pipe. So the cohesion is ignored the ignorance of the cohesion has because one is that the reason for ignoring the cohesion is that a considerable time would have to elapse before the cohesion could develop. And second thing is that it is ignorance of the cohesion leading to a conservative assumption as no cohesion would yield the pipe would be designed for that type of load which is coming from the war burden soil above the pipe. And the soil density and the frictional properties are assumed to remain constant over the depth and the soil friction is assumed to vary in direct proportional to the active horizontal pressure of the fill against the trench bases. So in this particular slide a free body diagram of the ditch conduit is actually shown here a pipe of diameter external diameter BC and or a conduit of BC and internal diameter let us say small d and the width of the trench is indicated here as bd but from the terminology we can also write b is equal to bd. And so let z is the depth from the natural ground surface so this is the ditch conduit condition. So Marston load theory is based on the concept of the presence of soil in the trench that impose a load on the pipe. So h is the height above the center of the pipe and b is equal to bd and so these are the side wall presumes and this is the portion of the soil above the soil within the prism which is above the pipe. And this is called the bedding of the pipe and then consider the per unit length. So the Marston load theory is based on the concept of a prism of soil in the trench that impose a load on the pipe. So this is basically for ditch conduit the weight of the war burdened soil transferred to the underlying soil with the due consideration to the soil arching action. So equating upward and downward forces we get cd into gamma into bd square. So the how we have got is nothing but we have taken a thin horizontal slice of having thickness small dh at a depth h below the ground surface or at a depth z below the ground surface. And when this the self weight of this the soil within the wedge is nothing but gamma that is the unit weight of the soil back filled into the trench after installing the pipe gamma into bd into dh. So bd into dh into 1 is the volume that is the per unit length into gamma is the weight of the slide weight force acting like this. Now at a given portion at a given portion say at a given depth here what you can see is that v is the vertical load acting or an area that is nothing but bd into bd into 1. So v by bd is the vertical stress into k that is nothing but the k is nothing but the coefficient of earth pressure which is ratio of lateral pressure that is horizontal pressure to vertical pressure. So k into vertical stress into dh acting over a small length that is the dh into 1 that is the force acting on the normal force acting on the side wall phase. So we can get the frictional force when this mass moves downwards k into mu dash or mu into v by bd dh is the countering shear resistance or shear force. So these are the from those two sides we actually have the shear forces and here there is a normal force acting at a given depth. So how we have got this one is nothing but this is nothing but v by bd is the vertical stress v by bd into 1 into k is the horizontal stress into dh into 1 is the horizontal force and then into multiplied by mu dash you will get the frictional force acting along the shear force acting toward this side action this side action and this direction. Now we taking the equilibrium of sigma fe is equal to 0 what we get is that we get a differential equation and then once we solve that one and for this condition then we get this v is equal to cd into gamma into bd square. So this is what actually is described here for sigma fe is equal to 0 taking upward vertical forces are equal to the downward vertical forces. So for equilibrium vertical force at the bottom plus shear force at the sides is equal to vertical force at the top plus weight of the element on the slice. So vertical force at the bottom that is the bottom of the slice and shear force at the sides is equal to vertical force at top and weight of the element. So that is vertical force at the bottom that is nothing but v plus delta v and shear forces at sides that is because the two sides are there it is multiplied by 2 into k into v by bd dh into mu dash. So two sides are there so multiplied by 2 is equal to v that is the stress acting on the top of the slice horizontal slice plus the self weight force that is gamma bd into dh by simplifying what we get is that 0 is equal to bd minus 2k mu v mu dash v by bd into dh by dv. So this is basically the solution of differential equation and the solution will actually yield to v is equal to gamma bd square by 2k mu dash into 1 minus e to the raise minus 2k mu dash by h by bd. So this expression for w v is equal to gamma cd bd square so this is given as the cd is nothing but 1 by 2k mu into 1 minus e to the raise minus 2k mu z by b where k is nothing but the coefficient of earth pressure and mu is the coefficient of friction for granular soil back filled ditch interface that is the coefficient of friction of the granular soil backfill and ditch wall interface and which can vary from 0 for a smooth wall and tan phi dash for a rough wall where phi dash is the angle of shearing resistance of the granular soil fill and gamma is nothing but the unit weight of the granular backfill and b is equal to bd is the trench width and cd is the load coefficient so this is the load coefficient. So this if you use v is equal to cd gamma cd bd square we are able to get then we can actually get the load on the pipe. So the cd it can be obtained by 1 by 2k mu into 1 minus e to the raise minus 2k mu into z by b. When z is equal to h then it will be on the top of the pipe that is the if you consider this portion when z is equal to capital h we will get the load coming at the stress coming on into this particular portion in kilo Newton per meter. Now let us look variation of cd with z by b for different values of k mu. So what we said is that this particular graph which is actually shown where the cd is plotted on the x axis from 0 to 5 that is the load coefficient and z by b which is z is equal to 0 at the top of the that is at the ground surface and for no arching no soil arching the smooth wall you can see that you know this particular curve tends to be somewhere here. But when there is k mu is equal to 0.1, k mu is equal to 0.15 and k mu is equal to 0.2 as the you know we can see that k mu is increasing there is a decrease in the cd value with an increase in k mu value there is a decrease in the k mu value there is a decrease in the cd. So in case of you know this variation of cd with z by b for different values of k mu handy and Spangler 2007 they suggested that k mu can be taken conservatively as 0.11 for saturated clays where it is a multiplication of k initially Marston proposed k0 ka that is the Rankine's active earth pressure condition for k this is originally proposed by Marston. Now it is actually you know convenient to consider k is equal to k0 where k0 is equal to you know coefficient of earth pressure at rest that is for like by using Jackie's formula it is 1 minus sin phi and where phi is the friction angle and it is also convenient to consider tan mu is equal to delta where delta is the interface friction angle between backfill soil and as well as the side wall soil in the side wall. So handy and Spangler suggested that k mu can be taken as conservatively as 0.11 for saturated clays and 0.19 for the you know the granular soils with all other soils they lie within this range that means that for most of the soils they fall within this range that is the k mu factor will actually fall in between 0.1 to 0.2 when the k mu is increasing there is a significant reduction in cd hence the load on the hence the load transferred to the conduit. So when cd you know decreases that is when k mu increases then there is a decrease in the cd value the significant reduction the cd value can be seen and that implies that the load on the conduit also reduces. So here what we have seen is that w that is v is equal to gamma cd you know vd square with that what actually we said is that when a smooth wall is there no arching is taking place then this is the situation. So when we have you know large width of the trench and when there is a no participation of arching then you know we may actually end up you know condition where no soil arching takes place. But for most of the soils where k mu falls between 0.1 to 0.2 where when k mu increases there is a significant reduction in the cd value hence the load imposed on the or applied on to the or transferred to the conduit also will be less. So in this particular slide the approximate values of ratio of lateral to vertical earth pressure k and coefficient of friction against the sides of the trench is given and here it is this is the Rankine ratio where k which is given as 0.33 for partially compacted damp topsoil and the coefficient of friction is mu then you know saturated topsoil it is 0.37 and the coefficient of friction is 0.4 the partially compacted damp clay is 0.33 k value and the coefficient of friction is 0.4 the saturated clay is 0.37 and coefficient of friction is 0.3 and dry sand it is 0.33 Rankine ratio k and coefficient of friction is 0.5 and wet sand it is 0.33 and coefficient of friction is 0.5. So when you have seen that in the previous equation when v is equal to gamma cd into cd into gamma cd into bd square where by substituting for in cd expression when substituting h is equal to h or z is equal to h we get the total vertical pressure at the elevation of the top of the conduit. Now how much of this vertical v load v is imposed on the conduit is depend upon the relative compressibility of the pipe and soil. So that we have discussed about the different types of the pipes where flexible pipe or rigid pipe for very rigid pipe the clay are concrete or heavy walled cast iron and so forth the side fills may be very compressible in relation to the pipe for very rigid pipe that is like clay or concrete or heavy walled cast iron and so forth the side fills may be very compressible in relation to pipe and the pipe may carry practically all the load v for that means that for most of the rigid pipes the entire load is transferred to the pipes. In case of flexible pipes the imposed load will be substantially less than v since the pipe will be less rigid than the side fill soil. So in case of flexible pipes the imposed load will be substantially less than v since the pipe will be less rigid than the side of the soil. So how much vertical load is transferred on to the conduit is depend upon the relative compressibility or relative stiffness of the pipe and soil that is the pipe soil relative stiffness. So we have understood now for very rigid pipes and the side fills may be very compressible in relation to the pipe and the pipe may carry practically all the load and the type of arching also comes into picture here is that passive arching and in case of flexible pipes the imposed load will be substantially less than v since the pipe will be less rigid than the side fill soils. So that is active arching phenomenon. So if you look into this the schematic representation of soil pipe contribution in load carrying apportionality, rigid pipe carries almost 80% of the load and only soil carries 20%. In case of flexible pipe the soil bears the entire load and only 20% of the load is actually apportioned by the pipe. So because of the active arching condition what will actually happen is that the flexible pipe attracts less load and it actually transfers the load to the surrounding non-evaluating portions and the favorable arching phenomenon is actually used in case of a flexible pipes. So this particular slide shows the strength contribution percentage in y axis and rigid pipe or flexible pipe the type of this thing. So this clearly shows that this pipe and soil contribution rigid pipe is actually very high the load apportioning capability is very high compared to flexible pipe. So effect of the ditch width on the loading on the pipe so here the width of the trench and width of the, suppose if you are having less width of the trench then the pipe actually attracts less load because of the participation of the trench and if width of the trench is actually more then the load carrying the pipe increases with the width of the trench so that width of the trench should be just enough for the compaction of the soil on the sides of the pipe. So here what we have tried to understand is that the effect of the ditch width on the loading on the pipe if you look into it if we are actually having the large width of the trench there is a possibility that the pipe attracts the more load. So we have to be careful in actually selecting the width of the trench such that it is actually just possible to do the compaction and so there should be an optimum width of the trench such that the pipe will not actually attract the more load. So this analogy is actually very similar to the slope stabilization technique by using piles. When we are actually having piles which are actually spaced closer we said that the participation of arching is very, very significant. When the piles are actually sparsed or spread along the slope and which are actually further if they are actually spaced at larger spacings then we said that the participation of the arching will be very, very less. So the effect of the ditch width on the loading on the pipe if you look into it the load carried the pipe increases with the width of the trench. So the width of the trench should be just enough for the compaction of the soil on the sides of the pipe. And similarly the mode of failure of the flexible pipes if you look into it when well compacted soil provides good lateral support to the pipes but poorly compacted soil provides reduced lateral support to the pipes. So the pipe actually undergoes the changes in its shapes particularly for flexible pipes one can see that the pipe undergoes deformations like this the elliptical tapes which will actually take because of the loss of the support on the sides. So the poorly compacted soil provides reduced lateral support to the pipes. If you are having well compacted soil and provides the good lateral support and the settlements will be within the allowed the 2% of the diameter of the pipe. So if you are having a mode of failure of a flexible pipe if you look into it the poorly compacted soil produces lateral support to the pipes. And because of the loss of the support the pipe actually undergoes changes and further increase of any external loading which we will be actually discussing in the next lecture with that what will happen is that the pipe can actually undergo distress. Similarly when you are actually having rigid pipes we said that rigid pipes attract the most of the load and the structural strength and wall well compacted soil provide good lateral support to the pipe and poor structural strength for example that if the pipe is actually subjected to certain you know the cracking this poor structural strength provides reduced lateral support to the pipes and the cracking can actually can endanger the pipe integrity. So what we have seen in this particular lecture is that we introduced ourselves to the different conditions one is called the projecting conditions one is trench condition or ditch condition then we said that different types of positive projecting projective conduits and negative projecting conduits and I also discussed about the imperfect projection condition. Then we also discussed about that we did to distinguish between rigid pipes and you know flexible pipes and in a way we actually brought active arching and passive arching phenomenon and we said that active arching phenomenon is actually predominant in flexible pipes and passive arching phenomenon is actually prevalent and predominant in rigid pipes. And then we also discussed about the Marston's load theory for trench condition that is the ditch condition wherein we said that in case of a flexible pipe and depending upon the as the stress is actually portioned by the side wall friction the load transferred to the pipe will be reduced and we also said that how the load can be computed by using v is equal to cd into gamma into bd square and we said that if the breadth of the rigid trench is actually large and then pipe can actually attract the more amount of the load. And then we also have seen finally the different typical modes of failures of flexible pipes where flexible pipe failure basically undergoes large deformations and the deflections or deformations such that the shape of the flexible pipe will actually change. In case of rigid pipe they can undergo you know the stress cracking and this is because of the poor inferior strength of the pipe and which actually can reduce the cross section because of that what will happen is that the lateral support to the pipes will be lost.