 Welcome to lecture series in the course advanced geotechnical engineering, we are in module 6 lecture 2 on buried structures. So in the previous lecture we introduced ourselves to Marston's load theory and then we have understood about the definitions about the different trench and projection conditions. Now in this lecture we will try to look into the chief differences between flexible and rigid pipes and Marston's load theory for different trench and projection conditions. Before looking into that let us look into what are the significance of buried pipes or buried conduits and what materials are usually transported through these buried pipes. The materials transported through buried pipes are basically crude and refined petroleum and fuels such as oil, natural gas and biofuels and some power plants can also have the cooling water pipelines and the waste products in a fluid state including sewage, slurry and industrial waste and some pipes are also buried pipes are also used for water, transporting water for drinking and irrigation and in some cases hydrogen gas and highly toxic ammonia have been reported to be transported. So pipelines that are buried underground are required to sustain other loads besides their internal fluid pressures. So pipelines that are buried underground are required to sustain the other loads like first sulphate and then due to vehicular loads or due to or due to some seismic loads. That is they must support the soil war burden, groundwater and loads applied at the ground surface like vehicular traffic and forces induced by seismic motion in seismically prone areas. That means that these buried pipelines must support the soil war burden, groundwater and load applied at the ground surface due to load due to vehicular traffic and forces induced by the seismic motion in seismic prone zones. So buried pipelines are therefore the structures as well as conduits for conveying fluids and the special design procedures are required to be adopted while designing buried pipelines to ensure that both the functions are simultaneously fulfilled. That means that it will serve as a structure as well as a conduit. So the special design procedures are required to be adopted while designing these buried pipelines and to ensure both the functions are simultaneously satisfied. As the pipelines are buried more land space can be utilized for construction above the ground. So that is one of the attributes of using buried pipelines. And buried pipelines ensure minimum number of bends and crossing along the length of the pipe as they can pass in a relatively straight manner below the ground. So the buried pipelines ensures minimum number of bends and crossings along the length of the pipe as they can pass in a relatively straight manner below the ground. in turn ensures the minimum losses because if you ensure with less number of bends it ensures minimum losses due to bending and joints. And as the pipes are placed below the ground they are relatively safe from the sabotage point of view. So that is another important issue nowadays as because these pipes are placed below the ground they are required to be ensured the safety has to be ensured from any sabotage point of view. Remember these buried pipelines are necessary in case of pipelines carrying sewage oil or hazardous end products of complex industrial process in order to prevent contamination to the surrounding environment. So the functions of the buried pipelines are hydraulic and structural if you look into it. The hydraulic requirement is that designed to carry fluids carried by public water systems, sewers, drainage facilities and many industrial processes. In the structural they are basically designed to carry the weight of the ground and any load acting on it. So from the structural point of view the pipe need to be designed or buried pipe need to be designed to carry the weight of the ground and any load acting on it. So before looking into the Marston's theory let us look into the different types like again ditch conduits and projecting conduits. So we have introduced ourselves like buried pipes are divided into two main categories one is called ditch conduits and projecting conduits. In the pipe is installed in a if the pipe is installed in a narrow trench generally less than the two times the diameter of the pipe in undisturbed soil then the backfill to the natural grounds level then this is called you know the ditch conduit. The examples of this type of conduits are sewers, drains, water mains and gas mains and buried oil pipelines. So if the pipe is installed in a narrow trench generally the trench width less than two times the diameter of the pipe in the undisturbed soil then backfill to the natural ground surface and this type of conduit category is called ditch conduit. And the projecting conduits are basically two types one is called positive projecting conduit and negative projecting conduit. In the positive projecting conduit basically it is a conduit or a pipe installed in a shallow bedding with the top of the pipe cross section above the natural ground surface. So if the top of the pipe cross section is above the natural ground surface then it is called as a positive projecting conduit. Generally for highway and railway rail road culverts the positive projecting conduits are installed. A positive projecting conduit is a conduit or a pipe installed in shallow bedding with the top of the pipe cross section projecting above the natural ground surface. So highway and railway road culverts are basically installed in this way. And negative projecting conduit basically is a conduit installed in a relatively narrow and a shallow ditch with the top of the conduit below the natural ground surface that means that here this is the natural ground surface. The top of the conduit surface is below the natural ground surface. So there is a reason why it is called as negative projecting conduit. And the ditch is then backfilled with loose soil and embankment is constructed. So basically the ditch is then backfilled with loose soil and then embankment is reconstructed and here the embankment is with the compacted soil. So this is effective in reducing the load on the conduit especially if the backfill above the conduit is loose soil. So negative projecting conduit is a conduit which is installed in a relatively narrow and a shallow ditch with the top of the conduit is top of the conduit below the natural ground surface. The ditch is then backfilled with loose soil and the embankment is constructed. So this is basically effective in reducing the load on the conduit especially if the backfill above the conduit is constructed. So this is basically effective in reducing the load on the conduit especially if the backfill above the conduit is a loose soil. Then another case which is called as imperfect ditch conduit condition where it is similar to lead to embankment condition but more favorable from standpoint of load reduction in pipe used in very deep installations. So basically this is difficult to achieve for large diameter pipes and this type of construction is called imperfect ditch conduit or induced trench conduit condition where the top of the pipe is above the natural ground level and which is actually filled with a compacted fill and above the pipe it is actually placed with loose soil. So this actually process will invite seepage into the area and which actually endangers the stability of the pipe. So because of that this is not recommended for the wet areas and this type of category of installation of the pipe is called imperfect ditch conduit or induced trench conduit condition. So then we also have discussed about the active arching and passive arching. Then we said that this active arching occurs predominantly in flexible pipes and active arching occurs when the structure is more compressible like let us say a flexible pipe than the surrounding soil. So if the structure deforms uniformly on the plane above and below the pipe then the stresses on it tend to be lower toward the edges and due to the mobilized shear stresses in the soil. Because here in the case of active arching what will happen is that the pipe settles more than the surrounding soil. So in the process what will happen the load which is transferred to the pipe is actually less. So if the structure deforms uniformly on the plane above and below the pipe the stresses on it tend to be lower toward the edges and due to the mobilized shear stresses in the soil. So the stress distribution is actually shown here this is at the location of the pipe this is the diameter of the pipe this width is the diameter of the pipe and this is the one of the edges of the pipe and this is one of the edges of the pipe. So the stress distribution across the plane above and below the pipe and passive arching basically this is predominant in rigid pipes the soil is more compressible than the structure that means that the pipe will not undergo any settlements the surrounding soil undergoes settlements. As a result the soil undergoes large settlement the mobilized shear stresses which increase the total pressure on the structure while decreasing the pressure on the edges and ground. So here assuming that the structural deformations are uniform the stresses are highest at the edges and lowest at the center. So the stress distribution across the plane above and below the pipe is obtained like this assuming that the structural deformations are uniform the stresses are highest at the edges and lowest at the center line. This is because of the participation of passive arching particularly for rigid pipes where the soil is actually more compressible than the structure. So here the effect of soil settlement on flexible and rigid pipes is shown here. In case of flexible pipe it can be seen here the pipe undergoes more settlement than the surrounding soil. In case of rigid pipe the surrounding soil undergoes more settlement than the pipe. So in this case the this is the external prism on the left hand side and this is the external prism on the left right hand side and this is internal prism can be seen that this shear stresses are acting downward and imposing the load on the pipe. In this case what is happening is that because the pipe is settling more than the surrounding soil there is a resistance actually offered from the this external prisms on the left hand side and right hand side in case of a flexible pipe. So the actual load on the pipe is less than the load on the central prism due to the you know the direction in which the shear stresses are acting. Similarly here actual load on the pipe is more than the load on the central prism due to the direction in which the shearing stresses are acting. So here the actual load on the pipe is more than the load of the central prism. So the actual load on the pipe will be more than the load of this central prism that means that sulphate of this prism in which this is because is attributed to the direction in which the shear stresses are acting. So in this case you can see that the surrounding soil is settling more than the surrounding soil is settling more than the structure under consideration that is nothing but a rigid pipe. So this is once again which is actually shown in a pictorial way the arching effect in underground conduits particularly for flexible pipes this actually acts like a this is the direction of relative movement and here in this case this is the direction of relative movement. So here the for rigid pipes inverted arching action takes place. In case of flexible pipes the favorable arching action takes place like this. So the direction of movement is this like this and this side and this side there is a resistance. So we can see this arch which is actually formed which is favorable like this for flexible pipes. In this case you know the shape of the arch is like this and the direction of relative movement is downward here and this is the inverted arch action for the rigid pipes. Now let us based on the previous discussion which we have done in the previous lecture let us look into this example problem. So we need to calculate what is the maximum load on a very rigid pipe in a ditch excavated in sand and the pipe outside the diameter pipe of the pipe outside diameter that is nothing but the outer diameter of the pipe is 0.45 meter and the trench width is about 1 meter and the depth of the burial is 2.5 meters and the soil unit weight is 18.4 kilo Newton per meter cube. So the soil unit weight is about 18.4 kilo Newton per meter cube. So first of all in this solution what we need to do is that we need to find out what is CD. So here in this case if we assume it as a wet sand the rank and ratio k is obtained as 0.33 and coefficient of friction which is nothing but 0.5, coefficient of friction mu is given as 0.5. So the k mu is nothing but k into mu is about 0.165 that is 0.33 into 0.5. So first we need to determine CD and the CD can be obtained like this. Now by using this particular curve variation of CD with the z by b for different values of k u so we can see that as we have got for wet sand with k is equal to 0.33 and mu is equal to 0.5 k mu is equal to 0.165 so it will be somewhere here. And with that what we can actually get is that the z by b is nothing but h, h is nothing but the 2.5 meter is the which is given as the depth of burial is 2.5 meter. So 2.5 meter divided by b that is 1 which is nothing but 2.5 and for h by b that is is equal to 2.5, k mu is equal to 0.165 for z by b is equal to h by b is equal to 2.5, k mu is equal to 0.165 we can read CD here and CD works out to be 1.4. So with that what we can get is that w is equal to CD into gamma b square where in 1.4 into 18.4 into 1 square. So with that what we get is that it is nothing but 25.76 kilo Newton per meter. So normally if you look the weight of this prism above this pipe which is nothing but if you say the unit weight is say 20 kilo Newton per meter cube and the height is 2.5 so it is nothing but gamma h is nothing but that is 2.5 into gamma that is 2.5 into 20 is about 50 kilo Newton per meter square. So w is equal to CD gamma b square where 1.4 into 18.4 into 1 square. So with that what we can get is that 25.76 kilo Newton per meter. So in continuation of this discussion the Marston load theory particularly for embankment conditions need to be highlighted we have done for the previous case for the trench conditions. So in conjunction with the positive projecting conduits Marston determined the existence of horizontal plane above the pipe where the shear forces are 0 and if this plane is called the plane of equal settlement. That means that in conjunction with the positive projecting conduits which are actually where the top of the pipe is above the natural ground level the Marston defined the determined existence of the horizontal plane above the pipe where the shear forces are 0. So this plane is actually called as the plane of equal settlement. So above this plane the interior and exterior prisms that means that interior prism is nothing but the prism which is right above the pipeline and exterior prisms are the prisms on the left and right sides of the pipe. So above this plane the interior and exterior prisms of soil settle equally and the condition where the plane of equal settlement is real is called an incomplete or incomplete projection or incomplete ditch condition. If the plane of equal settlement is imaginary and that means that the shear forces extend all the way up to the top of the embankment then it is called a complete ditch or complete projection condition. If the plane of equal settlement is imaginary and the shear forces will extend all the way up to the top of the embankment then it is called as complete ditch or complete projection condition. So there are two cases which are actually considered for the positive projecting conduits. In case one the ground at the sides of the pipe settles more than the top of the pipe. So that is for a rigid pipe the ground at the sides of the pipe settles more than the top of the pipe. In case two the top of the pipe settles more than the soil at the sides of the pipe this is for the flexible pipe. So case one the ground at the sides of the pipe settles more than the top of the pipe. In case two the top of the pipe settles more than the soil at the sides of the pipe. So in case one was called the projection condition by Marston and is characterized by positive settlement ratio that is gamma that is rsd, rsd is nothing but positive settlement ratio. This is for a where this is for called by the case one is called by the projection condition by Marston and is characterized by positive settlement ratio. The shear forces are downward and cause a greater load on the buried pipe for class one. The shear forces are downward because the ground at the sides of the pipe settles more than the top of the pipe because of that the shear forces are downward and causes a greater load on the buried pipe for case one. And case two is called the ditch condition and is characterized by the negative settlement ratio that is rsd will be negative. So the shear forces are directed upward in this case and resulted in a reduced load on the pipe. This is the shear forces are directed upward in this case is resulted in a reduced load on the pipe. So in the positive projecting conduit conditions derivation according to Marston's load theory the two cases were considered on case one is that the ground at the sides of the pipe settles more than the top of the pipe and case two is the top of the pipe settles more than the soil at the sides of the pipe with case one and case two and case one where the positive settlement ratio gamma rsd is used and where case two where the negative settlement ratio where the shear forces are acting upward in this case and results in a reduced load on the pipe because case two the top of the pipe settles more than the soil at the sides of the pipe that is for a flexible pipe case. Now in this particular slide a case one is actually shown where h is the height from the top of the embankment to the top of the pipe and he is nothing but the height of the plane of equal settlement. So this is the plane of equal settlement and this is the interior prism and this is the exterior prism one on the left hand side exterior prism on the right hand side where in this particular case h is greater than he where h is greater than he and sf plus dc. So sf plus dc which is nothing but sf is nothing but the side field and tc is the deflection of the pipe is actually less than that sm plus sg that is the sm plus hg. So here not all pipes are installed in ditches and therefore it is necessary to treat the problem of pipes buried in the embankments and embankment is where the top of the pipe you know is project above the natural ground and this type of installation is defined as the positive projecting conduit and this condition where sf plus dc is less than sm plus sg and h is greater than he is called as a incomplete projection condition. So we are actually discussing about the embankment conditions as the not all pipes are installed in ditches therefore it is necessary to treat the problem of pipes buried in the embankments case also that is the reason why we are actually trying to reduce the expression by using the Barstons load theory so that we can determine what is the load imposed on the pipe for the different conditions which are actually are going to be you know evolved. So in this case the case 2 where h is greater than he and sf plus dc that is the settlement of the fill and you know dc that is the deflection of the pipe is greater than sf plus sg. So this case 2 is called the ditch condition and is characterized by a negative settlement ratio that is rsd the shear forces are direct upward that can be seen here the shear forces are actually acting upward in this case and result of this results in a reduced load on the pipe. So this particular condition of h greater than he and sf plus dc greater than sf plus sg is called as incomplete ditch condition. In case of the previous slide where it can be seen that the shear forces are actually acting downward the shear forces are acting downward so this actually contributes to more settlement on the you know below the critical plane here the critical plane is a plane at the top of the pipe. You can be seen that this is the critical plane so this particular portion is sf plus dc and this particular portion is sf plus sg. In this case also the critical plane you can see that the pipe actually settles more than the critical plane and the surrounding soils only settles this much sf plus sg. So this is sf plus dc where the settlement is actually very high because of the flexibility of the pipe because of the flexibility of the pipe. So here what is sf, sf is nothing but the compression of soil at the sides of the pipe and sg is nothing but the settlement of the natural ground surface at sides of the pipe. So sf is compression of soil at sides of the pipe, sg is nothing but the settlement of the natural ground surface at sides of the pipe and sf is nothing but the settlement of foundation underneath the pipe and DC is nothing but the deflection of the top of the pipe. So SF is nothing but the compression of soil at the sides of the pipe and SG is nothing but the settlement of natural ground surface at the sides of the pipe. That means that these settlements are due to this particular, this is the settlement of the foundation and this is the settlement of the fill below the exterior pipe. There is the sides of the pipe. So this is actually what is being discussed and this is Pbc, the P is actually called as the projection ratio, P is called as the projection ratio. So SF is the settlement of foundation underneath the pipe that is the settlement of foundation underneath the pipe and that is here also it is shown SF as the settlement of the foundation of the underneath the pipe and SG is the settlement of the natural ground surface at sides of the pipes. That is settlement of the natural ground surface at the sides of the pipes that is actually shown here. How much the natural ground surface is settling at the sides of the pipe and SM is the compression of soil at the sides of the pipe and settlement of the natural ground surface at sides of the, SF is nothing but settlement of the foundation underneath the pipe and DC is the deflection of the top of the pipe. that is the DC is the deflection at the top of the pipe. So after having defined the terminologies the settlement ratio is actually defined as SM plus SG minus SF plus DC by SM. So for flexible pipes the settlement ratio is negative because SF plus DC is actually greater than SM plus SG. So the ratio of SM plus SG minus SF plus DC by SM is defined as settlement ratio and critical plane of settlement is nothing but SM that is the strain inside soil plus SG ground settlement and settlement of the top of the pipe is equal to SF plus DC which is nothing but the SF conduit settlement and plus DC which is the vertical pipe deflection due to the property of the pipe. This is the pipe deflection is merely due to the property of the pipe. So in conjunction with the positive projecting conduits the Marston determine the existence of horizontal plane above the pipe where the shearing forces are 0. So this plane is called the plane of equal settlement. So above this plane the interior and exterior presence of the soil settle equally. So in conjunction with the positive projecting conduits the Marston determine the existence of horizontal plane and above this pipe where the shear forces are 0 and this plane is actually called as the plane of equal settlement. Above this plane the interior and exterior presence of soil settles equally. So for the positive projecting conduit in continuation of our discussion the condition where the plane of equal settlement is real that is located within the embankment and is called as incomplete projection or incomplete ditch condition. If the plane of equal settlement is imaginary that is the shear forces extend all the way up to the top of the embankment this is called as a complete ditch or completely projection condition. So for this the Marston load equation for positive projecting conduits is given and this is given by for complete condition for complete condition wc is equal to cc into gamma into bc square, cc is nothing but e to the raise plus or minus 2k mu into h by bc minus 1 divided by plus or minus 2k mu. So please note that there are plus or minus signs are there in expression for cc. So this is for complete condition where wc is equal to cc into gamma bc square where cc is equal to e to the raise plus or minus 2k mu into h by bc minus 1 divided by plus or minus 2k mu. So the minus signs are for the complete ditch condition and the plus signs for the complete projection condition. So in this particular expression the minus sign is for the complete ditch condition and plus sign are for the complete projection condition. For incomplete condition we have to use this one where cc is equal to e to the raise plus or minus 2k mu into hc by bc he by bc minus 1 divided by plus or minus 2k mu plus h by bc minus he by bc into e to the raise plus or minus 2k mu into he by bc. So where minus signs are for the incomplete ditch condition and plus signs for the incomplete projection condition where minus signs for the incomplete ditch condition and plus signs for the incomplete projection conditions and he is the height of the plane of equal settlement. So he is nothing but the height of the plane of the equal settlement. Suppose if you assume that the height of the embankment and height of the equal settlement that is the plane of equal settlement if they are equal. So when he is equal to he and this is actually converted into this case is converted into cc is equal to e plus or minus 2k mu into h by bc minus 1 divided by plus or minus 2k mu into this thing. So when we have h is equal to h mu this particular expression is actually converted into this one. So this particular portion will get nullified so what you have is only this one. So at h is equal to he the incomplete case becomes complete case when he is equal to he where that means that height of embankment is equal to height of plane of equal settlement then that case the incomplete condition gets converted into a complete condition. So what we have done is that by using this different for embankment conditions we actually have deduced depending upon the either flexible pipe or rigid pipe we actually have you know deduced the effect we have made efforts to calculate the load coming on to the pipe that is nothing but w is equal to wc is equal to cc into gamma into vc square. Now this cc basically is a function of the ratio of the height of the cover to the pipe diameter so h is nothing but the height of the cover to the pipe diameter and the product of the settlement ratio gamma sd and the projection ratio that is gamma sdp that is nothing but the that is the settlement ratio product of settlement ratio gamma sd and the projection ratio p and k is nothing but the Rankine constant and mu is nothing but the coefficient of friction. So the value of the product k mu is generally taken as 0.19 for the projection condition and 0.13 for the ditch condition. So the cc is a function of h by bc and the gamma sd into p k and mu. Now we actually have you know the four conditions have been classified according to Spangler based on the installation the complete projection condition is nothing but the top of the conduit settles less than the critical plane and the height of the embankment is less than the theoretical height of equal settlement that is called complete projection condition. In complete projection condition the top of conduit settles less than the critical plane that is the rigid pipe and height of the embankment is greater than the height of equal settlement and in case of complete ditch condition the top of the conduit settles more than the critical plane and the height of embankment is less than the height of equal settlement which is imaginary case. In complete ditch condition where the top of the conduit settles more than the critical plane and the height of the embankment is more than the height of the equal settlement. So we have complete projection condition and incomplete projection condition. The difference between complete projection condition and incomplete projection condition is nothing but the top of the conduit settles less than the critical plane and the height of the embankment is less than the theoretical height of equal settlement. So in the case of incomplete projection condition the top of the conduit settles less than the critical plane and the height of the embankment is greater than the height of the actual settlement. In the case of complete ditch condition the top of the conduit settles more than the critical plane and the height of the embankment is less than the height of the equal settlement. The incomplete ditch condition the top of the conduit settles more than the critical plane and the height of the embankment is more than the height of the equal settlement. So these conditions are actually shown once again here wherein the complete projection condition, the complete projection condition where height of the embankment, height of the embankment that is this is the embankment and this is the natural ground level and this is the top of the conduit is actually above the natural ground surface, this is a positive projecting conduit and where the height of the plane of equal settlement which is above the height of the embankment that is h less than h e and sf plus dc, sf plus dc is less than sm plus sg that means that here the surrounding soil actually settles more than the, you know this more than the, more than more than the soil at the foundation below right below the pipe and the deflection of the pipe. So sf plus dc is less than sm plus sg so this condition is called complete projection condition whereas so if you look into this in order to reduce this for getting this w is equal to cc into gamma bc square and for getting the different load conditions which we have discussed the forces which are actually nothing but the shear forces are acting like this, in this case it is downward, they are acting downward and these are the normal stresses acting like this and so what we have taken is that we have taken on the small element and integrated to the entire area so that we could actually get the entire load acting on the buried concert. So in this case you know the plane of equal settlement is less than the height of the embankment you can see that h greater than he so h is actually greater than he so this is the critical plane so this is the critical plane also in this case where sf plus dc is less than the sm plus sg, sf plus dc is less than the sm plus sg so this is incomplete projection condition where incomplete projection condition is here. So here initial elevation when h is equal to 0 when initial elevation when h is equal to 0 that is shown here and elevation when h is equal to he elevation when h is equal to he and after settlement and elevation at completion of the field which is actually shown here. So incomplete projection condition where incomplete projection condition means the top of the conduit settles less than the critical plane and height of the embankment is greater than the height of the equal settlement that is what actually has been shown here height of the embankment is actually more than he and the pipe actually settles less than the that is sf plus dc settles less than the sm plus sg that is again here the shear forces are acting downwards. So the both this incomplete projection condition incomplete projection condition which are actually shown the both are for the rigid pipes only. So in order to compute this cc value by considering the factors which are actually there for different possible conditions which can be possible the diagram for the coefficient of coefficient cc for positive projecting conduits is actually given here where this curve is for values of the coefficient of cc on the x axis here they range from 0 to 10 here and values of h by bc for different values of h by bc they are given here and this side is actually for k mu is equal to 0.19 and this side is for k mu is equal to 0.13 and at this along the central line that rsdp is equal to 0 at the along the central line rsdp is equal to 0 when rsdp is equal to 0 that means that cc coefficient will be equal to h by bc whatever h by bc will be there and that is equal to cc. So this condition when rsp is equal to 0 is nothing but you know h by bc so when you say that wc is equal to gamma into cc into bc square then cc is equal to h by bc so what we get is that the load on the pipe is nothing but wc is equal to gamma into h into bc. So here the negative settlement ratios are given that is minus 0.1 minus 2 minus 0.2 minus 0.5 and minus 1 so here the minus settlement that rsp ratios rsp term which is in minus is given here and positive is given here and along this line this is the complete projection condition and this is the complete condition and this is for different incomplete conditions and this is for incomplete projection conditions. So in this slide the diagram for the coefficient cc for positive projecting conduits is given wherein this is after the Spangler and Handy 1982 wherein for different values if you look into that rsp is equal to 0 it gets converted into cc is equal to h by bc with that the wc is equal to gamma into h into bc that is actually about. So this is for complete ditch condition incomplete ditch condition incomplete projection condition and complete projection conditions they can be obtained. So the settlement ratio rsd is difficult and if not impossible to determine even empirically from direct observations. As we discussed here when rsd p is equal to 0 cc is equal to h by bc with that wc is equal to gamma hbc that means the presumed load that is the weight of the soil above the top of the pipe when rsd is equal to 0 that is nothing but the plane at top of the pipe is called as the critical plane settles as the same out as the top of the pipe that means that if sm plus sg is equal to sf plus dc that means that the side soil and the sf plus sg and sf plus dc then they settle equally then it is called as rsd is equal to 0. If sm plus sg is equal to sf plus dc then settlement ratio will be 0. So the different values of design values of settlement ratios are given rsd for the rigid culvert on foundation of rock or underlying soil where the settlement ratio is taken as plus 1 and rigid culvert on foundation of ordinary soil where the settlement ratio is given as plus 0.5 to plus 0.8 and rigid culvert on foundation of material that yields with respect to the adjacent natural ground where the settlement ratio is given as 0 to 0.5 plus 0.5 and flexible culvert with poorly compacted side fields where the settlement ratio is nothing but minus 0.4 to 0 and flexible culvert with well compacted soil is nothing but minus settlement ratio is nothing but minus 0.2 to plus 0.8. So the design values of settlement ratios are given here for rigid as well as the flexible culverts. When the pipe is installed in a narrow shallow trench with the top of the pipe level with the adjacent top of the pipe level matches with the natural ground then the projection ratio p is said to be 0. The projection ratio p is said to be 0 that means that when the pipe is installed just at top of the pipe matches with the natural ground surface then the projection ratio is said to be 0. The distance from the top of the structure to natural ground surface is distance from the top of the structure to the natural ground surface is represented by BBC. So the load transfer pattern in flexible and rigid pipes is actually shown in this slide. It can be seen that the load transfer in a rigid pipe is actually internally borne by the pipe itself. In case of a flexible pipe as the load increases the vertical diameter decreases and the horizontal diameter lateral diameter increases. So it derives support actually from the side supports. So here in case of flexible pipes as the loading increases it actually the vertical diameter decreases and lateral diameter increases. So the distinguish should pattern of the load transfer mechanism is actually shown here. In case of rigid pipe which is actually shown in this left hand side of the figure. In the right hand side of the figure wherein it is actually shown as the loading increases the vertical diameter decreases and horizontal diameter increases. And here the typical failure modes of the flexible pipes is actually shown here and wherein we have the wall crushing which actually can happen with the furlings here which is actually shown here. And here the buckling of the pipe which can actually can occur because of the non-informed stresses buckling of the pipe and excessive deflection of the pipe because of the material inherent property. If the material of the pipe which is actually used is inferior then the excessive deflection can actually can come. So that actually results in too much decrease in the vertical diameter and increase in the lateral diameters and excessive internal hydrostatic pressures. For example if you are actually using some cool water pipelines under the pressures of the order of 5 bar to 6 bar and because of that if there is excessive internal hydrostatic pressures the failure planes also failure modes can also can lead to the bursting of the pipelines. So the typical four difficult failure modes of the flexible pipes are shown here and wall crushing, wall buckling and excessive deflection and excessive internal hydrostatic pressure. So for flexible pipes most are proposed with the following expression where W is equal to gamma into BD into H. The B is nothing but the breadth of the pipe the diameter of the pipe and H is the height of the embedded depth of the pipe. And the importance of the proper bedding on buried pipes basically the bedding condition is very much important and the area of the contact between the pipe and soil is so small the stress is actually more and may damage the pipe. Suppose the area of the contact between the pipe and the soil is actually small the stress concentration which actually occurs here and lead to the damage to the pipe. So the area of the contact between the pipe and the soil is large so the stress is actually less. Suppose if you are having an appropriate bedding the proper bedding for the pipe then what will actually happen is that the stress concentration will get reduced here and will not lead to on toward the failures like buckling of the pipes and leading to the non-form stresses generation the pipes and this can actually happen lead to failure in flexible pipes. So this importance of the proper bedding on buried pipes for example when we have at the joints and if the joints are actually supported like this there is a possibility that this portion actually becomes unsupported and the load is concentrated at this zones and it lead to the formation of the cracks and the proper way of doing is that it needs to be properly supported or the proper bedding need to be given so that the load is uniformly distributed and the cracks can be or damages or distresses in the pipe can be prevented. So in this case here because of the improper bedding there is a possibility that the distresses in the pipes can be cause. So after having discussed about the different loading conditions of installation of the pipes as we have discussed that the pipes or buried pipes are subjected not only due to the sulphate of the soil but also due to some external loads and internal loads. The external loads are nothing but dead load or war burden pressure and live loads live loads nothing but if that area is actually having some vehicular traffic then the vehicular traffic needs to be considered and the seismic loads particularly as we have seen that in order to reduce the load on the pipe we actually use the loose fins and if they are actually saturated then in the case of seismic perturbance there can be possibility of the pipe flotation and liquefaction which we are actually going to discuss. And this the internal loads basically the internal pressure and vacuum and pipes associated pipe and associated contents. So the loading on the pipe lines is basically the two broad categories one is called external loads and internal loads and another external one of the other external load is also loaded load due to external fluid pressure the load due to external fluid pressure. So when you have the dead load or war burden pressure this is the pressure due to the weight of the soil and the water above the pipe. So here if suppose if we are having a let us consider four different pipes which are actually placed at different levels the pressure increases with the depth of the pipe that is the dead load of the pipe P1 is actually less than P2 less than P3 and P3 less than P4. So the pressure due to the weight of the soil and weight of the weight above the pipe so this is due to the dead load or war burden pressure. And the live load this is nothing but the static or dynamic load acting on the ground above the pipe and transmitted the pipe through the soil. So vehicles in case of railroad traffic the train loading and vehicle loading in case of highway embankment loading, aircrafts in case of air feed pavements etc they are the source of such loads. So magnitude of such loads get reduced at the depth of the embankment is increased. So what we have actually discussed in this particular lecture is that we have discussed about the different projection conditions and we defined you know the Marston's load theory for flexible pipe condition and where pipe settles more than the surrounding soil. In case of rigid pipe surrounding soil settles more than the pipe and when these two issues are there then we have actually defined a parameter called settlement ratio and depending upon the projection condition we actually have said that Wc is equal to Cc into gamma into BC square and then we said that the Cc is actually function of you know the H by Vc and the product of gamma Sd and P and K and mu and for different projection conditions we actually have deduced you know we can calculate what is Cc. So with that we can actually calculate for different projection conditions which are actually categorized and classified for pipe installation. We can actually calculate the prime and most important loading due to the soil and thereafter now we will actually try to look into how the external loading like loading due to some external loading and live load how they can actually affect and how these effects can be calculated and further we look into how these pipe deflections can be calculated and then subsequent to that we will try to see you know how the issues of pipe flotation and other issues can be addressed and how they can be what are the remedial measures to safeguard and prevent pipe flotation in case of a liquefaction attenuation.