 Welcome to lecture series on advanced geotechnical engineering, we are in module 3 and we are discussing about compressibility and consolidation, in particular we are dealing with methods for accelerating consolidation settlements. Then we have introduced ourselves, one of the prime methods is to preload the soil but in many situations it leads to huge amount of surcharges which are required to be placed and the timing periods which are actually required to wait are very long. In such situations one of the viable options is to provide drainage in the radial direction. So further we said that there are vertical drains which can be inserted into the ground at certain spacings and certain pattern and we said that it can be in the triangular layout or it can be in the rectangular layout or it can be in the square layout. So based on these considerations we are actually discussing about this radial consolidation. So in the process what will happen is that both radial and vertical consolidation they do occur at a simultaneously and then possibility of accelerating consolidation settlements are possible. So with that the site will be ready with by eliminating the primary consolidation settlements before the construction of a natural structure. But these increase in strength of the soil is limited to not more than certain magnitude because if we need to transfer the load to the soil then we have to resort to some other materials like some other techniques like granular piles or stone columns. So this is module 3 lecture 10 on compressibility and consolidation and we are actually discussing about the radial consolidation and theory behind this radial consolidation. Baron 1948 has given the theory on the radial consolidation by assuming that only the radial consolidation takes place and vertical consolidation is coupled after analyzing for the radial consolidation. So in this particular slide this we have shown already and in this it shows that pre-loading along with vertical drains. So what it precludes is that the vertical drains accelerate the settlements but do not reduce the final settlements but the vertical drains they contribute to accelerate the settlements. In contracts just now I mentioned about stone columns or granular piles they accelerate the settlements as well as contribute to the reduce the settlements because of the reinforcing effect induced by the highly stiff material like stone columns. So here in this particular the curve which is actually shown with broken line it shows that the settlements the so-called settlements occur in a less amount of time. So this implies that the site can be ready or the pre-consolidation settlements can be ready in a given period of time. So this helps us to accelerate the consolidation settlements and then prepare the site for the construction. So these are the different layouts of the vertical drains we have discussed, one we said that is a square layouts so wherein you see that the drains are actually placed in the center. So each drain will have an influence area and that influence area caters to the particular drain caters to the influence area. So if larger the spacing then the efficiency of the so-called vertical drains also decrease. So it started with sand drains and then have come to big drains and then in modern terms these vertical drains in the form of free fabricated vertical drains are very popular and their installation is faster and with that what will happen is that the drains can be introduced at as minimum spacing as possible is 1.5 meter to 1.5 meter. So for square layout wherein the radius the effective radius d the diameter d or d suffix e is equal to 1.13 s in case of triangular layout this comes out to be d is equal to 1.05 s. So the procedure for calculating these things were already discussed. Now let us consider a soft clay which is strengthened with vertical drains. So what you see in the blue color they are the blue color columns they are nothing but the sand filled drains or the drains which are have got certain discharge capacity. So here what will happen is that each and each and every level there is a radial consolidation or radial drainage takes place and then as well as the vertical drainage takes place depending upon the type of double layer or it has actually has got single layer. So because of that what will happen is that you have double drainage or single drainage in vertical direction or it will have radial drainage. So but what we see in the in case of the so called the sand drains a certain the broken line which is actually shown here that is the zone called smear zone and this possibly this can actually occur when the sand drains are actually installed particularly with closed pipe in soft clay. So this is actually happened in one of the projects wherein the sand drains were installed in with a closed pipe and that actually has produced the enormous amount of smearing and would make which actually has made the sand drains ineffective and in a process what actually happened is that when they subject this to the stage wise preloading and because of the inefficiency of the installed sand drains that the consolidation could not take place but from the theoretical point of view it says that the stage wise loading can be released. So because of this particular issues what will happen what is happened is that a collapse or a failure of a stage wise bund was actually observed in that particular site. So the governing differential equation for both vertical and radial consolidation is given here wherein which says that dou u by dou t is equal to CR. So C suffix R which is coefficient of consolidation radial direction it is also referred as CH. So we have discussed that the CH which is actually more than CV because KH is actually more than KV and n sigma h less than sigma v for normally consolidated soils. So with this dou u by dou t is equal to CR into dou square u by dou r square plus 1 by r dou u by dou r plus CV into dou square u by dou z square. So the first component of this equation it belongs to because of the radial drainage and the second component is conventionally this belongs to the vertical drainage. So u is the excess pore water pressure and r is the radial distance measured from the center of the drain well that means that from here center to this distance and CR is the coefficient of consolidation in the radial direction. So in order to accelerate the process of consolidation settlement for the construction of some structures the useful technique of building drains can be used. So when a surcharge is applied at the ground surface the pore water pressure in the clay will increase and there will be drainage in the vertical and horizontal direction. So when the surcharge is applied there will be increase in the pore water pressure and the drainage will actually happen in both vertical and horizontal direction and the horizontal drainage is induced by the vertical drains hence the process of dissipation of excess pore water pressure created by the loading that is hence the settlement is accelerated. So why the radial consolidation is effective because the horizontal drainage is induced by the vertical drains and the process of dissipation of excess pore water pressure created by the loading is accelerated. Now in going to the theory of the vertical drains there are two cases which are actually have been derived here one is for free strain case and another one is for equal strain case. So free strain case is basically for flexible loading like and embankments when the surcharge applied at the ground surface is of flexible nature and there will be equal distribution of surface load. That means that whatever the surface load will be there the reaction will be equal and opposite. So because of this what will happen is that no bending moment and shear forces will generate and because there is no the reaction distribution is identical to the distribution loading itself. So because of that no shear and no bending moment occurs. So because of this what will happen is that the stresses are actually same but in order to keep the stresses identical the settlements will be non-uniform. So the Baron assumes that no arching take places and that the shear strain caused by differential settlement do not redistribute the load induced stresses within the soil at any time during consolidation. So the first case is a free strain case where when the surcharge is applied at the ground surface is of flexible nature example and embankment construction. There will be equal distribution of surface load this will result in uneven settlement or differential settlement at the surface. So because it is actually shown here a typical flexible loading where the loading is shown here the reaction is shown. So as they both reaction and loading they are equal and identical and the only thing is that they are in opposite direction. So because of that what will happen is that no shear and bending moments generate. So further Baron assumes that no arching take place and that the shear stress caused by the differential settlements do not distribute the load induced stresses within the soil at any time during the consolidation. So this is a free strain case and which is actually for the flexible case. Both that of the cases if you are having a flexible condition and this condition will apply. And equal strain case when the surcharge is applied at the ground surface is rigid that means that if you are having a rigid foundation and the surface settlement will be same all over that means the settlements what you can see here this is the loading and settlements are same all over. And because the settlements are same in order to keep this the stresses will be non-uniform here. The reaction stress distribution will be non-uniform here. So this reaction stress distribution will be non-uniform here. So it presumes that the arching that it makes the arching actually redistributes the load. And the vertical strains at certain depth become equal. So these vertical stresses redistribute in a such a way because of the arching such a way that the reaction stresses they distribute in a such a way that the vertical strains at certain depth are identical. So this is a case of equal strain. Now a smear zone one more thing which we have to discuss in discussing about the radial consolidation is that smear effect. And this smear effect used to be very, very high for the sand drains conventional sand drains. Because of the reduction in the mandrel dimensions sometimes even in case of free fabricated vertical drains this so called smearing is actually possible even in case of prefabricated vertical drains. The smear zone in a sand or prefabricated vertical drain is created by the remoulding of clay during the drilling operation for building it. So this remoulding of the clay results in a decrease in the coefficient of permeability in the horizontal direction because the surrounding clay gets densified because of that what will happen there is a reduction in the coefficient of permeability. So the smear zone which is reported to have a hydraulic conductivity or permeability lower than that of undistributed soil away from the PVD installation zone. So this actually impedes the horizontal consolidation of the soft place. So because of this lower permeability because this occurs because of the you know the because the it will be a confining increase in confining stresses by driving the mandrel. So because of that what will happen is that the surrounding soil will get densified and then because of this densification what happens is that there is a reduction in the coefficient of permeability. So this reduction in the coefficient of permeability impedes the horizontal consolidation of soft place. So instead of happening in a short duration of time it may actually happen in a long long duration because the permeability in that zone is you know reduced in comparison with the permeability in the undistributed zone. So the smear zone also alters typically anisotropic initial hydraulic conductivity of place. The reduction of the rate of consolidation in the radial direction is at the smear zone is so the reduction of the rate of consolidation in the radial direction at smear zone is defined as the smear effect. So the smear effect which is expected to be dictated by number of factors such as sensitivity of the soil and installation process and the size and shape of the mandrel and basically they are not completely fully comprehended particularly the extent of the smear zone and its hydraulic conductivity. So it is not very clear because the smear effect is found to be a function of factors like sensitivity of the soil, installation process and the size and shape of the mandrel. And these are basically not completely you know comprehended particularly the extent of the smear zone what is the extent of the smear zone there are some approximations which actually have been put forward by several investigators and its hydraulic conductivity of that smear zone. So this is according to Basu and Prezi 2007 this is for you know PVDs and so in PVDs what the postulated is that it actually has got you know two zones one is you know three zones basically you can say surrounding the prefabricated vertical drain one is the disturbed zone that is away from the drain and transition zone and then you know smear zone. So there is a you know here for convenience these are actually demarcated as a rectangular way it can be seen that the smear zone is actually close to the drain which is installed PVD and then this rectangular portion is the transition zone and then this is actually is the hexagonal portion which is actually shown is the undisturbed zone. So the disturbance zone around the PVD consists of basically two zones smear zone and the transition zone and the smear zone is the completely remolded zone of soil immediately adjacent to the drain. So this is the completely remolded zone the transition zone is the zone in which there is gradual transition of soil properties with the degree of disturbance decreasing with increasing distance from the drain. So as we go away from the as we go as we go away from the this one the soil properties actually gradually they change and then the degree of disturbance is actually decreasing with increasing distance from the degree of disturbance decreases with increasing distance from the you know drain. So this is the hexagonal unit cell with rectangular distributed zone. So where in this slide depicts for the recent research where with the smear and transition zones are documented. So let us consider for in reducing the theory of the particular degree of consolidation in the radial direction for both equal strain and you know free strain consolidation. So here consider a conventional you know the theories actually were originally developed for sand drains but these theories can be extended to prefabricated vertical drains without much you know difficulties. So here this particular horizontal distance which is nothing but Re, 2 Re is nothing but De or as per our technology it can be capital D is equal to De is equal to 2 Re. So that is the total influence zone and this is the you know the zone which is actually called from the center to this one RS. So that means that when RS is equal to RW, RS is equal to RW means that no smear that means that installation actually has been taken up such a way that the no smear actually has taken place. So RS by RS is nothing but the center of from the center of the well to the you know further end of the smear are completely remolded zone and RW is nothing but the radius of the well. So this is actually the cross section at SS if you take a cross section this is how actually it actually is less this is one unit cell what we call for a typical triangular layout. So this is the plan view of a triangular layout of the sand drains. Now the theories for the free strain and equal strain consolidation are presented here with by assuming that the drainage takes place only in the vertical in the radial direction and the no dissipation of excess pore water pressures in the vertical direction takes place. So here we have decoupled in the sense that the theory of theories of free strain and equal strain consolidation are presented here with by assuming that the drainage takes place only in the radial direction and no dissipation of excess pore water pressure happens in the vertical direction. So free strain consolidation with no smear so this is first case we actually has been considered afterwards we consider with smear and for triangular spacing of sand drains the zone of influence of each drain is hexagonal in plan and this hexagon can be approximated as an equivalent circle of diameter DE where we said that DE is equal to capital D is equal to 2 RE and RE is the radius of the equivalent circle which is nothing but DE by 2, RW is equal to radius of the sand drain well and RS is the radial distance from the center of the drain well to the farthest point of the smear zone for no smear case RS is equal to RW. So these are the notations which are actually defined in the previous figure which is actually shown in the previous slide. Now the basic differential equation for the radial drainage this can be written as dou u by dou t is equal to CVR into dou square u by dou r square plus 1 by r dou u by dou r where u is the excess pore water pressure to be dissipated r is the radial distance measured from the center of the drain well and CVR the coefficient of consolidation radial direction is also equivalent to CH. So for solution the following boundary conditions are adopted one is that at time t is equal to 0 u is equal to ui that is nothing but the initial whatever the ui that initial pore water that will increase in loading delta sigma then u is equal to ui is equal to delta sigma. At time t greater than 0 once the consolidation commences u is equal to 0 at r is equal to RW that means that at the boundary of the well it is assumed that the pore water pressure is 0 because the permeability of the sand drain is many times 1 million times more than the clay. At time r is equal to rc that means that r is equal to re where farthest end of the influence zone the dou u by dou r is equal to 0 that means that no consolidation no hydraulic gradient happens here that is at this point when r by when at this point when you know here dou u by dou r is equal to 0 is assumed here that means that there is no consolidation happens there. So by adopting these boundary conditions the solution for the excess pore water pressure at any time t and the radial distance can be radial distance r is given by where it works out to be u with a summation alpha 1 to alpha 2 so on to alpha to infinity minus 2 e1 alpha u0 alpha r by RW divided by alpha within square bracket n square u0 square alpha n minus u1 square alpha exponential of minus 4 alpha square n square t r. t r is the time factor for the radial consolidation for the radial flow which is given by tcv r or tcr by de square so de square is nothing but now here we need to understand de square is nothing but the effective diameter. So n is nothing but re by rw or de by 2rw and then you know some vessels functions actually have been used and then the solution actually has been further simplified for the free strain consolidation with no no smear and the cvr is equal to that cr is equal to kh by mv gamma w. So by knowing kh by knowing mv and by knowing gamma w we can actually calculate what is cvr the r is equal to cr. Now the average pore water pressure u average throughout the soil mass may now be obtained as u average is equal to ui summation alpha 1 to alpha 2 to alpha infinity 4 u1 square alpha divided by alpha square into n square minus 1 and within square brackets n square u0 square alpha n minus u1 square alpha and square brackets close and into exponential of you know minus 4 alpha square n square t r. So the average degree of consolidation can be obtained in the radial direction as u r is equal to 1 minus u average by ui. So that u r the average degree of consolidation u r in the radial direction is given by 1 minus u average by ui. So with that you will get the u suffix r. So this chart actually which gives u r versus t r where in for different values of n that is n is equal to 5, 10, 15, 20, 40 like this for PVDs particularly because you know the PVDs they have the diameter of the well because the PVDs come with some dimensions finite dimensions like A dimension and B dimension are having certain breadth and thickness mostly the PVDs they come from the breadth ranging from 95 mm to 100 mm and thickness is ranging from 3 mm to 5 mm. So in the process what will happen these are actually approximated as the diameter of the well and because of that what will happen you actually tend to get the higher n values. So the variation of u r with time factor t r which is actually given here for different values of n where n is equal to r e by r w r e is that you know the effective radius and r w is the diameter of the well or drain. So with that we can actually calculate by knowing for that certain time we can actually calculate what is the u r the degree of consolidation. Now the equal strain consolidation with no sphere this is actually case for you know a rigid case what we assume and the excess pore water pressure at any time t and radial distance r is given by u is equal to 4 u suffix average d square function of n within square brackets r e square natural logarithm of r by r w minus r square minus r w square by 2 the square brackets close. So where f n function of n is nothing but n square by n square minus 1 natural logarithm of n minus 3 n square minus 1 by 4 n square. So we can see that by substituting function n and then u average can be estimated like this which is nothing but average degree of value of pore water pressure throughout the clay layer which is nothing but u i e to the power of lambda where lambda is equal to minus 8 t r by f n where t r is equal to t c r by d square. So the average degree of consolidation due to radial drainage is given by u suffix r is equal to 1 minus exponential of you know minus 8 t r by f n. So for r e by r w greater than 5 it has been you know note it can be noticed that it can be it is it has been reported and it has been obtained that the both free strain and equal strain consolidation give approximately same results for the average degree of consolidation. So for r e by r w that is greater than 5 it has been noticed that the free strain and equal strain consolidation the yield you know identical results for average degree of consolidation. Hence you know for the design purposes with especially for PVDs and this condition is very much satisfied. So because of that what will happen is that these equations are actually used in calculating the radial average degree of consolidation in the radial direction and based on you know for a given dimensions given layout whatever is actually has been obtained. So with that we can actually calculate what is the average degree of consolidation after having obtained the average degree of consolidation then you know we can actually calculate club with vertical consolidation and then calculate the u v r where which is actually for both vertical and radial direction. Now let us the previous deliberations whatever we had and the we have discussed that that is for the you know without smear where r s is equal to r w. Now assume that smear zone is actually is there because it is getting is possible because of the what we discussed about the sensitivity of the soil and then type of mandrel what we use and so because of that you know the smear zone or smear effect also have to be incorporated because we have discussed that the smear effect actually impits the degree of consolidation. So Baron 1948 also extended the analysis of equal strain consolidation by sand drains to account for the smear zone. So the analysis is based on the assumption that the clay in the smear zone will have an boundary one boundary with 0 excess pore water pressure and other boundary with an excess pore water pressure that will be time dependent. So here in this particular you know analysis the Baron 1948 assumed that the clay in the smear zone will have one boundary with 0 excess pore water pressure and other boundary with an excess pore water pressure that will be time dependent. So using the above assumption we can actually obtain u is equal to 1 by m dash u average square brackets natural logarithm of r by r c minus r square minus r s square by 2 r e square plus k h by k s into n square minus s square by n square and natural logarithm of s where s is the you know that smear that is you know s is nothing but the that which is nothing but s is equal to r s by r w when s is equal to 1 and this actually reduce to you know without smear. Now where you know k s is the coefficient of consolidation the smear zone and where s is equal to r s by r w r s is nothing but the radial distance from the in the center of the well to the farthest zone of the smear zone and m dash is given by n square by n square minus s square natural logarithm of n by s minus 3 by 4 plus s square by n 4 n square plus k h by k s into n square minus s square by n square into natural logarithm of s. For s is equal to note that s is equal to 1 that is the no smear and the average degree of consolidation is given by u r is equal to 1 minus u average by u i with that what we can get is that 1 minus exponential of minus 8 t r by m dash. So instead of function n now it is actually nothing but m dash where m dash is nothing but n square by n square minus s square natural logarithm of n by s minus 3 by 4 plus s square by 4 n square plus k h by k s into n square minus s square by n square into natural logarithm of s and k s is nothing but the permeability of the soil in the smear zone. In reality we know that the drainage actually happens both in the excess both in the vertical direction as well as in the radial drainage. So carry low 1942 he has proposed the following expression which actually nothing but u is equal to 1 minus 1 minus u v into 1 minus u r. So by knowing after having determined u r by equal strain consolidation with or without smear we can actually calculate u v this is by the conventional method whatever we have discussed and then we can club and get the u which is nothing but u is equal to u suffix v, r where this is actually the average degree of consolidation for both vertical and radial drainage. Now here the effect of the smear zone on the radial consolidation is actually given here and this chart is for left side chart is for n and this is for m dash so you can be seeing that for different values of s is equal to 1, s is equal to 1.2, 1.5. So this s is equal to 1 means that that is no smear then with that what you can see that how the m dash values change and this is actually only for k h by k s is equal to 20. So in this case you know for two values of n that is n is equal to 5, n is equal to 15 and for this is for s is equal to 1 with n is equal to 5 and this is s is equal to you know n is equal to 5 and s is equal to 1.2 and 1.5. So these actually charts actually show the effect of the smear on the radial consolidation and the different values of ratios of k h by k s are actually assumed and with that you know it is assumed that how the m dash actually varies with the values parameters which are varied. Now the so before you know discussing about the characteristics of the free fabricated vertical drains it is required to calculate or understand about the design required design requirements because with the preloading with the sand drains or PV drains. So here in this particular diagram which is actually shown here so what we need to calculate is that if sigma f is the you know permanent surcharge which is actually you know which is going to be there and we need to calculate you know what is that sigma s that is nothing but you know the additional surcharge which is required and what is the time period so many times so what will happen is that the time period is actually given like 6 months or 9 months. So in this 9 months you know for the if you wanted to complete say let us say certain degree of consolidation say 90% of degree of consolidation then we need to you know preclude calculate what is the level which is actually required and what spacing which is required to provide and what pattern which is required to provide. So we have to start with certain assumption and based on that we have to do if that assumption is actually proving to be over conservative it can be come to we can actually do the optimization so with that we can actually calculate what is the time which is required to you know to accelerate the consolidation. So the design basically it includes to determine the surcharge intensity sigma s that needs to be applied at the ground surface and the length of the time that it has to be maintained that is length of the time or the duration of the time which is required to be maintained. So for doing that you know what we need to do is that we actually have to calculate average degree of consolidation both in vertical and radial direction and we also have to calculate what is the radial degree of consolidation and vertical degree of consolidation then we have to combine by using curricular method and calculate. So here you know we need to calculate the average degree of consolidation both in vertical and radial direction in case of preloading only we have said that it is the degree of consolidation at the bit plane but in this case it need not be like that it can be calculated in the procedure which we are going to discuss now. So first what we do is that we assume that only radial drainage takes place because of the provision of the you know sand drains or PV drains so in this case by for the assumed layout let us say with certain spacing and certain pattern let us say we have got a square grid means DE is equal to 1.13s where s here is the spacing center to center spacing of the drains and DE is equal to 1.05s which is for the triangular layout. So by determining function n, n square by n square minus 1 logarithmic of n minus 3n square minus 1 by 4n square and with that what we can actually get is that TR is the time factor for the radial flow can be obtained as for a given time TCVR by DE square the CVR is nothing but the radial quotient of consolidation in the radial direction and by DE square. Many times what will happen is that in order to counter for the smear it is assumed that the CH is equal to CV and once the CH is actually is more than CV but in order to counter and in order to consider the possible effects due to smear in a simple way by not considering smear and then it can be taken that CVR is equal to CH is equal to CV with that we can actually calculate what is TR, TR is nothing but the time factor for the radial flow where TCVR divided by DE square and after having obtaining TR and Fn we can actually calculate degree of consolidation the radial drainage UR is equal to 1 minus exponential of minus 8 TR by Fn where n is equal to DE by 2RW which is nothing but DE by W and DE is equal to 1.13S for square grid and 1.05S for the triangular grid. So after having obtained that before you know for as far as the PVDs is concerned you know because the PVDs are actually having dimensions which are actually nothing but the diameter of the well which is approximated based on the drain geometry and the configuration. So after Hans von 1979 the diameter of the well is equalizer like this so DW is equal to 2 into A plus B by 5. So this actually has been obtained nothing but you know by equating pi DW that is nothing but the perimeter of the diameter of the well is equal to 2 into A plus B where A is nothing but the breadth of the PVD and B is nothing but the thickness. So A is nothing but the width of the band shape drain cross section and B is nothing but the thickness of a band shape drain cross section. So by equating you know 2 into A plus B is equal to pi DW we get DW is equal to 2 into A plus B by pi. There are also some other considerations where in you know based on the Farrington analysis many investigators actually have come forward and put forwarded different type of you know different equivalent diameter of the wells but however this proved to be universal in actually adopting in the design. So the above equation was found to be generally valid when the portion of the perimeter area of the band shape drain not obstructed by the drain core exceeds approximately 10 to 20 percent of the perimeter total perimeter. So for most of the PV drains this condition is easily met for most of the PV drains this condition will be easily met. So after having obtained the radial drainage what we need to do is that the average degree of consolidation due to vertical direction only for this you know what we need to for time T2 determine you know TV which is nothing but T2 into Cv by hdr square. So here being vertical drainage by knowing the thickness of the clay we can actually calculate you know what is the time factor TV and for time factor TV after having obtained time factor calculate the average degree of consolidation vertical direction by using either by using pi by 4 into uv by 100 whole square for u less than 60 percent and or for u greater than 60 percent TV is equal to 1.781 minus 0.933 logarithmic of 100 minus u percent for u greater than 60 percent. So after having obtained uv then using the Kerylov 1942 expression we can actually calculate uvr is equal to u is equal to 1 minus into 1 minus uv into 1 minus ur so with that we will be able to get the average degree of consolidation both in vertical and radial drainage. So after having obtained the vertical and radial drainage this thing so once we get the uvr and by knowing you know sigma f which is actually permanent load and sigma not dash is the effective stress at the mid depth of the clay layer we can actually calculate what is sigma s by sigma f. So here you know we need that previously you have said that u is equal to uf plus s is equal to here we assume now this u is nothing but uvr. So this with this actually what will happen is that the presence of drains because they accelerate the radial consolidation so you will have the you know the sigma s by sigma f values are very low values because the degree of consolidation will be high and the values are low. So this actually tells us that you know the pre-loading which is actually required with you know with the drains which actually is very low magnitudes and so that you know the stability issues and all other issues will not be there. So we actually have discussed while discussing pre-loading you know we have an example problem we discussed and we said that you know when we wanted to have accelerate the consolidation by using pre-loading only we said that there is a you know this fill which is actually required is about 207 kilo Pascal s and which actually indicates that you know require very high embankment for allowing the consolidation to take place. And then we actually have stopped saying that for adopting this solution and there can be some stability issues which actually can hamper the you know the process. Now let us see the same example you know by with the provision of the drains so that we will understand the efficacy of the drains in accelerating the consolidation and also bringing down the required huge heights of the pre-loading. So this is in continuation of the design problem solved with pre-loading only and at that time we said that the site to be ready in time 9 months we also assume that the same 9 months and assume that Rw is equal to 0.1 meters that means that 0.2 meter is the diameter of the drain and effective diameter is given as 3 meters and the CV is equal to CH is equal to CVR is given as 0.36 meter square per year and no spear is actually assume and from the given data the average degree of consolidation in the vertical direction UV 67% for TV is equal to 0.36. So based on that we can calculate first N and N is equal to DE by 2RW which is nothing but 3 divided by 2 into 0.1 so with that what we get is that 15. Now using the following what we get is that for N is equal to 15 you calculate what is function N. So function N can be calculated by N square by N square minus 1 natural logarithm of N minus 3N square minus 1 by 4N square after having obtained function N. Now calculate what is TR and TR which is TCV, T is nothing but 9 months when the CVR is you know given as 0.36 meter square per year DE square is given as 3 meters with that what we get is that TR after having obtained TR and function N you calculate UR. So UR what we get is that it is about 77% now. Now after having obtained you know this UV 67% and UR 77% by using the degree of consolidation both vertical and radial direction is given by U is equal to 92.4%. So U is equal to UVR is equal to 92.4% and from the given problem we know that delta sigma P by sigma 0 or sigma f by sigma 0 that is from the previous chart here sigma f by sigma 0 wherein it actually shows that it is actually 115 by 210 that is 0.548 and U 92.4. So we can calculate and see from the curve here 0.5 that is somewhere here and 92.4 we can see that the sigma s by sigma f which is actually required sigma s is the height of the preload which is required and sigma f is the permanent field height and which is coming to be around you know so this is 0.2 so this is 0.12. So what we get is that this comes to be 0.12 that means that you know what we require is about 14 kilo Pascal's. So it is about height of you know less than a meter if you provide it is actually sufficient for the given soil properties but actually has been considered. So very nominal preloading is required with the sand drains in contrast for the example whatever we have taken it actually evolves as 207 kilo Pascal's which is you know beyond you know the difficulties but you know as have been discussed actually that the considering the soil properties there are also some cases where people combine vacuum consolidation as well as preloading with that what will happen is then up to vacuum with the help of vacuum consolidation what we get is that you know the preloading height will further reduce and then there is a possibility that you know the consolidation can actually happen in a relatively shorter duration and here with that what will happen is that you will have both vacuum consolidation as well as the preloading effects. So let us take a problem where the oil tank is to be sited on a soft level deposit of clay and below the soft clay is a thick layer of stiff clay so that means that it is actually has got only one way drainage it was decided that the circular embankment with sand drains inserted into the clay would be constructed to pre consolidate the soil. The height of the embankment is 6 meters and the saturated unit weight of the soil comprising the embankment is 18 kilo Newton per meter cube and the following data are available the thickness of the clay is 7 meters and the coefficient of volume compressibility is 0.2 meter square by for by 0.2 meter square per Mn mega Newtons and the CV is 3.5 meter square per year and the CH is equal to 6.2 meter square per year and DW is the diameter of the sand drain is equal to 0.3 meter. So the desired degree of consolidation is 90% in 6 months so it has been given here the desired degree of consolidation both in vertical and radial direction is actually given as 90% in 6 months and determine the spacing of square grid of the sand drains such that when the tank is constructed the maximum primary consolidation should not exceed to DM. So here the condition is that we calculate the spacing for a square grid of sand drains or the so called vertical drains such that when the tank is constructed the maximum primary consolidation settlement should not exceed to DM. So now the square pattern has been given and the spacing need to be assumed so by we can actually calculate DE and by assuming certain spacing and then calculate Fn and then calculate TR and then calculate for different spacings you calculate and then based on once we get this one then we can actually see which layout is actually the optimum layout which actually gives the desired consolidation and the desired meets the desired conditions. So this solution is required to be done at your end. Now as has been introduced one of the alternatives to substitute vacuum preloading is vacuum surcharging or vacuum consolidation and this is actually gaining popularity in the recent past because of the requirement of the reduction the carbon credits and so equivalent to 4.5 meter fill basically is effectively can be obtained by putting this vacuum consolidation. So here the typical you know the preloading pore water pressure changes and effective surcharge changes due to preloading are actually discussed and here you can see that this is in case of a vacuum preloading so we can see that this is the hydrostatic power water pressure before any application of you know the preloading due to vacuum and so what will happen is the moment the vacuum consolidation actually takes place the suction which is get reduced to minus us here and minus us here so with that what will happen is that from that point onwards you know the soil tries to come to the radial consolidation so in the process what will happen is that the effective stress increases by this amount at any depth sigma v z sigma v dash z is equal to sigma naught plus u naught z minus us z so with that the effective stress increases and this is the portion at to be you know dissipated. So the u naught z is the hydrostatic pore water pressure profile and ut z is the excess pore water pressure at any time t and us z is the suction line. So this actually has got a possibility that you know it induces the suction pressure and then this suction once it actually gets you know transferred to the entire soil and the effective stress increases by the amount by which the suction has been induced that means that if you are able to induce about 80 kilo Pascal's of suction the effective stress increases by increase subsequently or a period of that time by about you know 80 kilo Pascal's. Now this particular slide actually shows the way typically a vacuum consolidation is done at the site so in this what actually has been done is that either with PVD drain prefabricated vertical drains it will be much more effective and faster and if you are actually having this drain which is actually connected with you know PVD's and what we need to do is that it actually need to connected to a pump where it actually the suction is actually maintained. So where the suction should not be subjected to any leak here so this is the hydrostatic pore water pressure and with any due to any possibility of some pre-roading which actually happened in the process and this will be the you know the you know jet and then this get reduced to US jet and over a period of time and then you know the effective stress increase by that amount. So we can actually calculate what is the average degree of consolidation here. Now another advantage from the stress path diagram we can actually look into it here when you take pre-loading or you know vacuum we can see that from P dash and Q dash plots where P dash is equal to sigma 1 dash plus sigma 3 dash by 2 and Q dash is equal to sigma 1 dash minus sigma 3 dash by 2. So it can be seen here that in case of a pre-loading there is always a possibility that AB is indicating AB you know the line actually traverses towards the failure line you know AP indicates the possibility of the slope failure and consolidation follow. So once you know if it touches here then there is a possibility that is that you know as we discussed in the previous problem if the heights are actually very high and when you do the pre-loading on soft clays there is a possibility of the you know instabilities which actually can occur and the base failures can actually come. So when the failure doesn't cause and then in the case the BC line indicates that you know subsequent surcharging consolidation but whereas in case of you know in case of so called vacuum consolidation that AD that is actually you know is the process of vacuum consolidation and where in the case of vacuum line the stress path simply follows AE line lies below the K0 line. So AE line is the you know line below the K0 line that is the K0 line is the at rest line. So as a result of isotropic consolidation because the pressure because of the vacuum consolidation is identical in both the directions increase in increase of active stress there is no risk of slope failure with the so the important you know understanding what we had from this slide is that the slope failure will actually not be arising with the vacuum consolidation but whereas in case of pre-loading there is a possibility of the you know slope failure because before you know commencement of the consolidation there can be a possibility of the failure. And in this particular slide after Chu and Yaon 2005 a case study which actually has been monitored so it can be seen that a major difference between the pre-loading and vacuum pre-loading is that pore water pressure change under the fill change and the fill surcharge the excess pore water pressure will first build up from initial state by the same amount as the surcharge and then dissipate gradually under the vacuum pressure the pore water pressure in the soil will reduce from the initially hydrostatic by the same amount as the applied vacuum pressure. So here you can see that you know when the pre-consolidation is actually applied from here it actually reduce it to this certain level and then over a period of 30 60 70 days actually it increases to that one that means that the effective stress as the you know soil actually transfers the effective stress to the water gets transferred to this soil transfer to the soil there is an increase in the effective stress. So with that what will happen is that under the vacuum pressure the pore water pressure in the soil will reduce by initially hydrostatic state by the same amount as the applied vacuum pressure. So in this particular pressure in the lecture what we discussed is that you know the theories behind you know using equal strain consolidation and you know for the radial equal strain and free strain consolidation and we said that for n greater than 5 we said that both equal strain and radial consolidation yield identical results hence especially for PVDs the equal strain consolidation is actually adopted widely. So then we also have discussed that design how it can be done with by incorporating particular prefabricated vertical drains so with this we will actually try to design you know a particular time which is actually required you know to accelerate the consolidation for a given layout of you know vertical drains is used.