 Welcome to course entitled advanced geotechnical engineering in module 5 stability of slopes. So this is module 5 lecture 5 on stability of slopes. So in the previous lecture we have understood about the effect of rainfall on stability of slopes and with an increase in rainfall intensity we have seen that the factor of safety decreases upon increase in the pore water pressure within the slope. And having understood about the different causative factors for this slope instability is in this lecture we will look into various methods for enhancing the stability of slopes. This means slope stabilization techniques what are the various means of ensuring that a slope is can be maintained stable. So this lecture deals with the slope stabilization methods or methods for enhancing stability of slopes. So we have various methods of slope stabilization in bulk. The slope stabilization methods are generally reduced driving forces and basically increase the restricting forces or sometimes both. So the slope stabilization methods generally reduce the driving forces which are causative forces for instability and increase resisting forces. The driving forces can be reduced by excavation of the material from the unstable portion what it is called is head portion and drainage of the water to reduce the hydrostatic pressures acting on the unstable zone. So we are able to do removal of the material from the unstable zone and the draining of water to reduce the hydrostatic pressure acting on unstable zone one way this can lead to reduction in the driving forces. The resisting forces can be increased by various means primarily the drainage that increases the shear strength of the ground the draining of water reduces the hydrostatic pore water pressures and thereby increasing the shear strength of the ground. And elimination of the weak strata or the potential failure zone. So elimination of the weak strata or the potential failure zone or in a way some of the techniques what we do is that we reinforce the unstable zone with a stable zone. And the building of retaining structures or other supports by retaining the soil with an appropriate retaining solution there is a possibility that the resisting forces can be increased. Another method which is chemical treatment to increase the shear strength of the ground one of the prominent method which is used being used is by lime stabilization that is called by lime columns or some lime slurry injection which results in a sort of hardening chemical hardening. So the slope stabilization methods generally reduce driving forces increase the resisting forces are both then the resisting forces can be increased by primarily by drainage that increases the shear strength of the ground elimination of the weak strata or the potential failure zone and building of retaining structures at an appropriate location. And there are also some techniques where if in case if there is a deep seated failure then you need to adopt a technique where the forces are transferred to the deeper strata and this is called piled slope stabilization and to retain the soil at the surface level there is a method which is used for called suppressor walls which are attached to these piles. And the fourth method what we said is that the chemical treatment which actually can results in hardening and to increase the shear strength of the ground. Unloading which is one of the method to reduce the driving forces within a sliding mass. Because if there is a portion which is being subjected to slide or then unloading is one type of a method where the slope stabilization technique to reduce the driving forces. So next this thing is excavation is a common method to increase the stability of a slope by reducing the driving forces that contribute to the movement. This excavation in the sense that this includes removing the weight from the upper part of the slope that is called head of the slope and removing all unstable or potentially unstable materials. All unstable or potentially unstable materials and flattening of slopes and benching of slopes. So flattening of slopes in the sense that the slope which is actually having a steep inclination can be made flattened and this you know reduces increases the you know also the stability of a slope. And benching basically the provision of benches or berms increases the stability of a slope. So with an appropriate design there is a possibility that the stability of a slope can be enhanced by removing the weight from the upper part of the slope or removing all unstable or potentially unstable materials and flattening of slopes and benching of slopes. The flattening of the slopes basically in this particular figure a cross section of a slope is shown the slope which is the profile which is actually shown here is a one which is actually steeper than the slope section which is shown here 2 is slightly flatter. Let us assume that with slope cross section 1 this is the potential failure surface L1 and with the slope cross section 2 that means that the slope of 2 is flatter than the slope of 1 and the slip surface or potential failure surface is L2. So the flattening of the slope not only reduces the driving forces but also tends to force the failure surface deeper into the ground. So the flattening of the slope not only reduces the driving forces but also tends to force the failure surface deeper into the ground. So one of the options is that flattening of the slopes but many times if any structures which are actually present close to the surface of the slope there is a possibility that the flattening option may not be viable. The another popular method to increase the stability of a slope at the downstream particularly is the provision of rock fill buttress. The provision of rock fill buttress is nothing but a simple method basically to increase the slope stability and this is basically to increase the counter reaction at the downstream of the slope that is a special at the toe of the slope and which counter force the resisting failure. So the rock fill buttress basically consists of with stones rather than soil. If you use this buttress material or a riprap instead of soil it is preferable because the frictional resistance between the these granular or stones is high and has the resistance again good resistance against the shear forces or disturbing forces and same time the slope is actually divide of water because of the free draining capability of these buttress materials which are nothing but the stones. So rock fill buttress is basically a simple method to increase the slope stability basically this is done in hilly areas where particularly when the slopes are actually constructed or highway or railway networks are constructed by using curtain fill methods in order to increase the stability of a slope at downstream end it is advisable to provide rock fill buttresses depending upon the necessity which way to increase the slope stability and to increase the weight of the this basically is to increase the weight of the material at the toe and which creates the counter force that resist the force. But this riprap which is provided is to be in the form of stones instead of soil because it has a greater frictional resistance to the shear forces and is also has excellent free draining capabilities. Then another very recent technique which is coming up we said that the removal of material from the head of the slope but if sometime we need you know a certain elevation to be maintained and but at the head of the slope we need you know the so called the slope profile has to be maintained the one of the you know very recent techniques is to use lightweight materials traditionally sawdust and other materials are used but the sawdust actually is prone for biodegradability so in view of that it is to be it has to be protected against the biodegradability up to some extent. So in the recent past there is a technology which is called use of geofoam in slope stabilization particularly EPS is nothing but expanded polystyrene which comes in blocks of different sizes and they can be placed one above the other like which is actually shown here and the film mass volume can be achieved and then this pavement system and other aspects can be provided. So here this you should have a cover which actually prevents you know the intrusion of any other into matter into this particular zone and these geofoam is actually available in different densities basically varies from 12 kg per meter cube to about 40-50 kg per meter cube and depending upon the type of application one need to select you know a particular type of geofoam and for this type of application reasonably as there are high compressive stresses it is maybe advisable to go for high density geofoam materials. But however there are by using this material it is a very advantageous to reduce the weight at the head of the slope and but it actually has got you know a particular failure modes at the failure modes can actually happen within the soil without interacting or interfering with the blocks. So there can be external instability internal stability and pavement system failures. So the major components of an EPS block geofoam system after Arinalo et al 2009 is actually shown here. So this is a portion this is the film mass which is with EPS blocks and a soil cover you can see that the soil cover and sometimes here also they provide the soil intermediate soil covers depending upon the requirement and this is the pavement system and this is the existing slope material which is regraded and provided and the pinching is provided to you know ensure the enough resistance along the this particular surface and this is the lower slope and this is the upper slope. So one way is actually used very widely nowadays with you know particular type of materials called EPS block geofoam but as we said there are different types of failures the first and foremost is you know from the you know localized failures within the you know soil that is the downstream failure and a global failure here and the upper slope failure that is above the this thing above the this portion where you know lightweight material is provided and this particular type of failure is actually failure surface passing through the blocks as well as the downstream surface as well as here also in upstream portion. So there can be you know possible potential failures so while designing this these failure surfaces have to be ensured have to be checked so that the adequate stability can be ensured. So while using for the stability analysis the appropriate strength properties of the geofoam need to be adopted and as well as you know the unit weights can be obtained and can be used in the analysis. The other failure which is in case of seismic instability the blocks can actually you know slide all together that means that if there is an excitation force because this the resistance is very low as the weight is very low there can be you know force which actually can make the blocks to slide. So involving the horizontal sliding of the entire embankment so this is one danger in case of you know seismic stability and this is an external seismic stability failure involving war turning of an entire you know geofoam portion. So this figure which is below in the slide you know ensures the external seismic stability failure involving the war turning of an entire vertical embankment about the toe of the embankment. Then in this particular slide there is this is a type of internal stability failure where internal seismic stability failure where there is a slippage between the blocks that means that the magnified version which is actually shown here where the car soil will be subjected to disturbance and there is a slippage which actually happens between these two blocks surfaces. So these need to be ensured and there are some means of in the field means of ensuring the stability by an appropriate anchoring from one block to another block so that entire system behaves like a the monolithic unit and this is another type of internal failure where local bearing failure of the blocks will happen and this is actually can cause when because the external load or stress which is actually more than the compressive strength of the blocks then there is a possibility that these bearing failure of the blocks can occur and which can lead to an internal stability failure as far as EPS geofoam slope system is concerned. In addition to that there can be some pavement cracking failures which can cause because of this. So these are the as a with the technology there is also the different modes of failures are discussed basically to you know make aware of the need of you know understanding which is required in this area. So then the another appropriate technique for enhancing the stability of the slope what we discussed is the drainage so that the drainage of water reduces the hydrostatic pressure and so the shear strength of the ground can be enhanced. So adequate drainage of the water is the most important element of a slope stabilization scheme for both existing as well as the potential slopes prone to failure. So the adequate drainage of water is the most important element of a slope stabilization scheme for both existing as well as the slopes which are actually prone for failure. So drainage is effective because it increases the stability of the soil and reduces the weight of the solid sliding mass and drainage you know the way of you know providing this draining of water from the slope there can be two ways one is called surface drains and subsurface drains. The surface drains can be through either surface ditches along the slope which is done in a highway embankments, shallow surface subsurface drains or surface drainage is especially important at the head of the slide basically so that the water is diverted where a system of cut off ditches that cross the head well of the head wall of the slide and lateral drains to lead the runoff around the edge of the slide basically they are effective. So the drainage techniques are basically the two types one is surface drainage another one is the subsurface drainage. The surface drainage can be through the either surface ditches or shallow subsurface drains the basical surface drainage system is actually at the head of the slide that is at the top where the system of cut off ditches that cross the head wall of the slide and lateral drains lead to the runoff and the edge of the slides basically they are effective. So the drainage techniques in continuation the factor of safety against any potential surface that passes below the periodic surface can be improved by a subsurface drainage. So that means that in the subsurface drainage the methods which can be used are the drainage blankets, trenches, cut off drains and horizontal drains. So factor of safety of any potential surface slip surface or a failure surface and that passes below the periodic surface can be improved by subsurface drainage. The methods that can be used to accomplish subsurface drainage are drainage blankets and in the form of trenches and cut off drains and horizontal drains and another method is called relief wells basically the function of these relief wells is to lower the water pressure in layers that are deep down in the subsoil. So primarily these subsoil layers where the water has to be drained cannot be reached by open excavation system. So in such situation relief wells is a type of subsurface drainage technique which is adopted. The primary function of this is to lower the water pressures in layers that are deep down into the subsoil and the another technique is the drainage tunnels or galleries when there is a requirement of substantial number of horizontal drains and this is actually substituted by a drainage tunnel to drain the water. So various subsurface drainage techniques or drainage blankets and trenches, cut off drains horizontal drains and relief wells and drainage tunnels and we said that the relief wells which are you know when it cannot be reached by open excavation for a particular subsoil layer then relief wells are installed and the drainage tunnels or galleries basically they are provided when there is a requirement of substantial number of drains horizontal drains then it is substituted by a tunnel or drainage tunnel or a drainage gallery. Then as I said another prominent and popular technique of retaining the soil is slope stabilization using retaining walls, the different types of retaining walls are used at the gravity retaining walls which is a common application. The other application is to use cantilever retaining walls or reinforced concrete retaining walls or sometimes when there is you know availability abundant availability of the stones locally then nowadays the gabion retaining walls are also very popular and very recently the soil tire retaining walls which are also becoming popular wherein the used or the tiles which are the waste materials like the tiles which are actually used in a fashion to with the tire material actually forms the facing of the wall and then they are actually used with a particular technique to basically to construct soil tire retaining wall to retain the slope or to you know ensure the slope stabilization using a particular type of retaining wall. So this provision of the gabion walls is nothing but it has actually has gabion baskets and they are filled with selected amount of stones which are actually shown here and which actually ensures you know because of the excellent drainage characteristics they ensure the drainage of the water and to prevent or to loss of the fine soil particles there is a requirement of provision of a filter layer which is right behind the surface. So that this ensures this is done in the form of a nano by placing a nano in jota steel fabric right behind the you know this surface so that this will not allow the water to take away the fine soil particles. So this is as I said another type of stabilizing the slope by using the retaining walls and these retaining walls previously started with steel reinforcements nowadays we have the steel reinforcement elements are substituted by the polymer strip reinforcements and which are high strength in nature and they are used for retaining constructing these retaining walls and in addition to that they are also now the polymer materials like polyester geogrids which are actually used particularly uniaxial geogrids are used to retain the slope mass and also in this technique there is a possibility that because of its flexibility that it does not require any nominal foundation because if the soil which is available is actually having adequate bearing capacity then it does not require it requires a simple foundation about a mud bed depth of about 1 meter and it ensures also vertical phase so that the additional roadway and there is a possibility of you know at the upstream and downstream and the right of way can be used for other applications at highway transitions and although we provide these retaining walls but if the failure surface is such that it is traversing beneath the retaining wall then the provision of this retaining technique may not be useful. So here after Grinmore modern mechanic handbook 2001 a gabion retaining wall which is subjected to excessive forces can actually see here the deformed gabion wall can be seen here this particular portion and this is the failure surface which is actually passing right below the retaining wall so in such situation the slope stabilization is using right retaining walls may be not an appropriate solution. So in such situations popularly techniques called the slope stabilization by using vertical piles is an option wherein it allows one to transfer the loads to the deeper status. So this particular technique by name slope stabilization using vertical piles so in this a typical cross section which is actually shown here a slope cross section having a certain slope and this is the toe of the slope and this is the unstable soil and this is the stable soil and this is the slip surface or a failure surface. So assume that the piles the question is that and the piles are you know placed in a row a row of stabilizing piles embedded within the slope which is actually prone for failure. So that is what actually this sketch is actually shown that distance between these two that is called the spacing of these piles and D is the diameter. So this technique works in a way it actually mobilizes the resting forces and by transferring the forces to the deeper status. So in such situations what will happen is that where the retaining structures survivability is questioned and this particular technique by using vertical piles is a viable option. So here it is very important to realize and to understand is that where to locate these piles particularly whether at the toe or whether at the crest of the slope or in the mid distance and another question which is required to be understood is that what is what should be the spacing. So some of the you know studies which we have actually carried out at IIT Bombay and the studies in the recent past studies which are actually being carried out at elsewhere indicates that a phenomenon called arching plays a trivial role in ensuring the mechanism of this pile slope stability technique. So here according to Eund and Ellis 2009 so here the similar cross section is shown here and this is the you know pile loading from the unstable so the unstable mass actually has exerted a loading and this is the stabilizing pile and this is resistance in underlying stable material. So if the piles are located at a distance yes from the center to center so here what actually happens is that the material flows between the piles and if the piles are close enough then there is a possibility that the development of arching can take place. So arching is nothing but the transfer of the loads from the yielding portion to the non yielding portions. So here what actually happens is that the arching across the gap develops in a way what actually happens is that the pile become active in supporting the you know or restraining the unstable soil mass. So here and another way of arrangement of these piles is also called as staggered arrangement. In this case what will happen is that you actually have two piles which are actually placed and then there will be a pile which is actually placed at the center so like this. So when this actually has a portion the arrangement which is done like this then what you then it is called as a staggered arrangement. So then it is not called as a discrete row it is actually called as a staggered arrangement. So there are two patterns one is called discrete row of vertical piles or otherwise a staggered arrangement of the vertical piles. And so here this is again a similar typical picture which is actually shown after Ashore and Adelaide 2012 where the load which is actually applied on the pile surface pile on the pile and the soil pile resistance which is actually shown here. So the driving force induced by this special soil mass about the sliding surface is actually shown which is used in the analysis to derive these things. So the next method which is also by the reinforcement or inclusion is the slope stabilization using anchors basically the this is basically the permanent grouted anchors have been extensively used to provide vertical and lateral supported natural and engineered structures during the past 6 decades. So the slope stabilization using anchors basically here a row of anchors when they are actually placed at certain inclination and the possibility that the slope actually can have you know can have a better stability. So in this case the grouted anchors with fixed length and free length and basically provide a vertical and lateral support to the natural and engineered structures during the past 6 decades. So the end type of anchorage where the tendon is grouted below the potential surface has been used to stabilize the dangerous slopes to a specified safety factor because of the significant technical advantage resulting due to substantial cost savings and reduced reconstruction period wherein it involves simply a driving of these making a borehole and driving installing a tendon and the grouting the fixed length portion and then making the anchor active by applying a desirable force so which makes the slope so this is basically an active anchor wherein it generates the you know resistance because of the embedded anchorage with the resulting due to grouting which actually done in the fixed length portion. Sometimes if there is a bedrock or a rock outrock is there then the soil is actually anchored to the rock outrock otherwise if it is done in soil it is mandatory to do a verification test particularly a float test at the float test of the anchor basically to check the anchor capacity for which is designed and available at the site is in order or not. So in this particular slide what we have studied is that we are try to introduce ourselves to the slope stabilization using anchors. Now here in this particular slide a cross section of the slope is shown wherein we have the inclination of the anchor which is at an angle theta with the horizontal and this is the potential failure surface which is actually assumed and this is an angle with alpha with horizontal and this is the particular slice having width to B and weight W. So here S is nothing but the soil shear strength and this is the normal reaction which is actually from the soil. Now here this particular force P which is nothing but the anchor force or tension in the anchor acts in this direction. So here there is the two approaches which are actually used one is that they are using the reaction of the resolving the forces like this P along failure surface as well as taking it vertical that means that P sin theta in this direction and P cos alpha plus theta along the surface in this direction. So there is also the another school of thought to you know instead of resolving like this resolving in the form of like resolving like along the horizontal surface that is P cos alpha plus theta and this is nothing but P sin alpha plus theta which is normal to the failure surface. So it actually P sin plus alpha theta minus n that is the net force which actually acting at the base of the slice. So this we will try to see one is conventional effect is vertical effect, vertical approach other one is the normal approach. So the safety factor of the slopes stabilized with the anchors can be calculated by the following two approaches what we said is that vertical effect approach conventionally used in practice and normal approach that is what resolving with slopes with resolving P one of the components is normal to the slope surface. So thus the safety factor for the vertical effect approach is can be given by factor safety one is equal to this is with the modified Bishop's method and m alpha is nothing but cos alpha plus sin alpha into tan phi by factor safety. So here where P is the axial tension per unit width and theta is the angle to indicate the orientation of the anchors. So we also required to know what is the position of this anchor from the toe of the slope and for a given slope inclination and what is the inclination of the anchor with the horizontal that is theta which is optimum which can ensure the highest factor of safety. So thus the safety factor for the normal effect with particular resolution like has been discussed is that P sin plus alpha theta P cos plus alpha theta where safety factor for the normal effect approach can be obtained by dissolving axial tension in the anchor into two components namely normal and tangential to the normal and tangential to the base of the slice where the slip surface intersects the anchor where the slip surface intersects the anchor. So at this portion where we actually dissolve the forces as normal to the slip surface that is P sin alpha plus theta that is the component of P and this is nothing but P cos of alpha plus theta. The tangential component of the axial tension was assumed to be have no influence on the normal force at the base of the slice where the slip surface intersects with the anchor. So the factor of safety of the slope which is reinforced with the anchors can be obtained by normal approach, by considering this particular type of resolving with considering the normal component of the anchor tension we get this expression like this where summation C B plus W tan phi plus P sin alpha plus theta cos alpha tan phi by M alpha. So here you can see that both resting forces and driving forces are getting modified. So this actually likelihood of giving the higher factor of safety. So the reinforcing mechanism of anchors in slopes can be explained using additional shearing resistance induced by the axial tension on the slip surface. The additional shearing resistance was given more rationally by the normal approach than the conventional vertical approach. So in this what we would like to impress upon is that the normal approach is actually is actually provides additional shear resistance induced by the axial tension than the conventional vertical approach. And for a type which is actually considered by Kai and Ugai 2003 type of slope with a type of material what they have used with one vertical one horizontal slope for that it is found that from their analysis that you know the stabilizing effect was optimal when theta is in the range of 7.5 to 22.5 degrees and the anchor position is 2 meter horizontally from the crest of the one vertical one horizontal slope. So in stabilizing the slope with anchors what we understood is that the stabilizing effect will be optimal and it found to depend upon the sloping inclination and the anchor position which is also need to be looked upon so that the optimum angle of inclination optimum position of the anchor along the slope surface ensures a better slope stability using anchors. So here this particular slide a typical application is actually shown and this is the sliding portion and this is the resulting with the anchor which is actually in place an active anchor and where you can see that the failure surface is actually subjected to a so called compressive force which actually makes the unstable force and to feel like tied with stable ground and this is the fixed length portion this is the free length portion and this particular anchor the this is actually a sheet and this is the anchor with spacers and this is actually shown so basically this ensures the stability by providing the frictional resistance at the you know beyond the failure surface. So mobilization through the pre stressing in case of active and relative displacement in case of passive so basically here there are active anchors and passive anchors the passive anchors in the sense that the soil movements make the mobilization of the tension in case of pre stressing or what you call is active is that without any soil movements where in what we tend to do is that we apply pretension to these anchors and lock the you know the arrangement which is actually shown here this is a concrete slab arrangement on the slope as a face this is the anchor after applying adequate pretension and these are locked here so this actually ensures these are the active anchors and this is something like to have both green vegetation as well as you know a support this is something like called grid type of beams which are actually used popular nowadays in number of applications. So this type of this thing is actually after wakai go 2012 where in this type of you know applications are actually happening in the practice. The another technique which is actually used for stabilizing slopes particularly clay slopes soft clay slopes or silty clay slopes where in so basically these stone columns is a replacement technique and which is used for improving the ground enhancing the load carrying capacity but there is because of the in the stone columns what will happen is that are a granular column or a the replacement of the soil actually happens and the soil is replaced with a stone charge or a granular material the stone columns are basically having a diameter of about ranging from 400 mm to 1000 mm to 1200 mm basically here what will happen is that the soil borehole which is actually play done with a drilling and sometimes vibro floats which are actually used and then the material is actually supplied. In this way the use of the if it is done on the downstream of the road the existing road can used for the placement of the rigs and then drilling the boreholes for installing the stone columns. So basically here in this particular slide a typical slope which is actually stabilized with stone columns which is actually shown here and this is the potential failure surface and it has to be ensured that this stone column traverses at a certain depth beneath the failure surface and it is also required to understand the lateral stability of these stone columns particularly when they are subjected to shear. So it will be interesting say for example to perform certain tests where in whether this particular you know the at this interface and whether it is adequate to have you know adequate restraining forces or not but in this also the different types of patterns of arrangements one is the like what we use in soft ground improvement the square layout or staggered pattern. So in the staggered pattern or what we say that the it is also called as the in plan they look like the stone columns are placed or a position at equilateral triangular pattern wherein what will happen is that you have S is the spacing and each side of an equilateral triangle is actually having a size of S. So with that what will happen is that actually equilateral triangular pattern and square arrangement out of this for as far as the soft ground improvement is concerned the equilateral triangular pattern was found to have superior performance. So here also for this slope stabilization using stone columns the equilateral triangular pattern may provide you know adequate resistance and then provide higher average shear resistance. So this is actually designed by using the similar concepts which are actually adopted that is the here this is the load which is actually portion by stone column as well as the ground. So this ratio which is actually called as the stress concentration and here in this portion the two methods one is the average shear strength method the other one is that to estimation of the shear strength by involving this you know the stress portion by the stone column and the ground. So this is the equilateral triangular pattern arrangement which is actually shown here and here this is the square arrangement pattern S is the spacing and here also S is the spacing and D is the diameter of the diameter of the stone column these are in the plan and this is in the cross section. So the another approach which we are actually going to discuss which is actually called the average shear strength approach which is the popular and once we get the average shear strength parameters based on the type of layout which is actually adopted for improving the stability of a slope then those average shear strength parameters can be used and then conventional slope stability analysis can be performed and which actually can be used for getting the effect of the stone columns on the slope stability. So there are the softwares which actually can you know adopt this scheme and then give the factor of safety one of the examples is the Talran which actually has got a facility to incorporate stone columns and then different layouts to induce to calculate the factor of safety of the effect of the stone columns. So here in this the stability calculations are carried out by using conventional slope stability analysis once we get the average cohesion and friction. So here C average is actually given as Cc into 1-AR plus Cs into AR where Cc is equal to 1-AR and Cs is equal to 0 for stone column so Cs is equal to 0 for stone column. So tan phi average which is nothing but 1-AR into tan phi C plus SR into AR tan phi S divided by 1 plus AR into SR minus 1 when SR is nothing but 1 plus SRv minus 1 cos alpha and gamma average is nothing but 1-AR into gamma C plus 1 plus AR into gamma S. So the different notations will look into it. So the notations are nothing but C average is nothing but average cohesion to be used for the treated soil and Cc is the cohesion of the in situ soil or the ground and cohesion of the stone Cs in case of a stone charge it is 0 that is what is actually told in the previous slide. So average is nothing but average weight shear strength within the area tributary to the stone column that is within the unit cell and SR is the stress ratio which is also indicated as small n appropriate to the orientation of the failure surface at the location and SRv is nothing but the ratio of sigma C to sigma S to sigma S the stress ratio are vertical stress in the stone column divided by that in the in situ soil. Using the strong material the stone column actually a portion higher portion of the stress than the surrounding soil and AR is nothing but pi D square where D is the diameter divided by 4s square for square array and then AR is equal to pi D square by 4s square cos 30 this is for triangular pattern of arrangement. Then we have the some other variables which actually we have used tau C is equal to shear strength of the in situ soil and tau S is the shear strength of the stone column then sigma S is nothing but the effective stress due to weight of the column and applied loading where gamma Z and S mu S and the various other parameters like vertical stress in the in situ soil and alpha is the angle of inclination of the failure surface from the horizontal and mu S is nothing but SRv 1 plus SRv minus 1 into AR where phi is the internal friction of the angle of the stone. Generally these stone columns are actually having a stone charge which ranging from 10 mm to 40, 50 mm size of the particles and then the basically in that gravel or you know in that particular range. So wherein it actually has got excellent friction angle depending upon the type of the stone so it actually can have friction angle ranging from 38 to 42, 44 degrees and the internal friction angle of the in situ soil phi C in case of undrained case where saturated then phi C is equal to 0 and then phi average is the average internal friction angle of the treated soil and gamma average is nothing but the average unit weight of the treated soil that is actually composite soil which is nothing but stone column reinforced ground unit weight and gamma C is nothing but the unit weight of the in situ soil and gamma S is nothing but the unit weight of the stone. So these were the this was the method where the average shear strain parameters are estimated by using the method which is described below and this allows us to perform the stability analysis by using stone columns with that there is a possibility that we can actually ensure and then one of the other attributes of using stone columns is that because of their excellent drainage characteristics the stone columns actually has you know a possibility of having you know allowing for excellent drainage. So this is a particular case study which was done by Keller India that Keller ground engineering India private limited in 2007 in one of the ports below the birthing structure particularly below the birthing structure when the what happens is that the slopes which are actually excavated to a certain depth basically these are excavated up to minus 15 minus 16 depending upon you know the requirement of the heavy duty ships to come close to the birthing structure. So with that what will happen is that the slopes basically of clay nature and very soft in nature so with that what will happen is that there is a possibility that the slopes undergo failures there are cases of failures which are actually reported in Kandler portress wherein in the 7th birth because of the instability caused due to failure of a ground which is actually beneath the birthing structure resulted in the failure of piles of diameter equivalent to about 1 meter and led to the you know the disturbances which actually have caused to the failure of the you know this particular slopes. So one of the viable options is to you know use this replacement method replacing a stone replacing the you know the clay soft clay with this particular stone charge so this is actually successfully used by here for a stabilizing a slope below the birthing structure and in order to ensure the stability here the rock field actually is provided that makes actually you know the ensures the adequate stress concentration actually applied to the stone columns and with that so here in this they actually provided the you know these are the birthing structure piles which actually have gone and this is the you know the stone columns which actually have been drilled basically to ensure the stability of a particular slope. The another technique which what we discussed is the slope stabilization by using lime slurry so here basically if the slope is actually having adequate suitable for this type of the soil in the slope is suitable for this type of technique then there is a possibility that the slope stabilization using injected lime slurry is used to pump the lime at high pressure. So this is one of the popular technique wherein if you are actually having the existing slopes with for example with black cotton soils which are actually having low sulphates then there is a possibility that you can actually use this injected lime slurry is one of the in situ method wherein we can actually use this technique to enhance the stability of a slope. So the more about the slope stabilization using lime slurry and lime the another form of you know inducing or improving the shear strength of the soil particularly for a marine clay using lime columns slope stabilization using lime columns and lime slurry is wherein the lime is injected with high pressure. So in this particular lecture we try to understand about methods for enhancing stability of slopes we have been introduced to different types of slope stabilization techniques and we actually have seen the mechanism of reinforcement by using anchors and in case a deeper failures do occur then we said that one of the methods which can be used is the slope stabilization using piles where the forces are actually transferred to the deeper strata and the piles here are used as retaining elements and used as a retaining elements to restrain the slope movements.