 In the last few classes, I have discussed about that, what will be the expression of shear force, settlement bending moment, deflection for infinite beam, then semi-infinite beam. Now, today I will discuss initial part that about the finite beam, what would be the expression, how to solve those expressions. Then, we will discuss some field application of this structure interaction or solid foundation interaction theories that I have discussed. Then, first we will go for that, how that beam on semi-infinite beam, so if beam it is a finite length. So, in the finite semi-infinite beam that I have discussed that, suppose this is the starting point and this end, so beam in extended in one direction only. So, there the solution technique was that, so suppose we are applying any UDL or any concentrated load there and because of this concentrated load and UDL, there would be a moment and shear force will generate at point A. Now, to make this point, so if it is a free end, then we consider one infinite beam and then we apply a moment and a shear force M 0 and P 0, such that it will produce a minus M G M A and minus Q A shear force and minus M A bending moment at this point. Because at this A point, because of this UDL and P there will be a M A moment and Q A shear force will develop. Now, to make this point as a infinite beam, then you can apply this moment P 0, I mean moment M 0 and load P 0, such that it will produce a moment minus M A and minus Q A shear force, so that the net moment and net shear force of this infinite beam at this point A is 0, such that we can make it is a equivalent to a semi infinite beam and then we express the, then we can derive the expression, those things I have explained in the last class, we derived those expression. Then from this expression, we can calculate what would be the amount of the P 0 and N 0 we have to apply, so that the net moment will be and net shear force will be at point P will be 0. So, then we will get, we can then convert this infinite beam to a semi infinite beam with the net moment and shear force 0 at this point P and because it is a free end beam. So, that was for the semi infinite beam. Now, if it is an infinite beam, so suppose we have a infinite beam and then we have to apply basically two forces and two moment, one is at, so then for the, this is for the semi infinite beam, next for the infinite beam. So, we will consider this infinite beam, then we will apply, then because of this loading and the shear force say P and U D L, Q J will be a moment M A and Q A will develop. So, M at this point A and this is B, these are the two points of this infinite, finite beams and then now we convert this infinite beam and then the moment M A and Q A, similarly moment M B and shear force Q A B will develop at point A and B respectively. So, at point A shear force Q A, M moment M A. Now, we convert it to and then to avoid these things, what we have to do for the, this finite beam. So, we have to apply a moment M 0 at A and force Q A B. P 0 at A, similarly force P 0 at B and moment M 0 at B, such that the, it will produce, so this will produce minus M B and minus Q B and this will produce minus M A and minus Q A. So, in this, so that means this as we know that P 0, M 0 and P 0 B, M 0 B or P 0 A, M 0 A, these are the in conditioning force. So, that must produce these two moment and two shear force. So, such that ultimately this can be converted into a finite beam of length L with same force Q is applied and P applied at any point on this beam. Now, we have to make sure that once you do this exercise, like the semi-infinite beam, then we can derive this expression for the finite beam also. So, this expression is M E plus similar to the semi-infinite beam that I have already derived, so similar to semi-infinite beam. So, for the finite beam also, we can derive this expression that is M E, similar way P 0 A by 4 lambda plus P 0 B by 4 lambda C lambda L. It is the length of the beam plus M 0 by M 0 A by 2 plus M 0 B by 2 D lambda L that is equal to 0. So, this is the expression, one expression. Then the next expression that we will get, this is one expression, another expression Q A minus P 0 A divided by 2 plus P 0 B divided by 2 D lambda L minus lambda L 0 A divided by 2 plus lambda M 0 B divided by 2. D divided by 2 A lambda L that is also equal to 0. Now, for the third expression in terms of M B also M B plus P 0 A divided by 4 lambda C lambda L plus P 0 B divided by 4 lambda plus M 0 A divided by 2 D lambda L plus M 0 B divided by 2 equal to 0. And the fourth expression that will be Q B minus P 0 A divided by 2 D lambda L plus P 0 B divided by 2 minus P lambda M 0 A divided by 2 A lambda L plus lambda M 0 B divided by 2 that is equal to 0. So, we have this four expression. So, from here we have to, from these four expressions we have to determine four unknowns that is P A 0 M A 0 P 0 B M 0 P. So, P 0 A M 0 A P 0 B M 0 B. So, these four unknowns we have to determine such that this amount of load or moment we have to apply these two ends. So, that we will get a finite length condition. So, now how to solve this four expressions? One method that now to solve these expressions we have to use two boundaries conditions at least for example, that if it is a hinge beam say. So, hinge beam with finite length. So, one boundary condition will be deflection at both end is 0, another moment at both end is also 0. So, we will get, so similar to other load N condition also we will get boundary condition. So, these boundary conditions we have to apply here. So, that is y 0 equal y equal to 0 M equal to 0. Another thing is that apart from these things we have to suppose we have two loading condition that we have this type of loading condition this is a hinge beam both end. So, one boundary condition is y equal to 0, another boundary condition is moment at both end also 0. So, in this condition if we apply P then we can convert this thing into two part one is symmetric loading. So, suppose this is P by 2 similarly from the same distance from this side that is also P by 2. So, this is at a distance some a. So, this will be also a and this will be also a. So, that is symmetric then plus anti-symmetric V wave also that is here P by 2 and here it is direction is opposite P by 2. So, this is the distance is same. So, ultimately so this is symmetric condition we can get the expression this is symmetric condition P by 2 P by 2 both acting downwards with the same distance from this edge or end condition a and b. Another one we will get the anti-symmetric boundary condition anti-symmetric loading condition this P by 2 acting downward direction P by 2 minus acting on upward direction. So, net force is 0 at this point and this is P which is the original force. So, we have to apply this loading condition then we can and as well as we have to apply this boundary condition and then we derive this expression so this portion also these from this symmetric condition also we will get moment and the shear force expression and this anti-symmetric condition also will get moment and shear force expression then the net expression will get if we add this two things. So, from the net expression we will get the moment and the shear force and that amount of shear force and moment we have to apply in this two ends. So, that we will get this end condition this this finite beam condition we can achieve and as well as we have to apply this boundary condition here this is the boundary condition depending upon the end condition also you have to change the boundary condition. So, these are the different this is the solution techniques or the expression which are available for the finite beam or beam with finite length. Now, then the next section that now we know that what are the Pasternak Winkler model and what is the Pasternak model and what is the limitation of Winkler model what are the limitations and how to overcome those limitations. So, those things we have discussed then we know that how to expression of the beam resting on elastic foundation it is a infinite beam finite beam or semi infinite beam and what is the general expression of the beam those things we have already discussed. Now, how will apply those things to solve a field problem. So, now the next section that I will discuss that we will apply these all theories and then we will solve a critical field problem and then we will determine the expression and the solving technique of those problem and that is the idea of this all soil structure interaction expressions or the models. Now, the expression that I will discuss a problem that problem is a field problem which is very important in geotechnical engineering application. So, that problem is a problem where suppose we will solve the problem that is embankment resting on stone column improved soft soil. So, now we will solve we will derive the expression for this type of problem suppose if embankment which is resting on the stone column improved soft soil. Now, this problem is important so that it is very important for the highway construction it is very important for the railway construction also because suppose if we have a embankment which have that this is the embankment. So, which is resting so this is the existing soft soil say this is existing soft soil. Now, on this existing soft soil we have to construct a embankment so that here we can construct the road we can construct the railway. So, it is very important for the railway and roadway or the traffic engineering. So, now suppose if this is the embankment then first we will place one granular layer on this embankment say suppose this is the granular layer. So, that the idea of this granular layer placing is so that we can make this ground workable. So, that the machinery which is which will use to construct the embankment which can easily stand on this soil because this soil is very soft. So, now this thickness of the of this layer it can be 0.3 meter or more generally up to 1 meter and so. So, before we place this so now this soil you have to improve. So, now this soil can be improved by the application of the so this is say bedrock. So, this soil can be improved by the application of the stone columns. So, we can install stone columns here. So, these are the stone columns which are installed here. So, this is granular layer granular field, this is embankment, this is stone column. So, once this ground is improved then you can construct the embankment over this ground. So, that means first this soil are improved soil this soil is improved by using stone column. We can use the granular field also and then above this granular field layer then we can construct this embankment. Now what is the stone column and what is the purpose of this stone columns? The stone columns are the columns because here these columns are inserted the stones are inserted into the columns into the soil. So, that means for we have to remove one method is we can. So, this we can remove or replace this soil and then we insert stone into that replace ground another we can make we can insert this stone stones within the ground that that is called that this vibro replacement. So, that means here we can remove this soil or replace this soil by either using the dry method or wet method. So, these are methodologies are available to construct this stone column. So, that thing we are not going in detail we are just giving the idea what is stone column? Basically this is a column which has a specific diameter. So, this is the diameter of the columns and then based on this desired diameter we insert the stone into the ground. So, that is the stone of basically these stones are gravels. So, generally these are gravels which are inserted into the ground and these two columns are placed into the ground by square pattern square pattern or we can place it into triangular pattern also. This is triangular pattern. So, this is the diameter of the stone and this will be the spacing of the column. So, here also this is the spacing of the column and this is the diameter of the column. So, D is the diameter is is equal to spacing. Now, so that means first these columns are placed into square pattern this is square pattern or triangular pattern sometime it can be used by hexagonal pattern also. This triangular and square most common pattern of the installation. So, we will install this stone into the ground. Now, what is the purpose of this stone columns into the ground? So, first as we know these are the gravels and these are the very soft soil. So, the stiffness of this soft soil or the strength of the soft soil is less very less compared to the strength of the stone column. So, it is it will act a composite ground. So, what will happen that this soil this stone and this composite the stiffness of the overall ground that will improve. So, once the the stiffness of the overall ground improve then the strength of the ground will also improve. So, that means this way it will increase the bearing capacity and reduce the settlement of the foundation. So, that means this embankment bearing capacity it will increase it will increase the settlement of this embankment and sometimes also the stability problem. If the embankment is not stable then this if the this slope line is passing through this soft soil zone then also if we install the stone column the stability will also be increased due to the installation of the stone column because here we will get the more strength. So, now that is the three purpose that explain one is the improving the bearing capacity, one is the reduced reduction of the settlement and one is the stability problem increase the stability of the structure or the embankment here. And the third one that is also very a fourth one that is also very important that we know that soft soil it has very low permeability. So, consolidation it will take huge time to consolidate the total ground. So, if we use the stone column so that basically these columns are stones and made up with stones and the void between the stones are very high. So, what will happen the water will try to enter into this column or this path and then it will go upward direction and then it will we can collect this water from here and we can remove this water. So, that the consolidation that is called the radial consolidation or so that means there will be the radial consolidation as well as there will be the vertical consolidation, but radial consolidation is much faster than the vertical consolidation. So, overall consolidation is mainly because of this radial consolidation or this stone column. So, that means it will increase the consolidation rate. So, it will first in the consolidation of the soil. So, soil will get consolidated very early stage of the construction. So, that the future problem if there is a consolidation process will going on for few years and future will get some problems because of the settlement. So, the most of the settlement if we get within a very short period of time so that in future we can avoid that type of problem. And once the consolidation is over then we will get the more strength into the soil. So, that is the idea. So, basically for purpose this consolidation stone columns are serving one is increase the bearing capacity, reducing the settlement, one is the increasing the stability, another increasing the consolidation. So, that means here we will, but here for this methodology that we will discuss we will not consider on the stability purpose that we assume that column is stable. So, that means stability there is no problem only other three things that we will discuss here. So, that means one problem is here solving is the increase the bearing capacity then increase the consolidation then make stability another check another is improve the settlement. So, that means the stone column increases bearing capacity, increases consolidation rate, improve the stability and improve the settlement. So, from these things we will discuss the bearing capacity improvement, consolidation improvement and settlement improvement. So, these three things we will discuss and how to model these things incorporate all these behavior here. So, now the diameter typical diameter of the stone column is varies from 0.6 meter to 1 meter and spacing generally varies from 1.5 meter to 3 to 3.5 meter. So, this is the so the s by d ratio s by d ratio that varies from 2 to 6. So, these are the typical diameter of general idea how this consider what is the diameter of the stone column what is the spacing between the stone column s by d ratio is generally 2 to 6. So, now another very important thing will come into picture that when we will construct this type of embankment on soft soil where we can use the stone columns here then what will happen that soil above this. So, that means soil this portion is very soft and this portion is steep steeper than this portion. So, that means there is a zone above this two portion. So, as the center portion between this column this is very soft zone then soil within that center portion will deform more compared to the column. So, that means we have this type of settlement pattern. So, we will get this type of settlement pattern here. So, that means the soil within this zone that will deform more compared to the column. So, what will happen the soil above this center zone that will also deform more compared to these two zones. So, that means the soil deformation within this zone will more compared to these two zones. So, what will happen in that interface this is the interface of this is the one zone and this is another zone this is the another interface this is another interface this is interface at this interface there will be shear and this shear is because of this differential settlement. Differential settlement means difference of the settlement between this point and this point. So, that the center of the top of the column and the middle of the column and the center of this two columns. So, because of this differential settlement there will be differential settlement on the embankment soil also. So, this embankment soil there will be differential settlement and the shear will develop at the interface. Now, because of the shear resistance the soil has its own resistance. So, this because of the shear resistance what will happen this soil of this zone will try to hold some portion of this soil of some portion of this soil of this zone of the center zone. So, that means what will happen so because of this reason some stress because this soil above the stone column region this embankment soil because of the soil resistance shear resistance it is not allowing the soil within this region to settle down. So, what will happen that some stress on this region will be transfer on the stone column because of this shear resistance. Now, this phenomena is called soil arching. So, now if I go further that what is soil arching. So, that means if we have stone column here this embankment. So, suppose with the including this embankment and this granular layer the thickness is thickness of the embankment or height of the embankment is h and the density is also h. So, that means one stress that will act here. So, that is q s soft soil and another one we can consider q c or the stone column. So, that means here the soil within this zone the total soils pressure the suppose at this zone the total at the base of the soil the soft soil the ideal load is gamma into h that is the q which is acting here if there is no soil arching nothing. So, that means the total force is acting here q into h, but because of this soil arching and if it is there is a surcharge over this zone and that value is q 0. So, this will be q plus q 0 if there is a surcharge q 0 is surcharge. So, if there is no soil arching that means the q s will be equal to q if no arching. So, now if I write on arching ratio that is q s divided by q. So, that means if rho is equal to 1 then we can write no arching and if rho is equal to 0 the low is equal to 0 means q s equal to 0. There is no stress is coming on the soft soil all the stresses are going to the q c for this embankment that is called full arching. So, it is clear that if q s equal to q that means all the load which is applied above the soft soil is equal to the all the stress which is coming on the soft soil then it is no arching no arching has been occur. So, the stress at every point is same on the stone column as well on the soft soil, but because of this soil arching because there is a differential settlement then the shear stress will develop. So, that means the soil above the soft soil region the embankment material above the soft soil area that will also settle less settle more compared to the soil above the stone column region. So, there will be a differential settlement. Once the differential settlement there will be a shearing between this two soil of this two zone one is for the zone above the stone column and one is the zone above the soft soil. And because of the shear resistance of the embankment soil or granular material the stress on this region will transfer stress on the soft soil above the soft soil region is transfer on the stone column region because of the shear resistance. So, that phenomenon is called soil arching. And if I express it in the soil arching ratio then this is the if rho is equal to 1 there is no soil arching all the all the stress are coming on the soft soil and if it is 0 then full arching I mean no stress is coming on the soft soil all the stress are transfer on to the stone column. So, that means the stress on the stone column and stress on the soil are not same because of this arching phenomena. So, now sometimes in this stone column also embankment we can place our reinforcement layer geosynthetic reinforcement layer here at the base of the embankment. So, that we can provide geosynthetic reinforcement. Now, because of application of geosynthetic reinforcement the settlement is further reduced all the total settlement and the differences settlement will be further reduced and the bearing capacity will be further increase. So, another thing that will add here is the geosynthetic reinforcement there is a tension T there is a tension T. So, now if I go for the load transfer mechanism of these things that is the load or stress transfer mechanism. So, there is a three ways by which the stresses are transfer on to the stone column from the soft soil region. So, this is the stone column this is another stone column. So, if I consider the reinforcement layer. So, if we provide the reinforcement so that will deform in this form. So, deformation on this region we consider the as this is the steep that reinforcement deformation will be almost same here and here reinforcement deformation is not same. So, there will be a tension that will induce at both the end and this is embankment. So, this is embankment. So, stress on the stone column if I consider Q C and stress on the soft soil or above the stone column or then the stress below the stone column there will not be same. So, if I consider Q S top Q S bottom now Q S top is not equal to Q S bottom. Generally Q S bottom is less than Q S T. So, why it is less than? So, that thing we have explained that because of this once there will be the settlement of the reinforcement. So, this is reinforcement. So, this is reinforcement. So, once there is a settlement of the reinforcement in this form because of this shear between the reinforcement and soil that tension will develop within the reinforcement. So, that I have explained that one is string effect another is confinement effect. So, tension is developed equal amount of the compression is developed in the soil. So, the soil will get compressed to get the reduction. Another one is the tension will develop and this tension component if I take the sin theta T, T sin theta and T cos theta. So, T sin theta component will balance this some stress. So, that means ultimately the stress below the reinforcement layer is less compared to the stress above the reinforcement layer. So, now what are the stress transfer mechanism here? One is due to the soil arching. So, now the q c and q s. So, that now stress and here the reinforcement deformation is almost same. So, there will not be too much of this stress variation. So, here we assume the stress acting on the above the reinforcement below the reinforcement are same because we will not get this type of deformation. Once we will not get this type of deformation this huge deformation there will not be shear or there will not be any tension. If there will not be any tension, a mobilized tension then we will not get any reduction of the stress. So, that means the q c consider to be same above and below the reinforcement layer. So, that means the stress on the stone column and stress on the soft soil they are not same. So, if I run one ratio n that is the stress on the stone column and stress on the soft soil or here you can layer it s b. So, this is always greater than 1. So, that is called stress concentration ratio. This is called stress concentration ratio. Now, so that means the reason why this q c is greater than q s b one reason is due to soil arching because due to soil arching the stress in this region will transfer on to the stone column region because of the differential settlement. And that is due to the stiffness difference between the soil and the stone column because here it is stiff it will deform less it is lesser it will deform more because this soil is very soft. So, that is then because of that reason the soil arching will occur. So, that is one reason is soil arching and second reason is stress transfer due to the reinforcement. So, that is means that this reinforcement layer also enhance the stress transfer process from soft soil to the stone column and that is reason is that because we will get this t component here at the edge of the column. So, if this is the t component tension component. Now, if I take the two components of this tension then this horizontal component this will produce and this will release the stress because this t component because of this t component some stress will act on the stone column. So, that means this the stress which is coming due to this interaction shear interaction between the soil and the reinforcement this t will develop and this t will release or this because of this tension reinforcement will release the stress on this stone column because here the tension will develop within the reinforcement layer and at this point this end or edge of the stone column that is t and if we take the two components from this side. So, that vertical one is balancing this stresses this horizontal one is release the stress on the stone column. So, what will happen this basically this reinforcement layer is basically transfer the stress which is coming from the soft soil region and then it will transfer the stress from soft soil region it is carrying the stress and then release it on the stone column because here there is a zone from here also it will again deform in this form. So, again you will get another t here. So, from here also we will get some tension or we will get some stress. So, that means it will release this stress which is taking this stress and this component will get one t sin theta and one t cos theta. So, if we get t cos theta. So, that t cos theta part is released or on the stone column. So, that is due to this process it will enhance the stress transfer mechanism that means stress is transfer from this region and it is released on the stone column region and because of the region is the t tension which is developed at the edge of this column and this tension some part this horizontal part is actually resting on this soil. So, that will released here and the next one is very important thing. Suppose we have we can use here we can use a single reinforcement. Now, we can use say n number of say multi layer of reinforcement say 2 or 3 layer of reinforcement 2 or 3 layers. Then what will happen? Once we increase the number of reinforcement layer the differential settlement will also reduce. So, that means if we increase the 3 layers here. So, this zone if we increase the if we provide the 3 layers this zone act as a stiff region. So, it will not deform. So, once this there is a n number of 3 number of layers. So, this zone deformation will be less. So, if the deformation of the soft soil region is less and that is and if the differential. So, that means the differential settlement will be also very less. So, if the differential settlement is very less and the settlement pattern is almost same then if there is no differential settlement no soil arching. So, they are not be any soil arching if that happens and another thing if there is no differential settlement will not get this type of deform shape of the reinforcement. So, once we not get the type of reinforcement shape then what will happen? They will not be any shear between the reinforcement and the soil. So, once there is no shear no tension will develop once no tension will develop no stress will be transfer on to the stone column. So, then the first two will not act and that case if there is no differential settlement. So, to have sufficient benefit of the reinforcement layer there should be much differential sufficient amount of the differential settlement. Differential settlement means settlement difference between the settlement of the center of the stone columns and the middle of the stone columns and if there is no differential settlement no soil arching. Still in that case also we will get more stress on the stone column compared to the soft soil and that is due to the third region that is stiffness difference. So, because of the stiffness difference more stress will be coming on the stone column compared to the soft soil. As the stiffness of the soft soil a stone column is more compared to the stiffness of the soft soil. So, what will happen? The most stress will come on the stone column. So, that is the properties of the composite material. If a composite material we have a stiffer material and you have a very soft material then stiffer material will take more load compared to the soft material. Here also at the stone column is stiffer material compared to the soft soil then it will take more load. So, ultimately the stress on the soft soil is less compared to the stiff stress which is coming on the stone column. So, if there is single reinforcement so because of three reasons one is soil arching one is reinforcement layer and one is stiffness difference. So, there will be more stress on the stone column. If it is a multi layer system there will be very stiff system there will be less amount of the differential settlement then it will transfer because of the third reason only that is stiffness difference. So, ultimately we will give the stress concentration ratio which is always greater than one in a stone column input soil. So, now in these things now we have to model this total mechanism because it is a complicated mechanism low transfer mechanism. Now we have to model this total system by using this soil foundation interactions model. So, how will model that that is very important because in most of the cases these models are available for the pile supported embankment. So, there is very few limited amount of the studies have been done on the stone column supported embankment. So, difference of pile supported embankment and stone column supported embankment in pile supported embankment there will be very less amount of the deformation of the pile. But if it is stone column supported embankment then there will be huge deformation of the stone column also. So, and it is one case it is assume this the soil arching will happen like a circular arch. So, this is circular arch and the stress below this above the circular arch will transfer directly to the soil. Someone has considered that arching is happen on the this interface of the this two zone total arching. So, but so based on these assumptions it is done, but in this model that I am going to describe in this model no assumptions done here. We have not assume any shape of the arching that is that will come by default and it will just transfer the stress and will develop the model such that there is no need to assume any type of arching deformation whether it will deform circularly whether it will deform arch will form circularly parabolic or the straight or vertically. So, it will form automatically. So, it is the advantage of this soil structure interaction model that we have used here. And then the next thing is that that once we model this thing is the consolidation effect because as we have mentioned that the consolidation will play a very important role in the stone column in ground because the consolidation is very important thing that is also incorporated in the model. So, in the next class I will explain how this complicated problem is been modeled by using this soils foundation interaction theories and then how it is solved and then how what are the expected results we have received or we have obtained. So, those things will be explained in the next class. Thank you.