 Welcome to lecture series in advanced geotechnical engineering course. So in this lecture we are commencing with module 7 lecture 1 on geotechnical physical modeling. Nowadays with an increase in the infrastructure development there is a dire need for understanding about behavior of the geotechnical structures before collapse or before failure and at failure. This will enhance to understand the response of these structures before failure and at failure. This module is divided with the following contents physical modeling methods and especially a technique which we are going to expose ourselves is called centrifuge based physical modeling and its relevance to geotechnical engineering with a number of examples. And we will try to see application of this technique for static and dynamic problems with numerous examples. So in this module we are going to discuss about what are the different modeling techniques and narrowing down to physical modeling methods and application of this centrifuge modeling and its relevance to geotechnical engineering we will try to bring out that and then with number of examples and its applications we will actually try to see this centrifuge modeling of geotechnical structures. The basic intention of this module to cover the scaling loss or what we call scaling relationships and modeling considerations for physical modeling in geotechnical engineering both for static and dynamic conditions with examples. As I mentioned earlier geotechnical physical modeling has become a powerful and versatile tool for studying geotechnical problems. A wide number of applications can be studied by using this technique. This helps to gain insight into the behavior of geotechnical structures. So coming to the word which is there in geotechnical centrifuge modeling or geotechnical physical modeling what we said is that modeling. The modeling if you look into the definition it can be defined as a representation of some aspect of real behavior. So it can be a real structure and a situation or a problem that is soil or foundation by a more abstract system. If you are able to do that for example you consider an embankment resting on a soil then the all aspects of the embankment need to be modeled and then represented. Similarly a foundation like a shallow foundation subjected to vertical load horizontal load nowadays people are understanding about the subjected to torsional loads to the shallow foundations and similarly let us consider a pile foundation subjected to vertical load horizontal load and a torque load these things can you know represented in a abstract system. So modeling can be defined as representation of some aspect of real behavior a situation or a problem by a more abstract system. So in modeling it is essential to recognize three fundamentals one is that we need to see all significant influences should be modeled in similarity. That means first of all we say that material to be modeled and configuration let us say an embankment a slope of an embankment and height of the embankment and its bedding conditions need to be simulated. Similarly all the effects not modeled in similarity should be proven by experimental evidence. So any parameter which is not modeled in similarity should be proven by experimental evidence and any unknown effect should be revealed or proven in significant by means of test results that means that experimental verification will help us to ensure that any unknown effect should be revealed or proven in significant by means of test results. Suppose if we are not able to achieve a similarity between because of in virtue of certain parameter and we have to see that this is proven in significant by means of experimental test results. Further a model can approximate simplification of reality so the skill in modeling is to spot the approximate level of simplification to recognize those features which are important and those features which are unimportant. So what we need to know in modeling is that the skill in modeling is to spot the approximate level of simplification and to recognize a parameter those features which are important and not important and this can lead to more understanding of the behavior of a structure being modeled. So it is often necessary basically for the sophisticated design and analysis procedures as in the way of the only way for the geotechnical engineers can analyze a complex physical system at a fraction of cost of physical or any other type of modeling. So modeling is often necessary in the present sophisticated design because of the complexity which is involved by involving with number of parameters and this can be done at a fraction of cost of physical or any other type of modeling. So what are the steps involved in modeling? First of all selection of problem or system of interest and postulate the principal characteristics of the problem or system in the second step. In the third step apply principles of mechanics like for soils stress strain criterion, effective stress principle, Darcy's law, continuity equation, compatibility condition, stress history etc to reduce the response of the model. So in the third step we apply the principle of mechanics for soils stress strain relationship and or stress strain criterion and effective stress principle Darcy's law, continuity equation, compatibility condition, stress history etc basically this is done to reduce the response of the model. Further comparisons of the predictions with measured values from the carefully conducted institute or laboratory tests. So comparisons of the predictions with measured values from carefully conducted institute or laboratory results. So in step 4 if the agreement is not proper go to step 2 to reexamine the postulation and repeat the steps from 2 to 5 that means in step 5 if the agreement is not proper whatever is predicted is not in comparison with observed results then go to step 2 reexamine the postulation and repeat steps from 2 to 5. So having discussed the and defined modeling in modeling and in geotechnical engineering modeling word is not new, modeling techniques in geotechnical engineering is a conventional approach where it is adopted traditionally. So we have different categories of modeling we can say that empirical modeling even today because of the complex nature of the soil in many cases we adopt empirical models and theoretical models these are today even today they are famous and we are using a numerical models with the advent of computers. The numerical modeling is gaining popularity and is also getting tuned with advanced techniques in modeling techniques which we are going to discuss in this module. Traditionally we have analog models which are very popular to state an example the spring energy model adopted by Terzaghi for explaining the effective stress equation stands as an example and lastly a physical model which is nothing but classified as three heads basically one is full scale this is at 1 is to 1 and that means that a construction of a full scale structure and if many situations or many places or many sites the feasibility is a question and controlled experimentation of a full scale model is next to impossible in a such situation then small scale model is one of the viable option and these small scale model in the sense the model reduced by n times that means that all the dimensions of a full scale model are reduced by n times and this can be done at normal gravity that is at one gravity or at present please take it as that ng that is called a centrifuge based physical model which we are going to introduce in this modules. So the physical model is subclassified as full scale and small scale at normal gravity a small scale head centrifuge model or at high gravities so why we do we need the small scale models at high gravities in what cases you know these you know techniques are applicable you know that we will be elucidating in this you know part of lectures on in this module. So empirical models a long history of the empirical modeling in geotechnical engineering is there and we have a range strength criterion in the range strength criterion considering you know number of cases of embankment failures 0 man 1973 recommended a correction factor lambda for a vane shear strength measured in the field with that it has been found that with an increase in plasticity index of the soil the correction factor lambda is going to decrease drastically. So that means that you know by applying this correction factor to the measured value then you know we can actually get the strength value which can be adopted in the design based on that this can be you know matched close to the field observations. So an empirical model was developed basically by considering the observations which are actually done in during the vane strength test particularly when the vane blade is rotating the surrounding soil is subjected to some sort of disturbance and that leads to you know to give the higher value of you know the vane strength or shear strength value. So in view of that and this is actually predominant when with an increase in the plasticity index. So you know a correction factor has been recommended for this you know vane strength correction factor or even today it is known as Zurum's correction factor for vane shear strength measured in the field. Similarly we have for you know estimating consolidation settlement and we have examples like CPT and settlement of footings on sand and for interpreting pressure meter test results pressure meter is in a device wherein we can actually measure the in-situ modulus of the soil you know during by placing an inflated rubber membrane in the balloon. So depending upon the you know the you know the intact necks and softness or hardness of the you know walls of the borehole the response of the test can be achieved. So here there is also some empiricism and empirical models exist they do exist and we use in the practice of geotechnical engineering. Theoretical model even today like for example a classical example for theoretical model is for studying study state seepage conditions for Laplace equations is one possible theoretical model. So once a theoretical model has been formulated there are two possibilities for its application either the boundary conditions of the problem can be massaged in such a way that exact analytical result can be obtained or a numerical solution is required. So one of the you know examples which we can state is what study state seepage conditions and this is can be defined by Laplace equation the study flow of an incompressible fluid through a porous medium is governed by a familiar partial differential equation called dou square h by dou x square plus dou square h by dou y square plus dou square h by dou z square is equal to 0. So wherein you know this h is the head drop and this if it this is in three dimensional direction and this is reduced based on the continuity equation and if you have say you know only two dimensional condition then the third term which is dou square h by dou z square is equal to 0. So in that case dou square h by dou x square plus dou square h by dou y square is equal to 0 that is per 2D seepage condition. If we are having let us say only one dimensional flow like in the constant head perimeter for example wherein we can actually have dou square h by dou x square is equal to 0. So one dimensional flow wherein we have you know dou square h by dou x square is equal to 0 but two dimensional flow of an water through an embankment or earthen dam which is dou square h by dou x square plus dou square h by dou y square is equal to 0 and for the three dimensional flow an example is you know flow of water into the well that is the three dimensional flow and this approximate change in parameter also describes the flow of current as well as the flow of heat. So if you look into the analogous you know representation of these theories we can say that this is used in study in flow of current as well as the flow of the heat that is the heat electrical analogy as well as in the heat transfer. Coming to the next you know set of modeling which we have said is numerical modeling and numerical or constitutive modeling. Basically a numerical or a constitutive model is governed by the equations which ultimately describes the link between the changes in strain that is delta epsilon and delta sigma that is in changes in the strain and changes in stress for any element of the soil because of you know the loading they are subjected to. The complexity in numerical model increases with the requirement of defining several material parameters from laboratory and institutes. So what will happen is that the complexity and the efficiency of the numerical model increases, efficiency of the model becomes difficult with the requirement of defining several material properties from laboratory and institutes. Moreover in a numerical model if you are actually having several materials with interacting with each other very difficult to you know define the stress strain relationship between those materials which are actually you know taking part in the behavior. So this particular technique even though lot of progress actually has been made many times actually this requires verification of the developed numerical solution. If this is done by using an appropriate technique if this validation is done yes numerical modeling can be considered as one of the viable option for you know studying number of geotechnical problems. So numerical modeling is the subject of many basic and applied research efforts in engineering and basically such efforts involve the use of finite difference or finite element and boundary element and are discrete elements in conjunction with sophisticated non-linear elastic, allostero-plastic or visco-plastic models. So what we are discussing is that this requires verification of the developed numerical simulation. Once the numerical model is verified then the contents in using this model increases. So these efforts basically doing this numerical modeling involve the use of you know methods like finite difference method, finite element methods, boundary element method and discrete element methods in conjunction with sophisticated you know models like non-linear elastic, allostero-plastic and visco-plastic models. We have said that traditionally in geotechnical engineering these analog models are famous. The analog models basically the analog model carries a similarity in which a law which model follows is analogous to the law with the real situational problem follows. Let us consider electrical analogy, flow of water. So this you know can be given with, defined with two laws. One is you know in case of real situation let us say that is Darcy's law wherein we can actually define q that is discharge is equal to k that is the coefficient of permeability and I that is hydraulic gradient and A area of cross section. So area of cross section through which the flow is actually happening where I is nothing but H by L. Similarly in case of electrical analogy when the flow of current when we are actually considering we have Hohm's law where it defines you know between using the resistivity and a potential drop over a certain length and you know the cross section area of the conductor through which the electricity is passing. So there is you know this analog model assembles with a law which model follows is analogous to the law which real situation or the problem follows. So this particular concept was you know extended to you know for explaining the Darcy's principle of effective stress that is this spring analogy model. In the spring analogy model what has been considered is that the real situation we have soil particles and occupied by the voids in between soil particles occupied by water and when it is subjected to loading then what we have is that you have a situation that you know the skeleton the soil skeleton as well as the water surrounding the voids as well as the surrounding the particles is subjected to loading. So this was actually represented by a spring analogy problem so wherein what has been considered is that a cylinder of certain diameter was considered and it is actually having a piston arrangement at the top and let us assume that it actually has got a valve which is actually possible to close or open then the piston which can move up and downwards and let us assume that the bottom of the piston is attached with a spring having certain stiffness k and let us see that the spring actually represents the soil skeleton and pore water is actually represented by water in the spring analogy model also and let us assume that the you know the piston is placed in position and the valve is actually closed and the load is placed a load of P is applied on the piston so then when the valve is closed initially what will happen is that the entire load is actually bound by the spring so that means that under unrained conditions also similar situation happens similar phenomenon happens in real situation wherein entire load is actually bound by the pore water then moment the valve is opened then there is a possibility for the spring to undergo compression so in a way what will happen now is that the load is apportioned by you know the spring which is nothing but a soil skeleton in the real situation and pore water. So based on this after certain amount of time what will happen is that the hydro static conditions will provide and in a process what will happen is that we can say that the total stress is equal to effective stress plus pore water pressure so to explain this effective stress equation sigma is equal to sigma dash plus uw the spring analogy was used and further the concept was actually extended for explaining the secondary consolidation that means under a constant effective stress when there is actually change in you know the void ratio which actually happens for a period of time because of the certain nature of soils like peat or marshy from the marshy lands or we have you know the man made material like municipal soil to waste it undergoes you know use amounts of secondary consolidation. So this spring and dash spot analogy was actually used to represent you know these you know particular phenomenon of secondary consolidation or a creep of a soil. Before explaining that let us try to look into extension of this spring analogy model you know previously we have said that for explaining the total stress is equal to effective stress plus pore water pressure we actually have used a single spring but in order to explain the consolidation behavior of a soil layer having you know two open or two sand layers sand layers at top and bottom where they are actually having high permeability at the top and let us assume that this is the water table at which you know the saturation below the soil is completely saturated and it is subjected to say certain increase in the load say delta sigma. So this situation is actually you know can be modeled analogous in analogous way by using spring analogy model. So here what you have done is that we have taken a long cylinder and the cylinder is assumed to be filled with water and has got top and bottom two walls and the diameter of the walls represent the you know the permeability of the soil. If it is you know large permeability the diameter is high and if it is small permeability the diameter is small and this is a piston which is movable and each at each compartment it is divided into here is shown as 5 compartments 1, 2, 3, 4, 5 and between each compartments there is a spring of stiffness k is attached and another spring of k is attached here another spring of k is attached here and fourth spring and fifth spring and all these compartments are interconnected so that means that water can flow you know in its freely. Now what will happen is that initially when we actually apply delta sigma so we have a situation that if delta sigma is applied you know then the pore water pressure which is within the soil is nothing but delta U which is you know what we can say is the first isochron which actually it gets developed and what a period of time let us say at time t is equal to 0 and when time t is equal to 0 when the load is applied then we can say that you know the first isochron is nothing but delta U it shifts by equivalent to delta sigma then over just let us say that when time t1 where t1 is actually greater than t then what will happen is that because of the previous nature of you know these two layers open layers here sand and at top and bottom that what will happen is that you know the thus the water transfers the stress directly to the soil and the pore water pressure drops to 0 that means that at this particular point the next level the isochron comes towards the you know origin the center from where the pore water pressure increase to delta U or delta is equal to delta sigma so at both top and bottom it actually gets you know to reduce to 0 then correspondingly what will happen still at the midpoint there will be a high amount of pressure pore water pressure to be dissipated or transferred to the effect transferred to the soil scale. So here what happens is that over a period of time what will happen the tendency of you know the isochron to flow towards the you know the initial condition tends to provide that is the initial condition is nothing but the original hydrostatic condition before the application of the load. So here the similar situation can be simulated with the if we are able to say open these walls simultaneously both upper wall and bottom wall then in line with you know what actually happened with two open layers which we consider in the left hand side of this figure we have the spring 1 and 5 will get compressed first then spring 2 and 4 will get compressed first and finally the third spring will start compressing slowly. So this process continues till you know the hydrostatic conditions for while in the situation. So here with this what we can actually explain is that the consolidation phenomenon of a soil having thickness h clay layer having thickness h can be represented by a spring analogy model with multiple springs attached in the fashion which is actually shown in the figure and as we have said that this analogy model was actually applied to you know to secondary consolidation of the creep of a soil also and two models which are can be discussed are that Gibson and Lowe 1961 and Barden 1965. So here this model by modifying the soil skelter response with the type so the soil skelter particularly what will happen is that under the application of a constant effective stress the because of the breakage of the soil particles are bending and breakage of the soil particles this phenomenon actually happens. So here this is represented by Gibson and Lowe in 1961 by a linear spring A1 here and the linear spring B1 and linear dashpot delta 1 and so this is actually represented and further this was actually modified by Barden in 1965 by replacing this linear spring and by putting a linear spring here and here what actually has been modified is that instead of linear dashpot a nonlinear dashpot has been used to go close to the creep or a secondary consolidation phenomenon in the real situation. So in this way the analog models were traditionally applied and particularly to understand about the electrical analogy and the spring analogy they are famous in as far as you know geotechnical engineering is concerned. Going to the fourth type of the last type of modeling which we have discussed is nothing but physical models. These physical models are basically performed in order to study the particular aspects of the behavior of the prototypes and these physical models help to postulate or portray the failure mechanisms and to understand about the behavior of a prototype before failure and let failure and a full scale testing is in a way example of physical modeling where all features of the prototype or full scale structure in field beings study and are reproduced at physical at full scale 1 is to 1 that means that all aspects are actually represented and that means that we have real soil, real ground condition and real ground motion and real loading and when we have this situation then full scale testing is in a way of example of physical modeling where all features of the prototype or full scale structure in field being studied or reproduced at full scale. This full scale testing is also quite common in geotechnical engineering practice like for example like a plate load test which is actually conducted to estimate the bearing capacity that means that from the through a field plate load test we can actually carry out in the plate load test can be carried out in the other side to get the bearing capacity. Similarly in order to get the axial load capacity or lateral load capacity there is a mandatory requirement of testing of piles which are actually being used in a project for we actually do the two categories of pile load testing one is to test up to the failure the other is to test only up to certain amount certain times of design load so that these piles can be used in the structure. So the one which is actually applied beyond the design load is generally done initially as a test where the piles can be abandoned so these full scale testing is in a way example of physical modeling where all features of prototype or full scale structure in the field being studied or reproduced at a full scale. Similarly the full scale models are full scale or full prototype which is you know at one is to one scale usually associated with performance of testing of complete geotechnical systems and can be used to use real geotechnical materials so the need of theoretical model of their behavior disappears and because as we are using the real geotechnical materials the need of theoretical modeling of these materials disappears and provide data for validation of analytical modeling approaches and can thus provide a basis for extrapolation of physical model to the geotechnical prototype. So with that you know there is a possibility that providing a data for validation of analytical numerical modeling and approaches which we have discussed earlier and can thus provide a basis for extrapolation of the physical model to the geotechnical prototype. And another added advantage is that instrumentation and monitoring geotechnical prototype can itself is a physical model serving this validation purpose so by instrumenting the you know the physical model and monitoring the geotechnical prototype itself be a physical model serving this validation purpose. So full scale testing is usually performed to evaluate the geotechnical process which is which it is believe it may be so dependent on the details of actual soil fabric that is imperative to use real soils. So for example in some type of problems like embankment construction and soft soil wherein we noted to monitor the you know the degree of ground improvement for example for a pre-loading of a soil. You know to do that you actually need to do the you know construct the embankment in let us say if suppose if the embankment is of 7.5 meters height above the soft soil and it has to be done in three stages let us say and then between each stage there should be a waiting period of certain way at period let us say two months or three months. And with that in order to you know set time the you know time for you know increasing next stage and going for next stage and all. You need to do the instrumentation wherein you measure the settlements you measure the water pressure or pore water pressure changes in the soil based on that above we can actually decide about you know the about the this thing. So here the full scale testing is imperative with real soil fabric conditions and similarly when we are actually try to accelerate we try to use the you know the drains like prepropagated vertical drains or sand drains. So in a way what will happen is that wanted to know the influence of spacing of the drains and you know the installation effects on the performance and all. One need to you know monitor and you know these full scale testing. So full scale testing is usually performed to evaluate the geotechnical processes which it is believed may be so dependent on the detail of actual soil fabric that is imperative to use at use real soils. So we can put the you know advantages of the full scale modeling are working with real ground conditions and real soils and real loads and real stress levels and real stress histories. This is very important particularly for when we are referring to soft clay history whether it is normally consolidated soil or over consolidated soil and so this you know these conditions you know are you know very very relevant and that is the benefit of you know they doing a full scale model test but you know we have you know number of types of structures as well as all these types of structures are subjected to different types of loadings. So many times you know simulation of these structures to climatic forces like say rainfall or earthquake and all those things are very very difficult. So these you know simulation of you know construction of a structure and waiting for you know a certain destructive force to come is you know difficult and next to possible. So these are the you know in the right hand side here it has been so whatever we have discussed in the previous slide as actually shown here that is the trail embankments where in the evaluate the process of ground improvement we tend to do the full scale testing or a full scale prototype behavior and use of different types of drains and spacing whether they are adequate or not whether the functionality is okay or not can we have to be checked by monitoring only. And disadvantage is in the sense that smaller scale model leads to much more rapid results purely because of the smaller size smaller the size the length required or small physical dimensions are small the amount of the requirement of material will be small. So you know the small scale model leads to much more rapid results purely because of smaller size. Construction of an embankment on soft soils and you know the stage construction for example if you look into it may take years to complete. So that situation you know is not advantageous as far as you know full scale or physical model testing in geotechnical engineering and cost will increase with the scale of the modeling that means that if you are having let us say one is to one scale it will be almost equivalent to the scale which is actually the cost will be equivalent to that of in the field. And difficult to perform parametric study and difficult to perform you know controlled you know full scale testing in and many times you know even with a lot of precautions the instrumentation and monitoring is also difficult but if you are able to achieve an appropriate instrumentation and monitoring and if you are able to you know do it with you know in a controlled way then full scale model testing, physical model testing is the you know in you know number one option to you know validate the number of you know different new concepts as far as which can be lead in the to study in the geotechnical engineering or to understand about the behavior of geotechnical structures but you know considering the difficulty of the you know performing you know parametric study and also involving the construction of the cost aspects as well as the feasibility aspects you know the small scale modeling turns out to be a preferable option. The small scale option in the sense that the model which is not you know tested at one is to one which is nothing but reduced by a dimension n that is n is nothing but a scale factor by which the model is reduced. So if you look into the physical models at small scale the key question concerned is you know with establishing the validity of the models whatever we have scaled down and ensuring the secured route to the extrapolation from the model behavior to the behavior we could expect at full scale. So the question which is required to be established is the you know validity of the models and ensuring the secured route to extrapolation from the model behavior to the behavior you know we could actually expect at full scale. So you know when we actually test the model at full scale you know we do not have these questions because we are actually doing at one is to one with real soils real ground conditions and real stress histories and real loads and all. But when we do this at you know scale which is different from a full scale which is smaller than you know by a smaller by a factor n the extrapolation from the model behavior is a very important aspect to be considered. So existence of the supporting theoretical models is thus even more important for interpretation of small scale physical models than for the full scale models. So existence of supporting theoretical models is thus more important for interpretation of small scale physical models than for full scale models. So physical models at small scale you know if you look into the merits you can say the greater advantage of small scale laboratory model is that we have full control over the all details of the model that means that all aspects of models under control in the sense that homogeneity you know or some requirement of the small quantity of soil and drainage paths if at all water is actually flowing thus are short so that test durations will be short. Possibility exist in performing many tests and performing with parametric variation is possible that means parametric study is possible and effect of varying key parameters can be considered and another important aspect is that the smaller the scale the cost will be low. So liberty in choosing the boundary and loading conditions of the model this is another important aspect where in we can actually have you know we get the liberty in choosing the boundary and loading conditions and smaller quantity of soil and drainage paths will be short so the test durations will be smart and but we can look into the size of the model is you know as advantages as well as the disadvantage it is not that you know we reduce by you know by greater factor n and say that we have done a small scale models always it is important for us to see that the whatever the small scale model is done is relevant as far as the particular phenomenon is being tested and see that these results represent the full scale response of a respective full scale structure. So the physical model plays a fundamental role in the development of geotechnical understanding and it is you know performed basically to validate theoretical and empirical hypothesis and also is done to validate you know see new phenomenon or you know to perform understand the behavior of the prototypes this is you know is the fundamental role in developing the it plays a fundamental role in development of geotechnical understanding and this physical model testing as far as in the laboratory is concerned this is we actually have two standard examples one is a casagrandes liquid limit test and the other one is fractional compression test. So let us look into the casagrandes cup test in which what we do is that we put a you know make a soil pad in a cup which is actually having certain curvature and it is placed on a you know a standard you know base and it is subjected to once that pad is actually formed by using the casagrandes tool what we do is that we separate and make you know two equal slopes at a formed at a you know this pad which was actually formed at a defined water content and then the tool is you know casagrandes tool separates the two slopes at the toe by a distance 2 mm and the height of this slope is about 8 mm and what we actually do is that you give by giving a tamping energy to these two you know this particular two separated portions we see that the water content at you know the number of blows at which you know these two you know separated slopes you know get close by about you know over a length of 13 mm and we say that you know there then we can say that you know that is that critical number of blows for that water content. So here what we are doing is that the slope height is 8 mm and the slope inclination is about 60.6 degrees also at both the both the sides and this is subjected to certain sort of tamping energy and the you know indication of these slopes moving close that means that internally there is a failure which actually happens as we can say that critical surface failure which actually mobilizes makes the slopes to move closer at the toe and you know then we can say that that is the point at which we take the number of the number of blows required to you know see that this portion closed by about 13 mm. So you know this in this what we are doing is that literally this is a type of physical model test which we are doing knowingly or unknowingly in order to arrive at the liquid limit of a given soil by Casagrande's cup test and similar example is the triaxial compression test what we do is that here if you are having let us say a sample which is collected at a certain depth and the sample before its collection is subjected to vertical stress and horizontal stress at elastic equilibrium the relationship between vertical stress and horizontal stress is say sigma h is equal to k sigma v basically for normally consolidated soils you know where k h k which is the coefficient of earth pressure stress will be equal to 0.5 so in that case sigma v is equal to point you know sigma h is equal to 0.5 times of sigma v and moreover upon this when it is subjected to loading that is the increase in loading what you know the sample at a particular level is subjected. So this is simulated in the triaxial compression test in order to understand the strength parameters of soil what we do is that here the sample is subjected to a shear by applying the increase in load that is nothing but what we call so this is simulated here what you can see that this confining stresses are simulated by using the water surrounding water placed in the triaxial cell and the deviator load which is actually applied that is nothing but the additional load sigma 1 is equal to sigma 3 plus p by a so sigma 1 minus sigma 3 is nothing but the deviator load is equal to p by a which is applied basically to see arrive at the deviator stress at the failure. So with that what we are getting is that the stress strain behavior of a soil and so that we can actually get the stress strain response of a soil to the given loading. So this is what we can say is the first in a physical model test which done in the laboratory to simulate the field or real situation condition. So coming to if the physical models what we have discussed is that the physical model can be done at 1 is to 1 or 1 is to n if the model is not constructed at 1 is to 1 scale that means that we need to have some idea about the way in which we should extrapolate the observations that means that if the model is not constructed at 1 is to 1 we need to understand about how these the model which is not tested at 1 is to 1 but it is tested at 1 is to n let us say reduced by n times how this model actually corresponds to a model which is you know which is should be at 1 is to 1. So the material behavior particularly if you look into this we have two categories one is linear and homogeneous for the loads applied in the model the other one is the non-linear if the identical structure possesses several materials which interact with each other. So in the second category what we see here non-linear and if the identical structure possesses several materials which will be interacting with each other. So in this case like you know for example for this is that let us say that we have got a cable is placed between you know two supports and let us say you know the cable is loaded at the center with let us say by load W. So here what we can say is that the span of the cable is say length L between two supports and by loading the cable undergoes a deflection by H. So in this situation what will happen is that the cable you know deflects or sacks by H in the center. So here if you wanted to get the relationship between you know P which is the tension and weight of the cable and then relationship between H the central deflection and you know the span can be obtained. So here details of model can be projectable to prototype stays you know but you know here in this case basically P by W which is nothing but the tension and weight of the cable to the H by L. H by L is nothing but H is the you know deflection and L is the span. For example if I wanted to project these results to something like you know P by W is function of H by L now where we can actually get P1 by W1 and H1 by L1, P2 by W2 and H1 by L2 so H2 by L2. So in the ways what will happen is that we will be able to do and postulate from one model to other model easily without much you know moderation about the scale factors or scale issues. But if you are having a situation of non-linear or if this geotechnical structure pauses several materials which interact with each other then you know what we need to know is that we require an understanding about the dimension analysis and development of the scaling relationship because particularly we have a number of phenomenal engineering let us say that we have got self-weight forces these are nothing but these body forces. So we actually have two types of forces one is called body forces and other one is you know the surface forces or contact forces. Because basically body forces are you know examples for body forces are nothing but weight forces due to the self-weight of the soil or you know seepage when the water flows to the soil the forces exerted by the flowing water on to the grains is nothing but a seepage forces in the process of dissipation of energy this happens. So seepage forces and you know self-weight forces are regarded as you know body forces and when the material interacting with each another material then the contact forces are actually developed. So but when the body is subjected to forces then it is subjected to stresses and if the body is you know a you know deformable body then body undergoes you know a change in lens then it is subjected to strains and because of this change in lens can also undergo or the strains can also be undergone by collapse of the particles or by you know breaking of the particles or crushing of the particles. So this you know this particular you know emphasis what actually requires is that in understanding of this dimensionalsis. So dimensionalsis is a technique which is actually used traditionally in engineering and which can be applied to geotechnical engineering to understand about the you know the several relationship between parameters which are actually describing a particular phenomenon that means that we will be having a certain main important variable and then it is actually influenced by several variables which are called as independent variables. So relationship between dependent variable and independent variables can be obtained by dimensionalsis and from there the similar shoot or similarity theories can be applied to deduce the schedule relationships this is one of the ways of deducing then we also have by using the concept of you know the differential equations or equations governing the phenomenon we can also deduce the scaling relationships.