 Welcome to in the course of advanced geotechnical engineering. This is module 7 lecture 5 on geotechnical physical modeling. So this is lecture 5 on geotechnical physical modeling and module 7. So in the previous lecture we introduced ourselves to number of centrifuge based you know the limitations which are actually involved with centrifuge based physical modeling. And they are basically listed here, non homogeneity and anisotropy of soil profiles which is difficult to model in centrifuge based physical modeling. And limitations of the modeling tools and variation of G level with the horizontal distance and depth of the model and subsequent errors and then we also have said that how these errors can be minimized by selecting an appropriate configuration of the equipment. And the boundary effects which are between the walls of the container and soil and the scale effects which is because of the our inability to not able to scale down the particle size then one of these scale effects is called particle size effects or grain size effects. And we discuss how these particle size effects can be averted by doing you know especially using the principle of centrifuge modeling we call modeling of models. Then after having you know once we introduce ourselves to you know the scaling laws then we realize that there are different scale factors for time for different types of forces like seepage forces or creep forces or viscous forces or weight forces then you know the inconsistency of the scale factor for time then we also discussed that whenever there is a motion of the body happens within the model then we said that you know it is going to subject to you know two types of accelerations one is Euler acceleration and second one is Coriolis acceleration the as far as the centrifuge modeling is concerned Euler acceleration is not very serious because there is no much change of angle acceleration because of that the Euler acceleration term is limited then but what we are having is that if you are releasing let us say you know sand particles onto the soft soil for enabling the construction of embankments on soft soil or releasing you know certain weight at a certain on to the acceleration gravity field. So these situations arise to cause you know Coriolis effect then also seismic perturbance which also can cause you know these you know Coriolis effect then we also discussed that for a Coriolis effect to be you know negligible we said that the velocities have to be as small as possible like either small as possible let this as small as 0.05 times V where V is the model velocity if the event is also very fast like a blast event or a projectile event wherein the ejecta is thrown at very very high speed in such situations also we said that you know the Coriolis effect is negligible in the sense that the ejecta will be thrown with very rapid speed and it goes and hits the periphery of the container walls. So we need to also when the explosion events are actually modeled in high at high gravity there is a need for you know putting sacrificial you know sheets along the periphery this helps in reducing or denting the container boundaries otherwise what will happen is that the container boundaries are subjected to you know serious errors due to denting. So when if the velocities of the moving particle within the model within say greater than 0.05 times the velocity of V with which the model is moving and less than 2 times V then we said that the Coriolis effect is cannot be ignored and need to be considered and for that we have to see and how you know the Coriolis effect can be minimized. So those models which are being tested with the velocities in the range of greater than 0.05 V and less than 2 V we need to check whether the model is free from the Coriolis effect or not. So the Coriolis force arises from any movement that occurs within the centrifuge model that is what we have been discussing and for example if you try to construct an embankment by raining sand in flight or if you drop a ball from the center of the centrifuge to study the projectile motion they impact on the soil or the simulation of rainfall. These conditions and these are actually nothing but the simulation of this climatic events or some construction process which actually lead to this Coriolis accelerations. So then that the moving object will have the Coriolis acceleration which what we said is that a suffix c is equal to 2 V omega and Coriolis force is nothing but 2 times m into V omega where m is the mass of the moving object within the model. So the Coriolis effect is 2 m V omega. Now after having discussed the limitations we have introduced ourselves to that are two types of machine configurations they are basically beam centrifuges and drum centrifuges and initially the beam centrifuges are known as balanced beam centrifuges and in the drum centrifuge we said that where you have a peripheral drum and then the central tool table we try to look into the details of these machine configurations. So the radial force acting on the central spindle should be minimized as sigma m r omega square tend to 0 and this if you are having that means that the mass balance is actually happening on the both the sides and whether it is with a beam centrifuge or whether it is with a balanced beam centrifuge we actually have to ensure that sigma m r is equal to 0 that ensures that the horizontal the model can be rotated within the plane of rotation and the second issue is that you know the bearings of the centrifuge will be you know unaffected. So a typical balanced beam centrifuge is actually shown in the slide where in you have the two baskets where in the models of almost equivalent weight or place and it is subjected to rotation about a vertical axis in a horizontal plane where r is the radius and if weight of the model at which this acceleration say for example if the weight is say 2 tons or 20 kilo newtons and if the if it is able to carry 20 kilo newtons at 100g then what we call is that as far as the capacity of the machines is balanced beam or beam centrifuge equipments are concerned it is called as capacity which is indicated as a g tons or g kilo newtons which is nothing but payload of the model that is 20 kilo newtons into at which the g level the 20 into 100 so it is something like called 2000 g kilo newton or 200 g ton capacity. So the capacities of the equipments are actually indicated worldwide and then you know various ranges of capacities of the equipments are there throughout the world. So we will try to look into the different centrifuges which are actually there in the world, one of the early centrifuges in the world which actually was you know put forward by Russians that is after the Moscow railway transport university where in we have got two swing baskets and then have you know the arm which is actually at the central the connecting to the central spindle. So the pictorial or you know artistic view of the centrifuge which was which was known earlier is the picture here wherein you have got two baskets and is called twin baskets and the central pedestal and arm or a beam of the centrifuge. Then what we see here is you know the Rur University Bokum centrifuge in Germany where the radius of the centrifuge is about 4.125 meter that means that the radius is measured from the center to the from here to here that is perpendicular distance. So it is 4.125 when the basket swings up and the maximum g level of the capacity of this machine is 250 and the maximum payload which can be mounted here is 2 tons. So the capacity of the machine is said as you know 400 g tons that means that at payload at 100 g it can carry you know at 200 g it can carry 2 tons that is the 400 g tons capacity what we are actually referring. So this is a typical balanced beam centrifuge where you know has radius of 4.125 meters and this is the turner beam centrifuge this is initially it was a restrained platform centrifuge afterwards with the restrained centrifuge was converted into with the pivot mechanism which is actually shown here and the basket can be detachable and where the model is actually mounted and equivalent weights are actually placed here and so that sigma MR is equal to 0 is achieved here and this also actually has a radius of r is equal to 4.125 meter and the maximum g is 150 and a capacity of this equipment is 150 g tons capacity and wherein you can actually see that you know the you know the it is a fin basket and this is the arm or a beam of the equipment. So this is in you know an university of Cambridge in United Kingdom and this is you know Hong Kong HCUST centrifuge and this actually has a radius of 4.2 meters and maximum g level is 150 and the capacity of this machine is 400 g tons capacity. So this is also a balanced beam centrifuge and this is the RPS centrifuge in USA the radius of this centrifuge is 3 meters the maximum g level is 160 and maximum payload is 1.5 tons and the capacity is 150 g tons at 100 g so you can see here it can carry 1.5 tons load at 100 g that is reason why capacity is 150 g tons and this is you know the balanced beam centrifuge where you have got a single basket or single swinging basket and this is the pedestal and this is the adjustable counter weight the counter weight is adjusted depending upon the weight of the model which is actually kept. So if the model is heavier then the counter weight you know adjust towards this side with the model is lighter the counter weight comes towards the center so that sigma MR is equal to 0 can be ensured. So this is F star centrifuge in Paris in France and this is the also known as LCPC centrifuge and where the radius of the centrifuge is 5.5 meters this is also a beam centrifuge and maximum g level is 160 and maximum payload is 2 tons and the capacity is 200 g tons capacity at 100 g so that means that at 2 tons it can carry at 100 g that is why the capacity is called 200 g tons. So this is the cased centrifuge wherein you know this is also a balance this beam centrifuge and what we look into all these cables and these things they are like you know for actuating earthquake actuator and also the data acquisition components and all and this is also equivalent to the radius of about 5.5 meters and the details are actually given in Kim et al 2013 and this is the k water centrifuge and which is having a radius of about 8 meters and this is again a beam centrifuge with a single basket. The basket area here in this case is about 2 meter by 2 meters and the maximum g level is 150 and the capacity of this centrifuge is 800 g tons that means that it can carry a payload of about 8 tons at 100 g, 8 tons payload it can carry at 100 g that means that very large centrifuge models it can actually accommodate so that the models can be tested and this k water centrifuge was actually developed for studying problems like the dam instabilities and levee instabilities and other you know the water relevant you know the structures. And this is you know the small geotechnical centrifuge at IIT Bombay was actually existing between 1994 to 2008 and this is very small equipment where the bios centrifuge was actually converted into engineering geotechnical centrifuge with the radius is 0.32 meters and maximum g is 200 the capacity is 0.4 g tons capacity. So this facility was actually used for geotechnical instruction as well as for some you know research problems which can be investigated within the you know the errors which can be allowed in these small centrifuges. So the payload which is very low is something like 2.5 kg which actually can carry. So here in this particular model which is actually shown and which actually has got the you know some mortar which gets activated and this you know the counter weight is actually used for supporting you know the weight of this package which is actually shown here. So this is the IIT Bombay's large beam centrifuge facility the radius is about 4.5 meters maximum g level is 200 and the capacity is 250 g tons and this is one of the indiscriminately built equipment and it actually has got regenerative braking system and in flight balancing and when compared to the contemporary equipments this actually has got high payload capacity and the g level which actually ranges from 10 gravity to 200 gravities in order to reach 200 gravities it can actually reach from 1g to 200g in 6 minutes. Similarly from 200g to 1g it can actually come down or ramp down in 6 minutes that is so it actually has got a 2 different ramping speeds one is 34 rpm per meter the other one per minute 34 rpm per minute the other one is 3.4 rpm per minute. So when you compare the capacity of the major balance and beam centrifuges in the world wherein some of the equipment which we have discussed then IIT Bombay equipment falls somewhere here between 100 g ton capacity to 500 g ton capacity and some of the new Korean centrifuge actually falls very close to you know 800 g ton capacity and UC Davis which they have the equipment which is very close to 1000 g ton capacity where in their arm radius is about 9.5 meters. So you know what we have is that you know this particular chart is actually for maximum acceleration and then this is the payload. So you can see that IIT Bombay centrifuge actually can carry at 2500 kg at 100 g that is why it is actually somewhere here it is actually located. So this is the hipster centrifuge very close to you know and then Cambridge centrifuge is somewhere here. So after having discussed different beam centrifuges and balance beam centrifuges now let us look into the drum centrifuges which is a typical drum centrifuge pictorial view which is at Tokyo Institute of Technology Japan and here as I have been told that there is a periphery which is actually exists and where in what it is called as the aspect ratio of the channel. So where in it is 1.2 meter in diameter 0.15 meter deep and 0.3 meter wide. So where in in this channel the models are actually made like this for example here an embankment model is actually made with after having made with certain clay then what is actually is done is that it is actually scrapped it is this portion is removed and this portion is removed then the shape is actually achieved and then you know then it can be tested for certain testing. So what we can see is that here a particular model which is actually mounted here a uplift capacity of a certain anchor which is actually embedded in soil is being tested and this payload capacity is actually 0.6 tons and EMAX acceleration maximum acceleration is 484 and the capacity is 290 g tons capacity. So the aspect ratio of the soil channel which actually said is that 1.25, 1.2 meter diameter 0.15 meter deep and 0.3 meter wide and here one advantage is that you know given stress history and number of you know the tests can be done. Then second thing is that also as I as been informed before that elastic off-space problems can be simulated because of the large extent of the you know areas from both sides of the model being tested. Then the preparation of the you know model in a drum centrifuge which is actually is trivial and this is how the sand models are actually constructed where the sand is actually allowed to drop onto a central rotating plate and both this as well as this rotates about the vertical axis in a horizontal manner in a synchronized way. Then what will happen is that these particles are actually thrown tangentially onto the periphery like how these the sand models are done and similarly some clay models which are actually done by feeding either sand this is either feeding the sand through nozzle so that the sand is actually dropped onto the you know the channel which is being constructed which is being used for constructing a particular model. So this is the one of the largest you know drum centrifuge in the world and this is the ETH jurid drum centrifuge and the pictorial view is actually shown here where in this is the model in which this be constructed and this is the periphery of the drum which is actually shown here and this is the central tool table and when the test is in progress where you will actually see that both the central tool table as well as the drum rotates in a way about the vertical axis in a synchronized manner. So this is actually about 2.2 meters in diameter that is about 1.1 meters in radius and wherein you can see that you know this is a particular case you know the investigation which is actually for stone column reinforced clay being investigated what you see the dots at the model stone columns. So this is the you know ETH jurid drum centrifuge. Now after having discussed about the relationship between the ramp angle I mean the different machine configurations and what we said is that we have got beam centrifuges or balanced beam centrifuges or we have you know the drum centrifuges. In the case of the drum centrifuge the model is actually prepared within the channel which is attached to the periphery of the drum and nowadays you know some of the drum centrifuges are also equipped with some baskets wherein the baskets are actually attached to the you know the periphery of the drum. So with that you know some confined models can be you know tested and now as far as the beam centrifuge or balanced beam centrifuge is concerned we need to understand what is the relationship between ramp angle that is the angle which is subtended by the basket when it is actually in the at 1g condition that is when the basket is in 1g condition the basket will be vertical and the weight will be acting downwards and when it you know rotation starts about vertical axis in horizontal plane let us say that this is omega which is actually rotating about vertical axis then there is you know the swinging up of the angle basket takes place. So at any point of time of t once it starts rotation of a you know the basket this swing centrifuge about vertical axis that angle be theta then you know assume that these are the radial axis and this is the y axis vertical and this is the frictionless hinge where you have got a tangential force acting like this and the normal force acting upwards and this is the model package and then with model package and the basket weight they assume that this is the center of mass that is the point P which is actually the weight acting downwards including the basket weight as well as this is actually acting downwards. Now by taking you know this angle is theta and this is mg now at this point from the distance from the hinge to the point P which is you know the distance is L so this distance is L sin theta so if this is radius from central of the center of the shaft to the center point of the hinge it is R suffix h plus you know L sin theta where R is equal to R h plus L sin theta. Now by taking you know sigma f r is equal to 0 and sigma f y is equal to 0 we can write sigma f r is equal to 0 we write you know what actually happens is that when the model rotates about vertical axis in horizontal plane there is a radial acceleration which actually developed and the radial force which actually developed the centrifugal force which is nothing but mr omega square has to be balanced by the t so t is equal to mr omega square. So we can write t is equal to mr omega square into for substituting R is equal to R plus L sin theta so the R is nothing but R h plus L sin theta which is indicated as R here R h plus L sin theta and by taking sigma f y is equal to 0 we can write n minus mg where is the mg and neglecting the vertical acceleration so if there is no angular acceleration of the model about its center of the mass p and so there can be no moments and we can actually assume that the resultant of t and n passes to the center of the gravity that is the center of mass point p. So the resultant of t and n that means that this angle n tan theta tan theta is equal to tan theta is equal to t by n and that angle tan theta is equal to t by n so we can write that t is equal to n tan theta now by taking n is equal to mg and then substituting from sigma f r is equal to 0 whatever the expression we got t is equal to m omega square into R plus L sin theta for n is substituting m so it is mg tan theta is equal to m omega square into R h plus L sin theta now what will happen is that m and m will get cancelled so this indicates that the ramp angle is independent of the mass so what we get the expression after simplification we get that g tan theta is equal to r omega square divided by 1 plus L by r sin theta so where the r is small r is nothing but r h plus L sin theta so what we can actually do what it means is that this equation once you know these configurations we can actually calculate what is the ramp angle that is theta is there in both left hand side as well as right hand side so we actually need to use the iteration process so that we will be able to get the relationship between theta and omega so as the omega reaches to certain value what we tend to see is that the theta becomes close to 90 degrees and then it remains constant then that means that that is the point what we can actually say that the desired gravity level is actually achieved once the desired gravity level is achieved either we do a test at constant gravity level or at the variable gravity level. So this is the swing in action of the IIT Bombay large centrifuge in action is shown and this view is actually obtained by a camera which is actually located in the center and wherein the camera gives the swinging up of the angle, swinging of the basket so what we can see is that the model actually is swinging up and it remains in that horizontal position as long as you know the machine is actually under rotation. So this is large centrifuge in action at IIT Bombay so once again the view is being shown. Now after having seen the how the machine when it is subjected to rotation about vertical axis it swings up and attains some constant you know the ramp angle. So now in this particular slide the variation of the ramp angle theta with rpm omega for a radius of 4.5 meter radius beam centrifuge is shown here. So what can be seen is that by using that equation this has been founded with Rh is equal to 3.3 meters and L is equal to 0.815 meters. So from the center of the hinge to top surface of the basket it is you know about 1.2 meters so this is actually assumed as 0.815 meters. So once this is calculated what actually it appears is that about 34 to 35 rpm you know then it actually attains so called the horizontality and you know the angle will be very close to 90 degrees but it cannot be 90 degrees why because there is a 1G gravity which is actually acting downwards. So even the what we have discussed is that radial acceleration field when it is actually happening the NG which is acting towards the model and then 1G is actually acting downwards. What we have is that the resultant gravity level which is the one which we need to consider this is nothing but root over N square plus 1 into G. So if you say that N is equal to 100 there are 100 square plus 1 root over which is almost equivalent to 100 point you know some decimals into times G so which is regarded as 100G. And this is for a you know typical 1.1 meter radius set diffuse where Rh is equal to 0.74 meters that is from the center of the shaft to you know the center of the hinge and L is actually assumed as 0.2 meter here. So this implies that the smaller the radius it actually takes longer RPM to become the platform to be horizontal so it takes about 210 to 220 RPM to actually acquire an angle of about 89 degrees or so. So it actually implies that if the smaller the radius the beam centrifuges can only be used above a certain omega value up to that time you know we cannot actually claim that the gravity conditions have been achieved and these are actually called as pseudo gravity conditions. So the one point we need to note down is that the mass is actually independent of that when we are calculating the ramp angle the mass is independent from the you know the discussion whatever we had. And second issue is that the beam centrifuges can only be used above a certain omega so where in we can actually you know get the desired you know RPM but beyond which we can actually can be used for example if radius of 1.1 meter is used so that means that you know the centrifuge test cannot be performed anything less than 200, 210 RPM. So you know then you know the scaling glass which we are actually going to discuss it cannot be applied appropriately. So before discussing about the scaling glass let us look into the you know these simple assumptions which are actually involved and the earlier the scaling glass were actually reduced by Crochet in 1988 and it is also based on the you know the definitions the fundamental definitions the you know the scaling glass have been reduced by Crochet. Some of the fundamental assumptions which actually have been put forward is that the soil can be treated as a continuum that means that different parts of the model are many times larger than the soil grains that means that if I have a length of the container the length of the container should be many, many times larger than the soil grains. Similarly if you are actually having a footing the footing should have reasonably large when you compare with the size of the grain. So that actually will you know the soil can be treated as a continuum this is one of the conventional assumption in soil mechanics and the soil properties are not affected by the change in acceleration because we said that in centrifuge basal physical modeling the we in order to acquire the identical stresses in model prototype we said that gamma m is equal to n gamma p. So where in gamma is equal to rho into g with that you know with gm by gp is equal to n if you are able to achieve with identical soil as that in the prototype what we said is that the gamma m becomes n gamma p. So that means that you know the soil properties are not affected by change in acceleration except the unit weight which is actually determining the result of the self weight. So the force of the explanation for this assumption was actually given by Schofield in 1980 it is said that the force of the gravity acts at the center of the mass of each atom and does not significant affect the electron shells which determine the all material properties other than the self weight. So only except the self weight the rest of the properties are assumed to be not affected. So this assumption you know ensures that you know identical soil properties like c and phi and that also we actually have deduced from the you know the dimensionals or the theory of the models that they should be identical in model and prototype. So these are the some fundamental assumptions as far as the scaring loss in centrifuge modeling is concerned. Now coming to the similitude in geotechnical engineering where in the like in conventional fluid mechanics where in we actually have the similitude definitions like linear similitude kinematic similitude and dynamic similitude. So where in the linear similitude in the sense that whatever the dimensions which are actually there in prototype they have to be scaled by 1 by n times and the factor is constant for all the dimensions that means that if I have a length and breadth and height then it is length in model and prototype which is also called as a length scale factor nl and breadth scale factor bm by vp and height scale factor hm by hp has to be reduced by 1 by n and if the length and breadth are actually reduced by 1 by n then area which is nothing but am by ap is equal to 1 by n square and similarly the volumes are reduced by 1 by n cube that is that vm by vp is equal to 1 by n cube. And so this is as far as the linear similitude is concerned and similarly kinematic similitude is concerned where in we have vm by vp is equal to nv that is nothing but the scale factor for velocity and similarly the accelerations am by ap has to be equal to na that is the as far as the scale factor for the accelerations concerned. So velocities and accelerations they also have to be scaled and then the similarity need to be maintained. Similarly for an identical soil in model prototype with rho m is equal to rho p the scale factor for mass and volume they are equal that is nothing but nm is equal to n suffix m is nothing but the scale factor for mass is equal to n suffix v is equal to 1 by n cube. The dynamic similitude is nothing but fm by fp is equal to nf wherein we can actually say that this in the case of dynamic similitude even if you have different forces like seepage forces or let us say you know creep forces or some viscous forces or if you are having some dynamic forces or if you are actually having some weight forces what are may be the time of type of the force what dynamic similitude says that the dynamic similitude says that it actually should have a constant scale factor for the force then only we can say that the dynamic similitude is satisfied. So for identical effective stresses in model and prototype and what we say is that the n sigma dash that is the scale factor for effective stress n sigma scale factor for tortuous stress nu the scale factor for pore water pressure where we have sigma dash by sigma p dash is equal to sigma by sigma m by sigma p is equal to um by up where u is nothing but the pore water pressure in model and prototype. So this pore water pressure is nothing but defined as gamma w into the height let us say h small h so you know when we have the gamma w in model is n times gamma w in prototype then even the with a reduced column of water height of water what we actually get is that identical pore water pressure as that in the prototype when you have the identical pore water pressure as that in the prototype and when you have the identical stresses total stresses in model and prototype are identical then the effective stresses are identical when we have the effective stresses identically is achieved then the soil stress strain behavior or the soil shear strength is actually simulated accordingly. So now here the scaling factor for the loss in centrifuge modeling as we have said that from the linear similitude point of view length and breadth and height are reduced by 1 by n. So in that case the displacements they can be due to settlements or a piled deflections or there can be you know lateral deflection of a wall and these displacements have to be 1 by n times as that in the prototype that means that if you are having a phi n and mm displacement and at 50g that means that it is about only 10 mm displacement it should record. So that means that the delta m is equal to delta p by n. So as we have scaled down the length dimensions of the prototype by a factor n in the centrifuge model the scaling for the settlement would be you know 1 by n. That means that as we have scaled the length scale factor by 1 by n times the displacements have to be also 1 by n. So the settlements in the centrifuge model are 1 by n times that in the prototype. So similarly the areas which are actually involved let us say a m by e p is equal to 1 by n square. So let us if you look into you know something like 0.72 meters by 0.36 meters area at 40 gravities. So this actually represents about 450 square meters of area in the field conditions. So this actually represents even the small areas when they are actually tested at higher gravity level. So there is a possibility that they represent the large areas under consideration. Similarly the volumes which are actually involved vm by vp is equal to 1 by n cube. Then the we have already proved that the stresses are identical in centrifuge in model and prototype. We said that the stress in the model and prototype are identical because of the enhancement of the unit weight. So with that sigma m by sigma p is equal to 1. And we also said that as because as we are scaled down the maintaining the linear similitude of you know length dimensions then the displacements or settlements have to be 1 by n times that of the prototype. So the strain you know which is caused the strain in the body is actually caused due to you know the displacements of the particles or the total displacement of a body. That means that you know it can be within the soil mass means this can be due to the crushing of the soil particles and you know the bending of the soil particles in case of clays. So the strain which is actually nothing but it can be due to some distortion or due to crushing of the grains or due to movement of a rigid body. So if you define the strain as dsigma by sigma and with now as sigma m by sigma p is 1 by n. So that small change in the displacement that also has to be small that is dsigma in model is equal to dsigma p by n. So with this what we can say that you know with sigma m by sigma p is equal to 1 by n and dsigma in model is equal to dsigma p by n. So epsilon p epsilon m and epsilon p is equal to 1. So this ensures that identical stresses and strains in model and prototype and this actually has got you know the relevance as far as the stress strain behavior of the soil is concerned. So this we have discussed that if you are actually having you know model behave the model 1g model test conducted at a small stresses. So initially the model actually will have you know one you know very low stresses because you know when we have the low stresses at low stresses the soil will actually have higher stiffness and though the scale factor from the dimensionals says that delta m by delta m is equal to delta p by n by the virtue of the similitude modeling but what actually you know we measure is that in the model test is that you know the less settlements in the for the structure being constructed. But when we look into the real prototype stress conditions where the stresses are high and as the strains are actually high the stiffness of the soil actually falls. So with that the real prototype conditions when it is actually there with the real stresses and then what actually happen is that the stiffness is low so the settlements are actually large. So this actually shows that you know the soil behavior is actually highly nonlinear and plastic. So in order to capture you know this you know the you know the identical behavior the creation of the full scale stresses and strains in the small scale models is very very important. This slide we have already discussed but you know to bring the relevance of the you know as we are discussing about the stress and strain modeling we actually trying to bring this discussion once again. Now let us look into different aspects of the scaling loss in centrifuge modeling. So here like we have a force, work and energy. Now the considering the basic definitions of Newton's second law of motion and we can say that a force F acting on the body we can actually define as F is equal to MA where M is the mass and then A is the acceleration. So force acting on the body in the direction of the acceleration. So in order to get this scale factor for force FM by FP is equal to M by MP divided into AM by AP and with M by MP is equal to 1 by N cube because we have used the same soil as that in the prototype with maintaining rho M by rho P is equal to 1 and reducing the volume VM by AP is equal to VM by VP is equal to 1 by N cube, the M by MP is equal to 1 by N cube and enhancing the acceleration by N times then it is AM is equal to N AP. So what we get is that FM by FP is equal to 1 by N square. So this actually has the practical relevance of this is that if you are having say 3000 kilo Newton of force in a 50 gravities centrifuge test the scales down to be only 1200 Newton's that means that a 3000 kilo Newton force in a 50 G centrifuge test scales down to only 1.2 kilo Newton's. So for example this is a case for example if you wanted to do the lateral load capacity of a large Cajun foundation then you need to develop a 3000 kilo Newton's in the sense that 6000 kilo Newton restraining capacity can clutch many times I know this type of situations are you know difficult to achieve in the field. And this is another advantage of a centrifuge test that as we can actually make actuators to load piles retaining walls or slopes or embankments or dams and forces that need to be applied by these actuators are relatively small. So you know these are another advantage of the merit of the centrifuge actually if you look into it. We can make the actuators to apply very small forces but these forces correspond to the forces which are actually you know correspond in the by applying the appropriate scaling conditions as per the gravity level being considered. So this is you know the score factor for the force is concerned. Similarly consider you know the work the basic definition of the work done W is nothing but the product of the force through a distance moving through a distance d that means that W is equal to f into d. So the work energy the work done in moving the moving the body by a force m through a distance d is given by W is equal to f into d where Wm suffix m by Wm suffix p is equal to fm by fp into dm by dp. Now having known with fm by fp is equal to 1 by n square and dm by dp is equal to 1 by n dm by dp is nothing but the distance from the linear similitude point of view it is scaled by 1 by n times. So with that you know when you substitute back what we get is that Wm by Wp is equal to 1 by n cube. So the scaling law for work done suggests that the work done in a centrifuge model is relatively small compared to that in the prototype. The work done is nothing but Wp by n cube that means that if the work done in the centrifuge model for the same energy is actually is you know is compared to is very small. So this is also another advantage as far as the centrifuge is concerned. So the scaling law for the work done suggests that the work done in the centrifuge model is relatively small compared to that in the prototype. Now after having discussed about the work as far as the definition of the units and is concerned from the physics definition is concerned work and energy are equal but consider the definition of the potential energy. The potential energy Pe which is equal to normally expressed as energy lost by falling mass m through a height h. So wherein Pe which is nothing but we can actually say that Pe is equal to mgh. So we can say that Pe in model and Pe in prototype which is nothing but we can actually write it like the product of mm by mp and gm by gp and hm by hp. So with mm by mp is equal to 1 by n cube and the acceleration which is n times that is gm by gp is equal to n and hm by hp which is equal to 1 by n. So we can write that the potential energy in model and prototype is 1 by n times. So here which actually says that the centrifuge model can offer a very effective way of investigating. See for example when you are trying to see the energy which is actually released due to some explosion or a blast load effect on certain geotechnical structures or buildings or earthen dams or dams or retaining structures without the need to conduct these at full scales many times they are expensive and they are damaging to the environment. So in such situations centrifuge modeling offers an effective way of investigating the effects of these explosions on these buildings wherein the energies are actually modeled which is you know which is nothing but 1 by n cube times that of the energy of the prototype. So PEP which is nothing but n cube times potential energy. Suppose this concept is something like if you are doing a type of ground improvement technique like you know dynamic compaction wherein a known weight is actually dropped over a height h with a potential energy you know h. So wherein what we actually get is that you know the 1 by n cube. So this concept of you know the energy modeling can be used for understanding some problems like the liquefaction mitigation measures or a ground improvement methods like dynamic compaction wherein we will be able to simulate with identical stresses and strain fields we will be able to see the response of a particular model applied with different energies where the different parametric studies can be done. Similarly the kinetic energy, kinetic energy which is also again said as half mv square and with that kinetic energy also is modeled as 1 by n cube wherein half mv square when this mass is actually is mm by nb is 1 by n cube and wherein velocity at present as far as the dynamic velocity or the motion is concerned which is Vm is equal to Vp and wherein with that what it actually tells is that the kinetic energy in model prototype is 1 by n cube. So with that what actually happens is that now the work energy and you know potential energy and kinetic energy it is actually modeled as 1 by n cube. So let us say that we are actually interested in studying impact load on the you know on a particular foundation. Suppose a ship impact load on the pile foundation wherein the certain weight of a model is actually the moving weight is released and to make it to hit the you know the foundation. So that actually marks the modeling of the impact on the particular foundation. So in this particular lecture what we try to understand is that you know different you know the scaling loss the basic scaling loss which are actually required for centrifuge modeling and then you know further we have you know different forces like seepage forces or we have some dynamic forces or we have some weight forces or with the weight forces are basically they are you know due to body forces or due to the structurally externally applied loadings that is nothing but you know application of a load to your footing or application of load to a lateral load to a foundation. Then in that situations this actually scales down you know we are required to understand the scaling loss. Then you know in this particular lecture we actually try to look into the basic you know parameters actually as for 11 to centrifuge modeling the scaling loss. Then we also have seen different machine configurations and we said that different type machines which are balanced beam and beam centrifuges which are some select machines are actually are you know shown. Then we also have deduced a relationship between ramp angle and omega that is the angular velocity of the model. Then we said that the beam centrifuges can be used depending upon the type of the configuration of the equipment. The smaller the you know radius then they can be used only up to the RPMs will be large to attain this you know the so called 89.5 or a 90 degrees horizontal plane. So you know that means that they can be used only beyond that certain RPM. Otherwise you know the pseudo graph up to that stage actually what it actually said is that the pseudo gravity conditions provide. Then we also have discussed that you know the typical beam set the drum centrifuge which are actually available and how you know the model preparation can be done by using sands in a drum centrifuge actually was introduced.