 Let me start our today's lecture for this NPTEL video course on geotechnical earthquake engineering. Currently, we are going through module 9 of this course which is seismic analysis and design of various geotechnical structures. So, within this module in the previous lecture what we have studied let me do a quick recap. So, in the previous lecture first we have discussed about seismic stability of finite soil slopes. In that first we started with basic concept of numeric sliding block analysis which was proposed in 1965. It is available in the journal Geotechnic that is how a sliding block mass can tend to slide through a sloping ground and how to determine the factor of safety considering this additional pseudo static inertia force. So, in terms of factor of safety once we equate it with respect to 1 then whatever the acceleration we are getting the seismic pseudo static acceleration that is referred as yield acceleration. So, that we have seen this is the equation to obtain the factor of safety against any movement of a soil block and how to estimate the yield acceleration coefficient in this fashion. From the yield acceleration coefficient why it is important we mentioned that if the slope is having factor of safety less than 1 in that case it is not a stable slope. That means it is going to get displaced and how much is the displacement of that slope that we can find out from this parameters that is whatever is the excess acceleration compared to that yield acceleration value that will be relative acceleration which is responsible for the movement of the slope when factor of safety is less than 1. So, to get this yield acceleration we equate this factor of safety to unity or 1 then we obtain this a y value and this a value is given at any particular site or wherever you are going to do the design. So, this is an input seismic parameter from this relative acceleration you can find out the relative velocity by integrating it over the time period for which this acceleration exceeds the yield acceleration and you can find out the displacement further integrating the velocity over that time interval. So, we have seen this is the variation of factor of safety for any finite slope of inclination 20 degree and different phi value of the soil 20 degree, 30 degree and 40 degree as we know for phi equals to 20 degree factor of safety will always be less than 1 because the slope inclination is also 20 degree. So, for stability we have discussed phi value must be greater than the slope angle. So, for higher phi values you will get the factor of safety more than 1 at static condition that is when k h equals to 0, but as k h increases the factor of safety keep on decreasing. So, that is the critical factor of safety and from this chart you can find out corresponding to factor of safety value equals to 1 what are the yield acceleration values for phi equals to 30 degree for phi equals to 40 degree and so on. So, beyond this point if acceleration exceeds you will get the relative displacement which can be calculated as I have mentioned just now. So, this is the way how to calculate the relative displacement as I have already discussed in previous lecture suppose this is your A y value. So, whatever acceleration is exceeds that A y value for that shaded zone in this figure whatever is the black shaded portion over that time interval you integrate it and get the velocity further you integrate it get the displacement only during that time interval and as displacement is additive it remains and keep on staying at that value. Then we discussed how to estimate the factor of safety of a finite soil slope using the vertical slice method and the pseudo static approach of seismic acceleration which is available in this A C geotechnical special publication by Chaudhary, Basu and Bray and these are the various soil parameters we have also noticed and learnt in our previous lecture that a phenomenon called shear fluidization can occur even in case of dry cohesion less soil which is mentioned by Richard's et al for that stability criteria phi value should be greater than tan inverse of k h by 1 minus k v and if it is a sloping ground then the slope angle also gets added to that that has been proposed by Sharma in 1990. So, this is the equation for a stability of a finite slope that phi value should be greater than beta plus tan inverse k h by 1 minus k v. We have also seen how this dynamic factor of safety varies with respect to not only the horizontal seismic acceleration coefficient, but also with the vertical seismic acceleration coefficient for different input values of soil friction angle and for a given slope angle. Then in our previous lecture we also discussed and learnt about the concepts for the seismic stability of tailing dam and what are the extra safety measurements we need to consider for tailing dam design that also we discussed compared to earthen dam design because in this case in the tailing portion you are storing mostly the waste material. So, it should not cause any environmental hazards also. So, safety of the dam needs to be fully ensured even under the seismic condition and this picture shows how the upstream method of construction and failure of dam is possible and what are the devastation area which it can get affected like this. Then as per the Indian seismic design code for IS 7894 of 1975 it suggests the pseudo static approach to be used for the seismic design of this earthen dam and there are basic two methods can be performed like one is circular arc method another is sliding wedge method. Sliding wedge method is nothing but the numerous block method and circular arc method is similar to what I have mentioned this modified swedish circle method or using the vertical slice equilibrium approach. So, this is the factor of safety for circular arc method using the pseudo static approach and as far as IS 1893 of 1984 is concerned based on the assumption of the portion of the dam and rupture surface the analysis needs to be carried out as outlined over here. Then we discussed thoroughly in our previous lecture about a case study this is for a real problem real field problem as well as another model problem for academic purpose and another is actual field implemented problem. So, this is the field implemented problem which was analyzed by Chakravarty and Choudhary in 2011 this is the ASC geotechnical special publication number 211 this tailing dam is proposed to be constructed at the eastern part of India under seismic zone 2 that is the most surface zone as far as Indians seismic zonation is concerned as per current IS 1893 2002 version. So, in two stages of height it is proposed to be constructed one is 10 meter another is 28 meter fast phase is 10 meter and it will be constructed by downstream approach or downstream method. This analysis is carried out using the finite difference based software FLAC 3D as well as the analytical method of pseudo static as well as pseudo dynamic method of analysis was carried out for seismic design of this tailing dam and these are all input parameters for the dam sections and the tailing portions. So, for the stability criteria we mentioned that we considered seven different possible combinations which can arise for this tailing dam these are all seven different combinations. So, for each of them we have to ensure the stability that this factor of safety should be greater than 1.15 or displacement should be within the permissible range. So, for that we used the tuft earthquake as the input motion in the FLAC software to provide this seismic excitation and after giving this seismic excitation of acceleration versus time history we get the what is the maximum displacement in static condition. So, this is under gravity loading and these are the under seismic conditions for the first phase of tailing dam for various cases or various combinations of loading as we have already discussed. This is the FLAC displacement contours for the static case that is under the gravity loading condition what is the displacement behavior at various portions of this tailing dam also under the seismic loading condition after the 30 seconds of earthquake shaking how much will be the displacement vector in FLAC these results are available for the first phase of the dam. And the output shows that at different height of this tailing dam like if we consider the dam base suppose this one at ground level then at various height that is 5 meter and 10 meter at different height we have obtained what is the acceleration versus time history. So, by doing the ground response analysis so by doing ground response analysis you can get acceleration time history in FLAC also like this and peak horizontal acceleration is found to be at 5 meter of height is this much whereas at 10 meter height it is obtained to be like this much. So, that automatically shows when the seismic earthquake acceleration travels through this height of this tailing dam there is an amplification of about 4 times from the bedrock motion or the at base level whatever excitation was provided. Also we need to calculate the fundamental time period of the entire system or the entire tailing dam to calculate that FLAC 3D analysis directly gives us the results of fundamental time period this is for the first phase of the dam this is for the second phase when it is 28 meter height what are the fundamental time period. Also IS code the BIS 1893 1984 version that also gives us this formula to calculate analytically what is the fundamental time period of any structure like tailing dam like this. So, for first phase this is the value for second phase this is the value which are very well comparable as obtained in the FLAC 3D analysis as you can see. So, when somebody is starting any design they should either you follow this IS code and in addition to that I will suggest they should do some numerical analysis rigorous dynamic analysis and some analytical solution also for the stability and dynamic behavior of the dam. This is under the factor of safety and yield acceleration values for the seismic slope stability analysis using TALREN 4 another software. So, FLAC 3D, TALREN 4 and slope W these are common slope stability software which can be used. Now, for seismic analysis these inputs k h value and k v value we have to provide to get the factor of safety because in TALREN and slope W you cannot do the dynamic analysis completely you have to do a kind of a pseudo static analysis that is why you need this input value of k h and k v otherwise in FLAC 3D you can directly do the dynamic analysis you need not to give this coefficients you can directly give the acceleration time history you can get the detailed dynamic response. So, in previous lecture we have also learnt another important criteria for the tailing dam design which is the liquefaction behavior or how this tailing dam behaves during a soil possible soil liquefaction during earthquake. So, the assessment of this liquefaction potential we discussed which is given by Chakravarty and Chaudhary 2012 paper. So, this is for the first phase of the dam this is for the second phase of the tailing dam and these are various points in the tailing portion why we have chosen the tailing portion as I said it is mostly in the unconsolidated state or in the loose state because you dump the waste material in the water. So, that is why there are high chances of getting this material liquefied and that is why we carried out the liquefaction potential not only that you have to do for the foundation soil also that you have to make sure. So, in terms of maximum value of pore pressure ratio if this pore pressure ratio R u equates to 1 means it is going to liquefy if it is less than 1 then it is not liquefying but we should not have a very close to 1 value. So, that is another criteria then there is always a chance that it may get liquefied if there is a input value of soil parameters or a site condition anything gets changed. So, we ensured that what are the values of possible liquefaction potential to make and recommend the safety issues of this tailing dam. This is the behavior of first phase as well as second phase of dam under liquefaction condition in flag 3D this is the typical output you will get for liquefaction analysis at different height and locations. Also, we had conducted the seismic slope stability analysis of this tailing dam using both pseudo static approach as well as the new method of pseudo dynamic approach we have elaborated in couple of previous lectures and this is a just assumed tailing dam section this is not the actual one the previous one was the actual one this is the assumed one. So, these results are available in the publication Chakravarty and Choudhury 2013 in the proceedings of national academia of sciences spring of publication this is the volume and page number. So, the factor of safety as we know is nothing but ratio of resisting force by driving force. So, resisting force expressions and driving force expression from that the factor of safety using both pseudo static and pseudo dynamic we will get the results and that if it is more than 1.5 then we will call it as safe it is if it is going to be less than 1 we need to find out the displacement. So, coming to our today's lecture we will start today with another subtopic on this module which is another important subtopic is seismic design of pile foundation. This is very important because for several high rises and very important structures and mostly in the urban areas where there is a scarcity of space and you do not have the luxury to get the buildings are new construction constructed horizontally, but you have to go vertically up and up like places like Mumbai plus is like New York plus is like Tokyo. So, all these places there is a space crunch. So, always you need to go in the vertically up direction and hence pile foundation is the only solution. So, now in the introduction as we know where pile foundations etcetera are going to get used. Now, for the soft soil and the supporting structures in liquefying soil where pile this pile foundations of the superstructure when they passes through some liquefiable layer then what happens when you are designing the capacity of the pile suppose you have taken the skin friction component also to calculate the capacity of the pile, but if that portion of the soil gets liquefied during earthquake condition obviously it is not going to provide any kind of frictional forces or any kind of resistance. In that case you have actually in the design calculated an overestimated value which is not exactly acting at the site that causes several time the failures of the pile. So, this is one of the major criteria why the pile foundations fail at several earthquake conditions around the wall in big earthquake even including the various Japan earthquake even our Indian earthquake also during Bhuj earthquake in Ahmedabad several buildings collapsed and the pile foundation got damaged even earlier when in the introductory lectures of this course I have discussed that for a Nigata earthquake for Kobe earthquake various earlier Japan earthquake several pile foundations got failure. So, those are basically or majorly due to the liquefiable strata which was lying in between and that reduces the capacity or strength of the pile which was estimated during the design. So, we should know how much extra bending moment how much extra displacement is going to come for those type of pile when you are going to design or when you are going to construct any pile which is a having a possibility to pass through some liquefiable layer. In addition to that what are the other possible way that pile foundation get failed because during the seismic loading the major load is nothing but lateral load as I have already mentioned mostly the lateral force comes into picture that is the major one of course there will be a vertical component as well. So, that lateral seismic inertia force that will cause extra lateral loading on this pile foundation and if your pile is not designed properly to take care of those extra lateral load obviously it is going to bend excessively and finally fail or it going to displace excessively and finally fail. So, these are another way or mode of failure for the pile during the earthquake condition. These two are the major things somebody is trying to design any pile foundation needs to take care of. So, let us discuss about the design philosophy etcetera. Before that let me show through this picture it is photo courtesy from Nisi like pile foundation of million dollar bridge of 1964 Alaska earthquake in USA. This is the Shoa bridge of 1964 Niigata earthquake in Japan which earlier I have mentioned in one of the introductory lecture. This is the pile tanks after 1995 Kobe earthquake you can see the failure of pile foundation can make several important buildings or structures to collapse. So, there is a paper by Madabushi et al in 2010 this paper discusses about the performances of pile foundations during various recent earthquakes that is before 2010 like in which cases piles performed very well. So, good performance and bad performance. So, these are the various case history of during Niigata earthquake some of the pile they performed well. So, they have done a study why those structures or why those pile foundations perform well and why others could not perform. So, this is an important paper one can go through and these are the examples where the pile provided or pile could not perform well. So, poor performance during Niigata earthquake and Kobe earthquake for different pile conditions. Also some more poor performance during Kobe earthquake including you can see over here as I have mentioned during Buj earthquake also like in Kandla port area, Harbour area several pile foundations they were totally devastated and damaged and also in Amdabad region several pile foundations were damaged. So, these are the examples of poor performance of pile during earthquake. Now, how this pile behave under lateral load and in in liquefying soil. So, this picture typically shows how to do a force based method of estimation for pile foundation design. This is actually proposed by J. R. A. Japan Road Research Association 1996 recommendation how to do the design of pile passing through liquefiable layer. So, this is an idealization of pile for the design in liquefied soil. Suppose you have a superstructure like this which is supported on various pile. So, this is the pile foundation group of pile and you have at this portion a non liquefiable layer. So, this layer is not going to liquefy. We know how to estimate the liquefiable potential of any particular soil that we all know about it and it is followed by some soft layer or some layer which is supposed to be prone to liquefaction under certain magnitude of earthquake. Then followed by another stiff layer which is non liquefiable. So, basically if you have a end bearing pile you will go and end it up to a strong or stiff stator or even rock, soft rock or dense stator, dense sand. So, in between this liquefiable layer will drastically reduce the capacity of the pile if somebody has considered the friction of this portion of the soil on the pile capacity in the design and the additional horizontal load will also come into picture on the pile due to the earthquake condition. So, how to take care of this? So, JRS proposed that whatever is the surcharge passive earth pressure is acting consider 30 percent of the over burden pressure is acting during the liquefiable layer. So, this is a design proposition, but later on many other researchers has mentioned this is not always correct and one needs to do a case specific analysis for individual ground response and the soil profile is concerned because this soil profile you should know through which your pile is going to get constructed and also the dynamic response at each layer. So, I will go through that detail very soon. So, the seismic analysis of pile foundations in liquefiable soil, the initial researchers they mentioned about Winkler type model as we know from the basic concept of any soil structure interaction code that soil structure interaction courses always discuss or start with basic concept of Winkler beam model. So, Winkler springs are considered for below a foundation analysis when we considered the foundation how it interact with the basic foundation soil. So, foundation with respect to soil we calculate through the Winkler springs. So, that concept of Winkler spring model is extended for the pile foundation also in liquefiable soil by various researchers like Winkler type model has been developed by Kagawa in 1992, Yawa and Nogami in 1994, Fuji et al in 1998, Liantha Perina and Paulus in 2005. So, for piles in the non-liquefiable soil, so when you do not have any liquefiable layer, but still it is under the earthquake condition or seismic condition you still have the additional horizontal load which needs to be considered. So, these are the researchers who used like Abrage and Chai in 1995 and Tabeshan Paulus in 2001 had developed pseudo-static approaches. As we know professor Harry Paulus a very famous pile engineer and professor and practitioner who developed the theory of pile foundation extensively as we read for even the pile foundation design and analysis even in the static condition the book by professor Paulus and professor Davis Paulus and Davis of 1980. He professor Paulus extended with his several researchers and students how to extend the static analysis to the pseudo-static concept of analysis under the seismic condition. So, these are very basic work and fundamental work in the area of earthquake engineering for the pile foundation design. So, one must go through these details to understand the basic concept of behavior of this pile foundation under pseudo-static seismic loading condition. Liam Patherina and Paulus in 2005 developed this pseudo-static approach which has two solution stages. So, carry out the ground response analysis that is their recommendation. So, you can see the importance of ground response analysis because at specific site you will get different ground response. You should have different amplification criteria, you should have different acceleration versus time history at different level etcetera should be known. And pile is analyzed as non-linear beam on elastic foundation considering the both kinematic approach and inertial interaction. So, now let me explain you when any pile foundation is subjected to an earthquake loading. There are two cases or two combinations of loading will arrive at the pile foundation. What are those things? One is kinematic condition, another is inertial condition. Inertial condition as we have already learned what is inertia when there is any seismic force, the seismic acceleration times the mass involved in it gives you the seismic inertia force. So, that is the inertial component which of course your pile when it is subjected to any earthquake excitation it is going to get experience. But what is the kinematic behavior like when the pile and soil they interact in between. So, soil also is subjected to some kind of earthquake acceleration. So, that acceleration in the free field we call when suppose there is no structure exist. So, that is nothing but free field condition. The soil is going to displace or going to move. So, that movement of the soil along with the movement of the pile how it is getting interacted that gives us the movement or kinematic interaction. So, this kinematic interaction along with the inertial interaction needs to be considered combined lead to get the combined effect of this earthquake forces on the pile foundation and in the design we need to consider both the aspects of kinematic as well as the inertial loading conditions in the design of pile foundation. So, let us look at this picture once again this is again from the j r a proposal like soil liquefies it loses its strength and starts flowing and dragging with it any non-liquifiable crust above it. So, this is a non-liquifiable soil then liquifiable soil then again non-liquifiable soil. So, when there is free field soil deflection. So, this is the free field when there is no structure available. So, that gives us the kinematic interaction as I was telling and on this inertial portion or the structural component whatever the seismic load is acting that gives us the inertial movement or inertial displacement or inertial bending. So, this is the deformed shape of the pile you can see there will be a difference between these two movement. So, that is why the combined movement needs to be considered and there will be a dragging because this portion of the soil when it gets liquefied. So, between two non-liquifiable layer two non-liquifiable layer if one liquifiable layer is there after liquefaction because of the slope of the ground etcetera it will start flowing. So, lateral spreading as we have already discussed after liquefaction after effect is immediate after effect is lateral spreading. So, once the soil entire things flow out the fluid portion flowed out what will happen there will be a relative movement between the upper and lower non-liquifiable layer. So, that is why it will try to drag the structure which is constructed already inside that liquifiable layer because when it is moving a from upper gradient to a lower gradient. So, that is why that needs to be considered when we are designing this pile foundation under earthquake loading and in liquifiable soil. So, the concept of pile failure under earthquake is given by Ishihara, Professor Ishihara in 1997. He gave the details about two basic concept one is called top down effect another is called bottom up effect. So, what is top down effect like at the onset of shaking inertia forces are transferred to the top of the pile and then it goes to the soil. So, this is one way of approach of analyzing it another bottom up effect is seismic motion had already passed the peak and shaking may still be persistent with the lesser intensity and therefore, the inertia force transmitted from the super structure will be significant. So, this is from the bottom up effect under such a loading condition the maximum bending moment induced by the pile may not occur near the pile head, but at a lower portion at some depth and this is referred as the bottom up effect. So, this picture once again the JRA method as I have already discussed. Now, let us look and understand what are the basic failure theory of this pile foundation under earthquake loading. Let us look at this slide. So, this failure theory of pile foundation was proposed by Tokimatsu et al in 1998. Professor Tokimatsu gave this details like if you see this is your super structure which is constructed on pile. Now, during shaking before the soil gets liquefied that is before your soil gets liquefied when just the earthquake came. So, inertia force is acting right. So, pile is getting extra bending moment because of extra lateral load. Now, during shaking after liquefaction when the soil gets liquefied there will be a displacement of the soil also. So, this will be still inertia force will act and soil gets displaced. So, what will be the combined effect? So, lateral movement after earthquake and liquefaction will be something like this. So, it will have both this bending moment due to the extra loading lateral load due to earthquake as well as it will have extra bending moment due to this ground movement for the liquefaction of the soil and movement of the soil. So, those aspects needs to be considered which are discussed over here. So, if you are interested you can go through this review paper also Choudhury et al 2009. It is available in the journal, Proceedings of National Academy of Science Springer Publication section A physical sciences. So, in this paper this is the February issue of this 2009 issue 2. In this paper all the review and discussion about how this philosophy failure theory etcetera holds good for the pile foundation under earthquake is discussed thoroughly. Now, let us come to a case specific design. So, why I am telling case specific design for pile foundation under earthquake condition? As I said already it is also recommended by various earlier researchers like Professor Poulos that for pile foundation design a ground response analysis is a must because you should know how that local soil is going to behave under an earthquake condition which are supposed to come or from the past history earthquake data you can provide at that soil condition knowing the local soil site condition. So, it is not a generalized case it should not be used for important structure. Of course, small structure you can use it or less important structure you can use it, but for important structure you should never use a gross design approach you should go for case specific individual design approach. So, that I am going to discuss now. So, let us go back to the work of Dr. V. S. Phonikant who did PhD at IIT Bombay 2011 he completed his PhD I am referring here his PhD thesis he did his PhD under my supervision at IIT Bombay. So, we have already discussed that for various borehole data at different sites at Mumbai city like Mangal body site this basic soil information data soil layers SPTN value were collected and the dynamic soil properties later on were obtained. From that you will get the equivalent linear ground response analysis to do that the important input parameters you require is the modulus reduction curve right which is nothing but G by G max versus cyclic strain. Also the damping curve you require damping curve versus cyclic strain for your equivalent linear analysis. As I have said we used deep soil software deep soil version 3.5 was used that time with all these input values like shear wave velocity is important parameter needs to be given unit weight also needs to be given layer wise damping ratio reference person strain with which you start and then converts the solution we have discussed these things already in the equivalent ground response analysis earlier in one of the module. This is a snapshot of the ground response analysis how it is done in deep soil like you can go for either linear analysis or equivalent linear analysis or non-linear analysis. So, here equivalent linear analysis has been chosen also you can either use total stress type effective stress type etcetera and you can give your input soil parameter layer wise. So, this is the basic layer wise information which you are getting from your borehole data. Once you get and do the analysis you have to give the input earthquake motion. We have already mentioned for ground response analysis we use this Bhuj earthquake motion as well as the Loma Prieta Kobe earthquake motion. And finally, the output were obtained for different borehole locations at different level like what is the amplification. So, this amplification compared to the bedrock level to ground surface how much this horizontal acceleration is going to get amplified we had also seen that like this is another picture of different acceleration versus period spectral acceleration response and the Fourier acceleration spectra versus frequency curve for different borehole data. Also this is the depth wise variation of the amplification right that also we have already discussed earlier in one of the module. And this gives us the acceleration time history at ground surface for different borehole location like Mongol Bari, Valkeshar, BJ Mar. So, how the model study has been developed. So, this is an analytical method which has been proposed by Dr. Fannikant in his PhD work. So, this work suggest this publication is getting available now Fannikant et al 2013 hour publication which is available in international journal of geomechanics published by ASCE USA. So, you can search in ASCE currently it is available online very soon the paper will be published it is published accepted available online, but the hard copy issue will come very soon. So, this is the single pile model he did the analysis only for single pile he started with. So, a single pile which is passing through suppose a typical layered soil and among that layered soil let us say this one is non liquefiable layer, this middle portion is a liquefiable layer and again a non liquefiable layer. So, they are corresponding length or depth of the soil layers are l 1, l 2 and l 3. So, l 2 is the liquefied layer depth of the soil. So, that depth of l 2 has been compared or the variation or parametric variation of this l 2 over the entire length l of the pile has been done in our analysis for various soil site at Mumbai using the local soil condition and different inputs seismic acceleration how this analysis is obtained for the single pile foundation design in terms of bending moment and deflection. As we know basic design parameters for pile foundation we should get bending moment profile along the depth also the deflection profile of the pile along the depth. That is what we do in the static case also since seismic case also similarly we need to find out. So, this is the soil pile analysis considering the ground deformation using finite difference technique. He used the final difference technique like he subdivided the entire pile into a number of small segments in number of segments and different node points. So, there will be two imaginary node upwards here and two imaginary node downward here as we know in the finite difference concept it is used. So, based on your assumed ground deformation where from you get this assumed ground deformation or you can get an actual ground deformation from your free field analysis of ground under a subjected to some earthquake acceleration. So, you know the ground displacement profile. So, from that you can find out what are the behavior of individual sections and how to get that let me show you the basic governing equation which needs to be solved for the basic differential equation for laterally loaded pile in a liquefiable zone. So, all of us are aware about this finite difference approach of pile analysis which is available even in the book by Professor Poulos and Professor Davies. So, Poulos and Davies 1980 book pile foundation it is available the finite difference approach, but it is available for static case only. So, we have extended that in the case of seismic analysis. So, how the extension was done? This is the basic governing equation as we know E i d 4 y by d z 4 equals to minus k h d times y minus y g. So, what is extra here you can see in the basic pile equation considering as a beam. This y g comes into picture under seismic condition in static case only this much portion is remaining. So, y g comes here for this seismic case y this y g is nothing, but your ground displacement that is free field motion and this d is diameter of the pile and k h is the sub grade modulus and y is the lateral displacement of the entire pile. So, compared to your ground how much is the relative displacement of your pile that interaction needs to be considered as you can see z is the depth from the ground surface and E i is the flexural rigidity of the pile as we know and how to consider this k h value that is another important criteria under seismic loading or this liquefiable zone condition. Under static loading condition this k h value or sub grade modulus value we know from the codal recommendations we can get, but this sub grade modulus also drastically reduces or changes under earthquake condition and even in the liquefiable condition as given by Tokimatsu et al. in 1998 that this sub grade modulus changes to k h n with a scaling factor or the reduction factor due to this liquefiable condition which is proposed by Ishihara and Krubrionovsky in 1998. These are the factors range of this SF is 0.001 to 0.01 you can see. So, how much reduction of this sub grade modulus will take place under this liquefiable condition for the soil when there is no liquefaction compared to that. Now, this is the result which Dr. Fonikant in his PhD thesis work he got as I said these are the authors with three authors. Dr. Fonikant myself as the supervisor from IIT Bombay and Dr. G. R. Reddy from BRC the external supervisor because he is a scientist at BRC they should have another external supervisor. This is our international journal of geomechanics paper ASC 2013. This is the profile of the bending moment the result along the depth of the pile. So, we have chosen some pile length and also different length of the pile was chosen as 10 meter radius of the pile as 0.25 meter this is the E value of the pile material. You can see different input motions were given like Bhuj earthquake, Loma Pretha earthquake, Loma Guillory earthquake and Kobe earthquake motion which we have also considered for the ground response analysis. Also you will see these data are only for the only one particular borehole data that is what type of soil is present there like Mangalbari site borehole number one. Like if you want to do for another site obviously the behavior will be completely different even though your input seismic motions may be same. So, any one of this is changing and suppose for this Mangalbari site even this for MBH 1 if I use another earthquake input motion say Taft earthquake or El Centro earthquake or the Sikkim earthquake motion even the analysis will be different. So, you have to take care that this case specific that is why I was so particular about mentioning whenever we are doing any important pile foundation design this case specific analysis needs to be considered, but the approach remains same of course. So, these are the values of this SF you can see SF value 1 means there is no liquefaction. So, under no liquefaction condition these are the bending moment curves see this solid lines or dark lines and this dotted line shows the bending moment under the liquefied condition when SF value is considered as 0.01. So, you can see there is a significant increase of the pile bending moment. So, when you have provided the pile reinforcement and designed the pile probably you have done only for this non liquefiable condition and under liquefied condition it is subjected to several times of that bending moment. So, obviously this is the reason why the piles automatically fails when the soil gets liquefied. Can you see that huge changes in the magnitude of the peak bending moment and the bending moment profile with the depth. So, unless we know about a site condition and from that from expected or earlier historical analysis or seismic hazard analysis if you do not do this liquefaction study for a possible liquefied zone you will end up with getting excessive pile bending and final the failure like this because of this sudden increase or excessive increase of the pile bending moment clear. Similarly, if you look at the pile deflection results and remember these results which I am showing of Dr. Funnican's thesis work these are for combined. So, you can have also the pile deflection in terms of kinematic as well as in terms of inertial that means what are the individual component of soil movement and the pile movement under the seismic inertial condition and the soil structure interaction between pile and soil considering that Winkler beam approach and things like that. So, you can see again here under different depth of liquefiable layer what has been varied over here same borehole has been used same soil has been used same earthquake motion has been used then what parameter has been varied same horizontal load seismic lateral load is used, but the change in parameter was made the thickness of the liquefiable layer. Can you see this L 2 by L ratio that is thickness of the liquefiable layer is 20 percent of the total length of the soil layer or the pile length and whereas here it is or entire 100 percent is your liquefiable layer. So, that is why you can expect when it is entire 100 percent liquefiable layer that total deflection of the pile occurred that is why this line can you see here whereas in other cases you get a pile deflection like this. So, for 20 percent you get a deflection like this for 40 percent there is an increase in the pile deflection like this and so on clear. So, depending on your thickness of that liquefiable layer compared to the entire depth or entire length of the pile you will get how much your pile foundation is going to deflect and at what different depth the deflection will occur like this. So, these are some more results you can see over here these are the combined values and these are the inertial values. What it shows the deflection for this particular case that MbH borehole 1 under this input motion mostly the combined deflection is because of the inertial component. Can you see that? That means here the kinematic component is not that significant, but it is not always true it depends on your local borehole soil condition and input condition also. So, this is a type of understanding that at what site you are going to address which problem like suppose in this site somebody suggest like as a civil engineer you have to propose the remediation technique also. What will be the remediation technique in this case? Suppose some unknown person unknown in the sense those who are not so conversant with the geotechnical earthquake engineering or the pile foundation under earthquake condition what they will propose they will say improve the soil go for ground improvement dynamic improvement etcetera which we have discussed in our previous another course on soil dynamics that is possible that is one way possible. But suppose with that what you are doing you are rectifying the possibility of liquefaction in the soil, but lateral load whatever is coming the inertial component still remains. You are not controlling on that and whereas for this particular site of Mumbai which we are analyzing here Mongol body site here the major component of deflection comes from the inertial portion. So, do you think that anybody suggest here a ground improvement will help too much? No, in that case you have to go for better pile design itself. So, that it can takes care of this additional inertia force, but if you have a soil where the kinematic component is more what you can do probably by going for ground improvement technique or dynamic compaction etcetera you can reduce that component of chances or amount of deflection due to kinematic portion for the pile. So, as an engineer you should give a remedial measure also because people will not only stay at this analysis level or design level they have to implement it. They should know what are the way out they know that it is going to fail, but how to protect it. So, these are the guidelines these are the recommendations one should give when they are going through this type of rigorous analysis analytically as well as numerically and there can be also some validation through some experimental method, but as you know under earthquake condition it is very difficult to carry out the experimental method and reliable value because in field it is not possible to do earthquake experiment on pile because whenever you are going to measure by the time it is all damaged. So, what people do? People use the centrifuge test on this pile under dynamic loading condition. So, dynamic centrifuge test not the static one. So, under dynamic condition what are the extra bending moment what are the displacement available. So, many researchers like professor Bollinger at UC Davis he and his research group has done extensive work in this area professor Tarek up down at RPI he and his research group has done extensive work in this area. So, their publications one can see how the behavior of pile and pile group under earthquake loading condition their bending moment their displacement profile how it has been observed through the dynamic centrifuge test. So, now we can see over here the results you can see the comparison of pile response in liquefiable soil as well as non-liquifiable soil. Non-liquifiable we know all about it. So, under when the soil is there is no chance of liquefying it under different ground motion condition these are your top deflection and peak bending moment. But when the soil gets liquefied see the amount of increase in the deflection see the amount of increase in the bending moment. So, if we take this ratio of deflection of the pile in liquefiable soil to non-liquifiable soil we can call it as a kind of deflection amplification due to liquefaction of the soil and similarly we can call peak bending moment amplification due to liquefaction of the ground you can see there is huge amount of amplification. This amplification of displacement as well as amplification of peak bending moment can occur when your soil condition just change from non-liquifiable to liquefiable under the same input earthquake motion condition. These details are available in the publication by Choudhury et al 2013. This is a keynote paper at Bandung Indonesia conference a pile 2013. These are some more results you can see over here this is the combined effect of deflection versus depth pile depth and this is for combined bending moment versus depth. So, with this we have come to the end of today's lecture we will continue further in our next lecture.