 Let us start our today's lecture for NPTEL video course on geotechnical earthquake engineering. For this video course, currently we are going through our module number 9, which is seismic analysis and design of various geotechnical structures. Before starting our today's lecture, let us do a quick recap what we have learned in our previous lecture. In the previous lecture, we started with the subtopic seismic design of pile foundation, which is one of the most important area in the civil engineering design and practices concerned for the pile foundation design under the seismic loading condition. In that, we have already discussed as you can see over here, that is piles, that is a superstructure founded on this pile foundation group of piles. They are subjected to extra lateral load due to this seismic effect, so seismic inertia forces etcetera. Also, if it passes through some liquefiable soil during the earthquake, that will create additional problem on the pile foundation, which we have mentioned that through several cases the examples of failures of piles are available under the seismic condition in liquefiable soil as well as due to extra horizontal lateral load, which probably was not considered in the design. So, this is the basic recommendations as you can see on this right side by JRA 1996, idealization for pile design in liquefying soil, that is when the piles are going through some layers initially non-liquefiable layer, then liquefiable layer and finally, again non-liquefiable layer. Now, Tokimatsu et al. in 1998 proposed the failure theory of this pile foundations under earthquake loading condition and liquefaction how it occurs. So, this is the first step, that is the behavior of the pile and this is the superstructure during the shaking and before the soil liquefaction occurs. So, piles are subjected to extra bending moment because of this extra inertia forces due to the horizontal load or inertia seismic load right. And in the next stage, stage 2 what happens the during shaking after the soil gets liquefied the ground will get displaced because soil gets liquefied. So, ground displacement occurs and in the third step both of them gets combined that means this inertia force and this is called the kinematic response of the soil that gets combined and the combined effect on the pile will be its pile displacement and bending that is lateral movement after earthquake due to liquefaction due to this inertia forces. So, the details about these responses or failure theory of Tokimatsu et al. is available in the review paper by Choudhury et al. 2009 in the proceedings of national academy of science springer publication section A physical sciences. Many other researchers also had carried out the research as I have already explained to you in the previous lecture in addition to that like professor Shubhamay Bhattacharya. He also did his PhD work and also currently he is working with various of his students who are working on this behavior of pile foundation under earthquake loading condition not only due to this lateral load they considered the vertical load also in their analysis and consideration. Also, professor B. K. Maheshari from IIT Roorkee they are also doing analysis of this earthquake conditions pile foundation using FEM approach that also are available in the literature. So, coming to case specific design for pile foundation under earthquake condition why case specific I have mentioned in the previous lecture because as recommended by Pulas and in various other researchers that whether at particular soil location how much ground amplification site response soil behavior are going to affect the pile design and seismicity needs to be considered that is why case specific design is important. For that I referred in our previous lecture itself that the PhD thesis by Dr. V. S. Funicant this PhD thesis 2011 at IIT Bombay he did PhD under my supervision at IIT Bombay. So, initially he carried out the ground response analysis from various soil site at Mumbai city this is Mangalbadi soil site borehole number 1 typical soil input data we are getting from the borehole data that has been used under different earthquake loading conditions using different input motions like Kobe input Prieta Loma Guillory and Buja earthquake motion to find out how much amplification from the bedrock level to ground surface through various layers of the soil how much amplification of the soil is going to get how much amplification of the seismic response or acceleration is going to occur due to the soil properties through the ground response analysis using deep soil. Now, when we were discussing the pile analysis Dr. Funicant had chosen a single pile analysis behavior of single pile which is passing through suppose these three layers one is non-liquifiable layer this is liquifiable layer of thickness L 2 and again non-liquifiable layer of thickness L 3. So, this parametric variation of the ratio of this L 2 over the entire length of the pile L has been taken care of in this analysis using the concept of finite difference method. So, in this finite difference method the entire pile section has been subdivided into n number of section from the assumed ground profile using Winkler's beam theory. The basic concept of basic governing equation was used for pile in liquifiable zone proposed by this equation the details are available in this journal paper Funicant et al 2013 in the international journal of geomechanics of ASCE USA. So, how this subgrade modulus is changing under the liquifiable condition of the soil that also has been proposed by Tokimatsu et al 1998 and values of this scaling factor are proposed by Ishihara and Kubronoski in 1998 which have been used for the analysis. And finally, the bending moment of the pile this black solid line shows the bending moment under non-liquifiable soil that is when soil is not liquefying and this dotted lines are showing the bending moment under the liquifiable condition. So, we can see there is a huge increase or significant increase in the bending moment of the pile along the depth of the pile from non-liquifiable soil to liquifiable soil and these are the reasons why many cases in practice we see failure of piles after an earthquake is occurring and the soil gets liquefied. So, the details will be available in this paper also the pile deflection we can see as a ratio of this liquifiable layer to the total length of the pile it is changing or increasing from 20 percent to 100 percent you can see the pile deflection is also increasing like this. Now, in today's lecture let us start with another subtopic on this area which is combined pile raft foundation which is in short it is called CPRF combined pile raft foundation under earthquake condition. So, before we address this combined pile raft foundation this is a very advanced topic and very recent one it is still getting developed around the world and people are doing research in this year of 2013. So, this is a very present hot research topic across the world. So, first to understand what is the pile raft foundation let us go through this this pile raft foundation is nothing but also called it is a composite foundation because the pile along with the raft that is most of the time as we know pile above there will be pile cap if you design that pile cap properly to take a share of the loading and the response behavior coming from the soil because of the soil pressure below the raft. So, that combined action of the raft and pile is nothing but the pile raft foundation CPRF. So, its settlement occurs through interaction and load sharing between the two sections differential settlement like raft provide the stiffness again load and it is very economical why it is very economical because in this case you are considering the effect of raft also which is neglected in case of only pile foundation that is we always put the pile cap but we never take the effect of pile cap. So, that is why using this pile raft combined design approach you can get the reduced number of pile which are required for a particular loading. So, that is why it is very economical and one of the pioneering work in this pile raft foundation as you can see in this picture this is nothing but the mesoterm tower in Frankfurt am Main in Germany. So, this foundation is this founded on pile draft foundation again this work has been carried out professor Katzenberg et al Katzenberg in his research group they are doing this work since two decades and you can see over here this construction of this mesoterm tower was done in the year 1988 to 1990 foundation was CPRF height of this building is 256 meter then pull us et al in 2001 here and his research group also has explained a number of idealized soil profile and founded that soil profile consisting of relatively steep clays and relatively dense and may be favorable for pile draft foundation you should remember there are conditions which are required for this CPRF it is not that in all cases you can use CPRF there is a particular ratio of the stiffness of the soil layers from the raft level to the pile bottom level it should be within some range then only you can apply this combined pile draft condition. Coming to next slide you can see over here in Germany in Frankfurt am Main there are several high rises which are founded on the pile draft foundation and the entire work was done by professor Rolf Katzenberg who is the professor at technical university Darmstadt in Germany and he and his research group are working as I said since last two decades more than two decades in this area of combined pile draft foundation and he is one of the pioneering and well established researcher who works extensively and worldwide in this area of CPRF. Another example like Deutsche Bank head office which is again in Frankfurt am Main in Germany that is also there are two towers upper tower and lower tower portions that is also was going through excessive settlement and that was stopped using this hydraulic jack technique etcetera. Then why for Messertram tower this CPRF technology was recommended if you see suppose if somebody uses the only raft type of foundation for this soil remember in Frankfurt am Main typically it is a soft clay it is called Frankfurt clay up to a depth of it is varying between 15-20 meter to up to even 100 meter also. So, this is the range of the stresses going to get established if somebody uses the raft foundation and settlement will be huge that boils down to use of the combined pile draft foundation. So, combined pile draft foundation as I was mentioning a raft is not possible because it gives us huge amount of settlement. So, to reduce the settlement you should go for deep foundation which is pile foundation and what is the pile foundation the problem is it will be very costly because number of piles when you are using more number of piles it will be costly. So, somewhat in between solution is always this combined pile draft if your soil profile and site condition is preferable. So, those are very important guideline and criteria which is discussed in this publication Kazzenberg et al. 2009. I am also a co-author in this paper we did extensive research in this area as well during my visit as a Humboldt fellow at Technical University Darmstadt to work with professor Kazzenberg and his research team. So, to reduce the settlement this excessive settlement and to reduce the cost the intermediate solution or best solution I should say is the combined pile draft foundation for a chosen soil condition. So, this is the basic concept of bearing what it occurs in the CPRF. So, the detail of this analysis will be available in Kazzenberg et al. 2012 this is again another paper where I am also involved as a co-author. So, this paper in this conference paper is available. You can see over here the interaction this is the raft portion this is the raft and soil interaction pressure and these are out external loading loading and moment etcetera. This is a pile you have the pile response from skin friction as well as end bearing. Now, interaction between the CPRF and soil how it occurs you have this portion soil response and this portion soil response. So, you are now considering this raft portion pressure as well as the pile portion pressure. So, in pile foundation design only pile foundation this part is neglected the part of the raft is not taken care of. So, what will be the total capacity of the loading that is nothing but sum of all this responses coming from individual pile. So, if you are using m number of pile you are getting m number of responses and this is from the raft. Also for each pile you are having two components one is end bearing one is skin friction and for raft how you are getting it? You are getting it by integrating it over the area of what this stress in the x y plane fine. So, what are the interactions occurring this zone 1, zone 2, zone 3 and zone 4. Zone 1 represents the pile soil interaction this interaction zone 2 this represents pile to pile interaction one pile to another adjacent pile. Zone 3 represent this zone raft to soil interaction and zone 4 this one represents pile to raft interaction. So, this is a nothing but a soil structure interaction problem or I should say soil pile soil or soil structure pile interaction problem in a complicated manner involving both raft as well as pile through this process. So, in this case simple Winkler beam approach will not work properly. So, this is the analytical study as proposed in way back by Kazenbach et al in 1998 they suggested that designing this combined pile draft foundation CPRF requires qualified understanding of the soil structure interaction using these steps. And now we are going to introduce one parameter which is called CPRF coefficient what is CPRF coefficient this is nothing but the load shared by the total number of pile divided by total load. That means how much fraction of the pile is taking the entire load it will be best designed if it is 50 percent or 0.5 if CPRF coefficient is 0.5 means you have designed it best. That means your raft is sharing 50 percent load and pile is sharing 50 percent load. So, we suggest for better design the range of CPRF is given 0.45 to 0.55 that is the typical range of the CPRF is set always for a better design of the CPRF foundation. Other researchers also recently have taken up this important topic of research as I said you can see the combined pile draft foundation has been modeled in abacus software the three dimensional analysis was done by islami et al in 2011 under earthquake condition. So, this is the node condition you can see the damper stiffener and the mass using these links and the dynamic loading response you can see over here as proposed by from their dynamic analysis with input acceleration of 1 meter per second square with input frequency there is a 36 percent decrease in the pile draft model compared to the pile group that means automatically that pile draft performs much better than a pile group under the dynamic loading condition as shown by these researchers also you can see here pile group bending moment and pile draft bending moment why it occurs because here the load sharing between the raft and pile is going to take place also in the dynamic case as well and these researchers mentioned that 54 percent decrease in the pile draft model of the bending moment using this CPRF technology under dynamic loading condition also islami et al 2011 they use the seismic loading condition also earlier was dynamic loading for seismic loading the input acceleration and displacement was considered as given as these values and L central earthquake way very well known L central earthquake acceleration time history they have chosen as the input acceleration time history result you can see over here they mentioned from their results that there is a 34 percent reduction in the pile draft in terms of acceleration time profile or response compared to your pile group response and this is the displacement amount there is a 9 percent reduction in the pile draft compared to pile group even with L central seismic earthquake loading now we will discuss one case study for this CPRF technology which has been observed very recent 2011 Tohoku earthquake after Tohoku earthquake in Japan so very recently the researchers coming out with this case study result this is very important because from this case study we learn how this CPRF is functioning under earthquake conditions so you can see over here these researchers Yamashita et al 2011 they proposed the case study of pile draft foundation behavior during this March 2011 Tohoku earthquake in Japan this is the building located at the Japan proton accelerator research complex JPA RC the data is like this pile draft foundation this pile draft foundation this is the soil profile you can see over here profile of the soil with SPTN value shear wave velocity profile along the depth of that region so that ground response analysis everything was done so it was pretty safe so let us see the pile draft foundation how it behaves it was constructed on 371 PHC pile the diameter is given over here and earthquake occurred 44 months after the end of the construction of this JPA RC building and epicenter was about 270 kilometer from this site where this structure was constructed with this CPRF and ground acceleration value was this and this for horizontal and vertical direction and what are the recorded response that is they have used in that CPRF because even worldwide nowadays people are using in CPRF to instrumented piles so that they can get the behavior of the pile that is deflection and bending moment behavior from this instrumented pile at collected from the field data or field site condition they had instrumented pile like pile p1 p2 p3 these are instrumented pile earth pressure instrumented pile earth pressure cell and piezometer and settlement gauge everything were provided it at that site and this data shows the variation of results this is the day of 11th March 2011 that is the day when the Tohoku earthquake occurred can you see this data and after the earthquake occurred after about close to a month time that is on 8th of April 8 4 2011 this is the Japanese way of writing date they write year then month then date so this 8th of April 2011 this much displacement of the pile had occurred so increment is this much only can you see that means CPRF what was designed to take care of the earthquake loading at the site they perform very well at that huge damaging earthquake of Tohoku earthquake the data are available in the paper by Yamashati et al. 2012 IS Kanazawa conference paper so these are the responses as I was mentioning for pile 1 p1 this is the total rate response and this is the effective load response this was the time in days in the x axis and y axis shows the ratio of the load carried by the pile so at the end of the construction you can see it is almost saturated this is the time when the earthquake occurred March 11 2011 and this is the ratio of the load shared by another pile that p2 which is also an instrumented pile as I had already shown so from these things it showed later on by Yamashati et al. in 2012 paper that there is a decrease from 0.85 to 0.82 after the earthquake there is a very marginal decrease you can see by pile p1 after the earthquake the load sharing and decrease from 0.67 to 0.57 after the earthquake by pile p2. Now let me tell you very recently international society for soil mechanics and geotechnical engineering which we called ISS MGE this ISS MGE is nothing but international society for soil mechanics and geotechnical engineering we the technical committee TC 212 which is on deep foundations we came out with international guideline for this design of CPRF in very recently in 2012 during our committee meeting at IS Kanazawa in Japan in September 2012 this is the version of the ISS MGE combined pile draft foundation guideline which has been proposed by this technical committee of ISS MGE on deep foundations and present president of ISS MGE is professor Gianluia Brio and present chairman of this TC 212 deep foundations technical committee is professor Rolf Kazenberg from Germany vice chairman of this TC 212 is professor Sangyong from Korea and currently I am the secretary of this ISS MGE technical committee TC 212 deep foundations myself professor Dipankar Chaudhary from India so all our members together after our meeting we have finalized this international guideline that is whoever wants to design any combined pile draft foundation they can follow this design recommendations or codal recommendations or design guidelines the international design guideline given by this technical committee worldwide that these are the steps to be followed for design of a combined pile draft foundation in practice now let us go to next subtopic for this module which is seismic design of ground anchors and for this topic I will like you to note down this is the reference which I am using let us look at it this is the Ph.D. thesis of Dr. Sunil Rangari who completed Ph.D. at IIT Bombay under my supervision and professor Divaka's supervision at IIT Bombay in this year 2013 so from his Ph.D. thesis work we will discuss about seismic design concept of ground anchors now what are the ground anchors as we all know where we use ground anchors like to mitigate the effect of earthquake ground anchors can be used for structures which are subjected to uplift or pull out type of load so which are those structures like this transmission towers, chimneys tall buildings so these are always subjected to some kind of pull out load or tensile load or uplift load so to protect the foundation against those pull out uplift we provide the ground anchors so that it remains in its place when the anchors are provided so that is why these anchors need to be designed properly so that it can withstand the pull out load including the uplift coming from the earthquake forces so estimation of the uplift capacity of these ground anchors it is nothing but an application of the passive earth pressure theory so passive earth pressure theory as we have already explained in one of our previous lecture that for positive wall friction angle for the case of passive earth pressure behind retaining wall that is the example of bearing capacity factors for shallow foundations that we have also seen under the earthquake condition how to estimate the bearing capacity factors under seismic conditions now the same passive earth pressure theory but for negative wall friction angle case is nothing but the application for ground anchors so that is why when we are calculating the uplift capacity of ground anchor I can explain you through this basic diagram as I have earlier also mentioned suppose this is an anchor rod which is connected to an anchor plate something like this say this is this ground anchor is embedded up to a certain depth let us say this is the depth of embedment of the anchor H and this is the width of the anchor plate now there are various types of anchors of course so this is plate anchor and for plate anchor we can have shape of square plate anchor rectangular plate anchor circular plate anchor like that and even strip plate anchor also strip means when length is much more than this width so this anchor rod along this there will be some uplift load so this is the p uplift that gives us the capacity of the anchor now how to determine this capacity of a particular anchor which is embedded in some soil like this and there can be different types of anchors one is shallow anchor another is deep anchor shallow anchor that depends on the value of this H by B which is known as embedment depth so this H by B is an important parameter which is called embedment depth based on that value it is decided whether it is a shallow anchor or it is a deep anchor now what will be the typical surface for this type of anchor ground anchor when it is subjected to this uplift pull out like this so typically it can fail like this this is one of the typical failure mechanism or it can fail like this that is when we are trying to pull it out like this in this direction obviously the soil above it will push out like this ok so we need to consider let us say this we are considering as an imaginary wall imaginary retaining wall actually there is nothing it is within the soil only we are considering it as an imaginary wall now what will be the movement of this wall laterally if we consider this this wall is actually moving towards this direction when we are pulling it out it pushes this side ok so that gives us the condition of passive earth pressure that is why we said that this is nothing but a theory of passive earth pressure now why it is negative wall friction angle because in this case what happens your wall moves up and the surrounding soil this one moves down that is the relative movement between these two zones right so in this case of passive earth pressure whatever is happening here on this one this face that is acting on the other direction that is it will be a negative passive earth pressure not a positive one which is occurring for bearing capacity theory for shallow foundation ok so that is why it is called negative wall friction angle case so it can be either planar rupture surface or carpeted linear rupture surface depending on what type of passive earth pressure theory you are going to use now with this basic concept of ground anchor and its application using passive earth pressure theory concept now this problem becomes more complex under the seismic condition and the pictures which shows the practical applications of this ground anchors as I said we put the ground anchors here in this foundation support of transmission towers always at four corners so that it does not lift up like this but what are the causes of this uplift force or tensile force or pull out load like there will be always huge lateral load like wind load seismic load etc because of that it will try to pull out right in addition so in seismic condition also there will be some additional pull out load which will be acting on the superstructure which finally you need to take care in the ground anchor design so these pictures shows you can have steel helical anchor or concrete anchor so various types of materials can be used as anchor plate even for the submerged pipeline those pipelines also are always put with anchor support at different intervals of the length of the pipeline otherwise what will happen when there is an uplift force the pipeline will come out from the embedment and it will open up so nobody will like to see suppose some wastewater line which is going underground if the proper anchoring system ground anchors are not provided at some point of uplift pressure if it is more than the surcharge pressure at which depth the pipeline has been constructed it can come out right so that is why to provide the stability to maintain the stability of the entire system this ground anchors are very very important another application as we know for the sheet pile wall design for anchored sheet pile that is sheet pile to reduce its excessive movement excessive lateral movement we provide the plate anchors like this right so that the sheet pile is tied in such a way that it does not deflect excessively so now let us see what are the available literature on this anchor actually on ground anchors there are extensive research work was carried out by several researchers under static loading conditions so first I am addressing here static loading conditions in this slide we are mentioning only a very few of them there are many others which are not listed over here this is just to give an overall idea that what are the major research going on for horizontal strip anchor horizontal shallow strip anchors so the one of the pioneering work in this ground anchor area was done by mayerhoff and adams in 1968 using the simple limit equilibrium method and they considered the log spiral failure surface but no seismic analysis was done as I said this is the static loading conditions so in this column of seismic analysis for all of them it is no because it is a static loading cases I am referring to another classical work was done by professor rhoe and professor davis in 1982 using finite element as well as the experimental values these are available then mure and gedis in 1987 they also obtained experimental results as well as they used limit equilibrium method also they used limit analysis method kumar in 1999 used method of slices using log spiral failure surface to obtain the uplift capacity of anchors under static condition then merifeld and slowon in 2006 used limit analysis both upper and lower bound using a planar rupture surface or planar failure surface for anchor and recently deshmuk etal he is another phd student who completed his phd at iit bombay under the supervision of professor divaker and myself so he worked using limit equilibrium concept for the uplift capacity of anchor using planar failure surface under static condition only so all this literature review and the state of the art concept for a ground anchors you can find in this research paper rangari choudhury and divaker 2011 in ac's geotechnical special publication number 211 these are the page numbers of the paper now coming to the seismic analysis that is under earthquake condition who are the researchers who did research for the estimation of vertical uplift capacity of horizontal and inclined strip anchors under earthquake condition like kumar in 2001 that was one of the pioneering work to obtain the uplift capacity of ground anchors under earthquake condition using pseudo static approach so he used upper bound limit analysis using a planar failure surface and pseudo static method was used for seismic analysis then myself during my phd I worked under the supervision of professor k subara at isc scholar so we had publications choudhury and subara 2004 and 2005 this 2004 is in the journal geotechnical and geological engineering of springer and 2005 is the canadian geotechnical journal we used limit equilibria method one is for horizontal strip anchor another is for inclined strip anchor we used log spiral failure surface and also for seismic forces we use set of pseudo static then gosch in 2009 he used upper bound limit analysis using planar rupture surface but he proposed pseudo dynamic method to get applied for estimating this uplift capacity of anchor so he used our proposed pseudo dynamic model as I have discussed earlier for the uplift capacity determination of anchors also so and as I am currently mentioning the phd thesis work of rangari so rangari et al 2012 our one paper we discussed about limit equilibria method using planar rupture surface and here we used pseudo static method but later on we used pseudo dynamic also which I am showing in couple of next slides so you can see from these researchers results that there is a scarcity of research and design method for estimation of vertical uplift capacity of both horizontal and inclined strip anchors under earthquake conditions using both pseudo static approach as well as pseudo dynamic approach and if we talk about obliquely loaded horizontal strip anchors or inclined strip anchors also that is horizontal strip anchor is a very common application but inclined strip anchors also are used at many places as I have already mentioned at various technical sites we need to probably provide the foundation little inclined position or the anchors in little inclined position so inclined anchor plates are also very much used in practice so for those inclined plate anchors or the horizontal plate anchor many a time what happens the pull out load or the tie rod will not be vertical it will be somewhat inclined so in that case it is subjected to oblique load or inclined load right it depends on your the direction of the force which is coming on the foundation system so this obliquely loaded horizontal strip anchor even under static condition also there is very few researchers who worked in this area though there is lot of application of this problem as I have already shown like Meyerhoff again in 1973 he proposed limit equilibrium method as well as he has shown the model test results for limit equilibrium method he used log spiral failure mechanism and it was static case so that is why seismic analysis is no then DAS and SELE in 1975 used the model test for this obliquely loaded horizontal strip anchor and it is only for static loading condition so seismic analysis is no so the state of the art literature review you will get in this paper in detail Rangari Choudhury and Divayakar 2012 in ASC geotechnical special publication number 225 these are the page numbers coming to some more researchers results in this area of obliquely loaded and the inclined strip anchor like for inclined strip anchor Meyerhoff in 1973 limit equilibrium method as I said log spiral failure mechanism no seismic analysis Hanna et al in 1988 they also used limit equilibrium method using planar failure surface and no seismic analysis Maya et al in 1986 used empirical relation it is also for static analysis whereas Choudhury and Subbarao we did the research for inclined strip anchors as I have said using limit equilibrium method using log spiral failure surface and it was done for seismic analysis also using pseudo static approach then Choudhury and Subbarao 2007 again limit equilibrium approach log spiral it was also pseudo static approach and Ghosh in 2010 Professor Priyanka Ghosh of IIT Kanpur he worked in this area of upper bound limit analysis using planar rupture surface pseudo dynamic results so it again shows the scarcity of the research for obliquely loaded and inclined strip anchors under both static condition and it is not yet touched in the seismic condition because seismic condition obliquely loaded nobody has done these are the inclined strip anchors but not inclined strip anchors with obliquely loaded inclined strip anchor with vertical load now what Dr. Sunil Rangar is PhD thesis comprises of he used the basic quotas equation quotas equation it was proposed in 1903 this mathematical expression for defining the distribution of reaction on any curvilinear failure surface is shown like this so this is the basic quotas equation you can see over here this is the curved failure surface this is the tangent at any point normal to this curvilinear failure surface if this horizontal angle of this tangent with this horizontal is alpha this infinitesimal small angle is d alpha you have for passive case this direction for active case this direction so the basic equation governing equation for quotas equation which was proposed by quarter in 1903 is given by this dp by ds plus 2p tan of phi d alpha by ds equals to gamma times sin of alpha plus phi where this dp is differential reaction pressure on the failure surface ds is the differential length of the failure surface p is the uniform pressure on the failure surface d alpha is the differential angle over here alpha is the angular failure plane formed by inclination of the tangent at the point of interest with respect to the horizontal as I said here gamma is the unit weight of the material so in our case it is a soil and phi is the soil friction angle so this equation is valid for curvilinear failure surface so if somebody wants to use the planar failure surface it needs to be modified so for simple planar failure surface the equation that reduces to or simplified to this value because in that case this term will not be there because there is no variation of this d alpha by ds so you will get this dp by ds equals to gamma times sin of alpha plus phi so p equals to you can integrate gamma times sin alpha plus phi times s fine so this is for simpler failure surface so what sunil rangar is thesis comprises of he first carried out the research on the area of horizontal strip anchor horizontal strip shallow anchor under seismic condition so this is a horizontal anchor subjected to vertical uplift load see all these conditions and width of the anchor plate is b depth of the embedment of anchor that is h and this is strip anchor so length of this anchor is much more than this width so this pud that is the uplift load for this anchor under dynamic condition needs to be obtained he considered the planar rupture surface like this simplest planar rupture surface but use the quotas equation so when you are using quotas equation the advantage is you are getting the soil reaction on this side which is unknown earlier as I have mentioned to you in one of the previous lecture so from this equation you can get these values of this total reactions r1 on this side and r3 on this side fine is it okay based on the planar rupture surface now these two zones are different because at one instant of acceleration at direction obviously it will be one side will be larger zone another side will be a smaller zone of failure right it will not be symmetrically equal or it should not be symmetric as the case is in under static condition so that is why the static loading condition solution is comparatively simpler than the seismic loading conditions and these are the seismic inertia force in vertical direction in horizontal direction and this is the weight of this central block this ecdf and these are the imaginary retaining wall ce and df on which we are considering this seismic passive earth pressure which are acting at an angle delta and that passive earth pressure has to be obtained for this failure zone dbf and this passive earth pressure has to be obtained for this failure zone cea fine now if we consider the central portion this details of this proposed method is available in the journal paper by rangari et al 2013 you can see the details of this paper rangari chaudhary and divaika 2013 in the journal geotechnical and geological engineering springer publication this is the volume issue number and page numbers so what has been done once you get the passive earth pressure then these are the forces which are acting on the central block okay and you are considering this infinitesimal small element now if you are applying the pseudo dynamic approach as we have applied here so these are the horizontal and vertical seismic acceleration as we have already discussed for pseudo dynamic case that will give you the seismic inertia forces in this which you can obtain by integrating this inertia forces over the entire depth of this anchor plate okay so that is how you get these values of qh2 and qv2 that is horizontal inertia force seismic inertia force and vertical seismic inertia force right now once you get that then you consider the equilibrium of all the forces involved both in horizontal direction as well as vertical direction so satisfying the equilibrium what you will finally get you will get the expression in terms of pud which you want to find out that is gross pull out capacity under dynamic condition and we express them in terms of net seismic uplift capacity qud net in this form which is given in the common form of half gamma bf gamma d what is f gamma d f gamma d is called anchor uplift capacity factor like in case of foundation shallow foundation we have seen n gamma d which is called bearing capacity factor for anchor it is called f gamma or f gamma d as proposed by meyarov also it is known as anchor uplift capacity factor so this is the closed form solution of the anchor uplift capacity factor in terms of embedment depth in terms of passive earth pressure coefficient which is acting on the different side of this failure plane so finally the typical design charts are proposed like this which will help the designers to design any anchor plate in this seismically active region suppose if you have k h value for a zone 0.2g you have a particular value of k v say half of k h and 5 value let us say 35 degree you can go here depending on your embedment depth remember this value embedment depth of 2 you will get some value of f gamma d which will help you to get the net ultimate and gross ultimate values of this net ultimate seismic uplift capacity factor q ud net similarly for other cases also so this is using pseudo static approach and similarly using pseudo dynamic approach these design charts have been proposed in this paper which one can easily get now what is the validation of these results you can see the results have been compared and with previous researchers results using the ultimate seismic uplift capacity factor this f gamma i which is expressed as p ud by gamma b square in that non-dimensional form so these are non-dimensional factors as we have seen so for various input values so these are the different input values you can see over here for different values of k h gosch pseudo dynamic approach these are the results of this f gamma e kumar's result using pseudo static approach these are the values choudhury and super house pseudo static approach 2004 these are the values whereas the present study gives the pseudo static values like this and pseudo dynamic values like this so you can see the present study using this quotas equation gives the minimum value of this seismic uplift capacity factor f gamma e what does it mean it automatically shows it is showing the most critical design or the better design as far as the ground anchors under seismic condition is concerned why it is so because the advantage as I said in quotas equation you are getting you are satisfying the condition of soil reaction at the failure surface and you are considering that at each and every point of the curve so that is the important addition compared to the other researches what they have done and more so in the pseudo dynamic case now coming to the application for the case of inclined strip anchor so if the anchor plate is like this inclined say the inclination angle of the anchor plate is beta with respect to the horizontal these are the soil reactions which you are getting again from the quotas equation and these are the various seismic this is the weight of the central block and seismic inertia forces in vertical and horizontal direction this is the uplift capacity now remember this inclined anchor is again is loaded obliquely load is not perpendicular to the plane of the anchor plate can you see that so this is this shy is the angle of inclination of load applied it is quite possible that you have used an anchor plate inclined like this and your load of action or line of action of load can be inclined to the vertical on that plane ok so for that complicated condition how the reactions and the uplift capacity has to be determined that has been analyzed the details you can find in this journal paper rangari chaudhary and divaika 2012 in the journal disaster advances this is the volume number and page issue number and page number so here also the q ud net is expressed in this form half gamma b and f gamma d f gamma d is non-dimensional uplift capacity factor or the design uplift capacity factor under seismic condition and the closed form equation of that f gamma d is given by this expression you can see these are function of this embedment ratio anchor inclination k p gamma d which is again in terms of function of phi value and the seismicity parameters k h k v also in addition to this these are functions of k h k v delta value and the inclination of your load of action on the plane that is psi so the critical angle of failure critical angle of failure means the angle of failure at which this alpha 1 and alpha 3 will give you the minimum value of this p ud that is nothing but your critical angle that needs to be determined using optimization technique that is you have to use the optimization method to find out the minimum value of these failure planes so that you get the minimum p ud value again here also the design charts have been proposed for practical use by designers for a particular value of k h and k v for a known value of phi one can get the f gamma d value for given value of embedment depth anchor inclination and load inclination load inclination 0 means the load is acting exactly perpendicular on the inclined plane of the anchor so for different other angles also results are available in this journal paper here also the comparison of results have been made with other inclined researchers result there is only few research available like choudhury and sub-brows result 2005 is for inclined anchor that has been compared here with the present study results and the results with kumar and gosh with the present study for the case of horizontal anchor beta equals to 0 degree is nothing but inclination is 0 degree means horizontal anchor that has been compared over here so with this we have come to the end of today's lecture we will continue further in our next lecture