 Let us start our today's lecture for NPTEL video course on Geotechnical Earthquake Engineering. We are currently going through our module number 9, which is seismic analysis and design of various geotechnical structures. So, within that let me have a quick recap what we have learnt in our previous lecture, which was on the seismic design of retaining wall. So, to recap quickly in the previous lecture, we discussed about the displacement based design approach for the seismic design of retaining wall both under active state as well as passive state considering sliding as well as rotational stability is concerned. So, this paper, where we considered both the inertia forces acting on the wall as well as on the failure mass of the soil under active condition that details we have given in this paper. So, how to find out the weight of the wall required under earthquake condition to be designed that has been proposed through this dynamic combined dynamic factor, which is nothing but ratio of the weight of the wall required under dynamic case by weight of the wall required for static case with respect to sliding stability. So, in static case we know how to design it with factor of safety of 1.5. So, the same approach we will use and then the proposed design chart of this combined dynamic factor F w to be used for a seismic zone where the k h value or input seismic acceleration for the design will be known to us correspondingly you will get the value of F w once you get the F w automatically what you can do you are getting this w w value which further will help you to get the section of the wall for which will be stable with respect to that particular magnitude of earthquake and you your design will be complete under the active state. And we have compared our proposed value of pseudo dynamic approach of this combined dynamic factor with the available results using pseudo static approach as given by Richard and Elms as I said that this method is commonly used in Eurocode Richard and Elms displacement based criteria. So, you can see the comparison given over here. The similar approach has been extended for the passive state of earth pressure also for design of the retaining wall using pseudo dynamic approach as can be obtained in this journal paper of Nimbal Karan Choudhury soil dynamics and earthquake engineering journal. Here also the same combined dynamic factor needs to be obtained from the proposed design chart like this and finally, the section of the wall can be provided whereas this paper Choudhury and Nimbal Karan 2007 this is also in the journal soil dynamics and earthquake engineering. This paper talks about the rotational stability earlier paper talks about the sliding stability to how to obtain the stable wall section and this aspect tells us about how to find out the rotational displacement for a particular section wall section with chosen height and width etcetera. So, once you design your wall with respect to sliding condition next you need to check for the rotational condition in this manner that how much rotational displacement you are getting for a particular value of earthquake excitation or input value how much rotational displacement is occurring. If that value of rotational displacement is more than the acceptable limit or permissible limit given by the designers or the owner of the property or as a designer if you decide that I will not allow the displacement more than this that is performance based design if you want to carry out in that case you can restrict the design section displacement in that manner and redesign the section so that rotational stability is also ensured under that magnitude of earthquake input motion clear. Then we had also discussed in our previous lecture what are the recommendations given by various design codes. In that connection we have mentioned that our Indian design code IS 1893 though the part 1 has been revised in 2002 that is the latest version so far, but for retaining wall design its version is part 5 of 1984 is till the latest one where it suggests the use of pseudo static analysis of age old method of Mononobe Okabe theory. So, there is no scope for using the deformation criteria or displacement based criteria also the point of application is a thumb rule it says that it acts at the mid height only for the dynamic component the static component acts at one third from the base, but the extra dynamic component it acts at the mid height that is what the code suggests. And this is the equation how to obtain this active earth pressure coefficient under seismic condition using this Mononobe Okabe approach as per the IS code. So, this formula is nothing but Mononobe Okabe's equation of seismic active earth pressure coefficient. Similarly, for seismic passive earth pressure coefficient using pseudo static approach as proposed by Mononobe Okabe the same formula is given in the IS code also. Whereas, we have seen that European design code that is Euro code 8 of 2003 version that discuss more about the displacement based criteria though the analysis proposed the pseudo static approach, but they considered the performance based design criteria or the displacement criteria in terms of translational as well as rocking mode as given by Richard and Eons. So, how to calculate the values of k h and k v those input parameters of pseudo static seismic horizontal and vertical acceleration as per Euro code that is when you do not have the specific study or specific site response analysis or the seismic hazard analysis if it is not available for your site. In that case you can use this formula given in the code to obtain the design value of k h and k v then international building code IBC 2006. They classify the soil or the site into 6 major categories that we have already discussed when we discussed about the Neharp code of US based on shear wave velocity, SPT value, undrained shear strength etcetera. So, based on that the based on different seismic category it is asked to design the retaining wall which will be stable with respect to overturning sliding excessive foundation pressure and water uplift. So, with that we had completed our lecture on the previous day. Now today we will start with another subtopic that is seismic design of water front retaining wall. So, what is water front retaining wall? Let us see through this picture. Another name of the water front retaining wall is let us look at the slide water front retaining wall is also called sea wall. Why it is called sea wall? Because you can see in these two pictures this wall is nothing but to protect the shore from the sea that is whenever this water waves from the sea comes and hits the shore that place where we provide the retaining wall or the retaining structures which withstand this water pressure to protect the shore that is known as nothing but water front retaining wall or sea wall the same thing but different names. So, it is nothing but a soil retaining armoring structure generally it is massive in nature to defend a shoreline against the wave attack it is used designed primarily to resist wave action along high value of the coastal property. So, for example in India in marine drive we have in Mumbai this type of water front retaining wall or sea wall which is available. Now let us look at the research done to take care of different aspects of this design of water front retaining wall under static condition the design criteria of this water front retaining wall is very well known. We have to consider not only the earth pressure coming from the soil side but in addition we need to consider the wave pressure coming from the water side also and remember in this case the soil side also will have a water table. So, we need to consider the water pressure from the both the sides in addition to that wave pressure coming and hitting on the wall that needs to be considered. Now let us move it little further ahead for our course which is on geotechnical earthquake engineering to address the issue of stability of such water front retaining wall or sea wall under earthquake condition. Now let us mention that when any earthquake comes in the water body or sea bed we have already discussed there is a chance or possibility that tsunami may get generated it is not always occurring we have discussed in detail what are the reasons where it will occur where it will not occur etcetera in one of our previous module. So, if during earthquake the tsunami occurs and that water front retaining wall is subjected to both the tsunami wave pressure in addition to the static water pressure remember the tsunami wave pressure is much higher than the steel water level pressure right. So, when we need to design those water front retaining wall or sea wall which is protecting the shore from the sea we need to design against in addition to that tsunami wave pressure as well as under earthquake condition what are the extra dynamic pressure coming from the earth side and what is the extra dynamic pressure or hydrodynamic pressure coming from the water side or wave side. Now let us look at the literature which are available so far on earthquake we have already discussed these names I have already referred to you in one of our previous lecture that the there are several researchers who did work extensively for design of retaining wall under earthquake condition. Now separately there are several other researchers who did the design of water front retaining wall or sea wall under the tsunami or hydrodynamics, but remember they have not considered the effect of earthquake. So, what they have considered only the tsunami wave pressure they have considered and the hydrodynamic pressure they have considered, but not the seismic earth pressure which is coming from the backfill side or soil side or from the shore side. So, who are the researchers who did extensive work on this hydrodynamics or tsunami on the retaining wall like Westergaard's method is the most pioneering work or most popular method which was established in 1933 to estimate the hydrodynamic pressure on a retaining wall. Then later on Fukui et al in 1962 designed the retaining wall or sea wall with respect to tsunami wave pressure then Ebelling and Morrison in 1992 designed the retaining wall to withstand the tsunami and that is the one which is used by US Army Corps of Engineers so far for design of water front retaining wall this approach given by Ebelling and Morrison in 1992. Later Mizutani and Imamura in 2001 they also proposed how to design this sea wall with respect to tsunami and crater 2006 it is a code which provides in the US this is a US code which provides the guidelines how to design the retaining wall under the tsunami cases and few other researchers also worked on this. But what was the limitation of this available literature nobody considered the combined effect of this earthquake along with tsunami as I said, but people can always mention that this is this combined effect of this earthquake and tsunami on sea wall it is very very rare yes it is very rare but what I can state here from the example of March 2011 Tohoku earthquake in Japan we already learnt that major damages occurred because of the tsunami and remember the major earthquake was followed by several aftershocks. So, earlier people used to say that whenever we are designing a sea wall or retaining wall we have to design only for either earthquake or tsunami because both are extreme condition like for extreme condition we are designing it. So, it should take care of the combined effect, but the Japan Tohoku earthquake of 2011 proved it wrong it said or it showed through several damages that after the major shock there can be several other aftershocks which are also of prominent nature not of negligible magnitude, but of prominent nature which are occurring along with the tsunami wave. So, when we are designing this sea wall or waterfront retaining wall we need to consider the combined effect of this earthquake as well as the tsunami. So, to do that the research work was done by my second PhD student Dr. Sayyad Muhammad Ahmed who is now a faculty lecturer at the University of Manchester in UK. So, he did PhD at IIT Bombay he analyzed this sea wall or waterfront retaining wall for two major cases what are the possible two major cases first cases when the tsunami is attacking the wall when let us look at here suppose this is our wall and this side is the water. So, tsunami wave is coming from this side and this side we have the shore or the soil side or and structures etcetera over here. So, when tsunami is hitting over here wall tends to move in this direction towards the soil side that means it generates the state of earth pressure which is similar to passive state clear. So, that is tsunami hitting the wall we call another case occurs when tsunami comes and then it over tops also sometimes the wall because of its height as we know then finally through the weep hole or filters we know that finally the water will go back right. So, when the tsunami goes back or recedes back to the sea because tsunami wave it comes then finally after sometime it is recedes back or goes back to the sea again. So, when tsunami goes back it tries to drag the wall right because entire water pressure comes from here. So, it trying to drag the wall towards sea side that means wall tends to move away from this soil which is nothing but a state of active earth pressure clear. So, that is why you can see in this slide there are two major cases one is for tsunami attacking the wall this is passive case of earth pressure and tsunami receding away from the wall or going back to the sea that is the creating active state of earth pressure. So, for both the cases in his PHD thesis work of Dr. Sayyad Ahmed he worked for stability aspects with respect to sliding as well as overturning in both cases of passive as well as active. So, let us see what are the details first case let us consider that is passive state of case that is when tsunami wave is hitting the wall or attacking the wall and considering the pseudo static method of analysis. So, this is the typical line diagram you can see to consider the combined effect of earthquake and tsunami on this rigid waterfront retaining wall. This is the line diagram of rigid waterfront retaining wall this is the downstream side we have named here that is the shore side where soil is there this is the ground level and let us take this is the water table level. So, now earlier all the earth pressure problem we consider for dry soil now we are introducing this water table effect also. So, remember that. So, this height of the water table we have considered as small h w d h w is the height of water d means downstream side and capital H is the total height of this wall and on this side that is upstream side or the water side or sea side this height h w u is nothing, but the steel water level. So, s w l is nothing, but steel water level of the sea, but when tsunami comes there will be extra height on top of the steel water level. Let us say this is h t is the height of the tsunami wave. Now, if we see what is the water pressure acting from this upstream side on the wall from this static steel water it will be p static u means upstream p s t means p static upstream and where it should at a height of h w u by 3 because it is a hydrostatic pressure. Now, tsunami pressure p t needs to be calculated that tsunami pressure is acting on the wall in this direction now because it is attacking the wall. Let us say it is acting at a height of h t by 2 from this sea water level clear. Now, direction of wall movement is towards the earth that is why we said it is passive case. Now within the wall the pseudo static seismic inertia forces will be k v w w w is the weight of the wall section. So, this is the vertical inertia force this is horizontal inertia force k h w. Now, from the soil side what will be having p p e is the passive earth pressure under earthquake condition at an angle delta with respect to normal. So, we can resolve it into two components of cos delta and sin delta. Now, see here are two components p s t d means this height of the water what it provides the water pressure. So, it is a static water pressure p s t in the downstream side clear which will act at the one third of this height h w d. Now, there is another additional pressure remember which is p dynamic what is p dynamic that is nothing but hydrodynamic pressure. This hydrodynamic pressure look at the direction it should be in the opposite direction not like hydrostatic pressure acts in this direction. Here also hydrostatic pressure acts in this direction tsunami wave pressure is acting in this direction, but hydrodynamic pressure is acting in this direction why because this wave is giving the generating the dynamic pressure hydrodynamic pressure and giving on the wall. So, that is why this p hydrodynamic is acting in this direction acting at a height of this one clear. How to estimate that hydrodynamic pressure? We use the equation given by Westergaard as I said Westergaard's equation for hydrodynamic pressure is very well known and commonly used worldwide which gives 7 by 12 k h times gamma w times h w d whole square. So, h w d is the height of the downstream water table gamma w is the unit weight of the water k h is the horizontal seismic acceleration coefficient. So, that gives you the estimate of this p dynamic. What is p t? p t is nothing but tsunami wave force. So, tsunami wave force is given by this expression 4.5 times gamma w h t whole square. This is given by this code crater of 2006. So, this is p t and this is h t based on your tsunami wave height you can calculate what is coming your tsunami wave force. Now how to calculate this seismic passive earth pressure p p e? In this case p p e as we have mentioned we are using pseudo static approach. The similar equation of mononobiocabe is used here which gives half k p e is seismic passive earth pressure coefficient h square. Remember instead of gamma now we have defined another parameter gamma bar. Why gamma bar? Because now it is no longer a dry soil you need to consider the effective soil unit weight because of this dry zone as well as this saturated zone. And k v is the vertical coefficient of seismic acceleration and what is r u? r u is pore pressure ratio. Now what is this gamma bar? Gamma bar is calculated in this fashion h w d by h whole square gamma saturated plus 1 minus h w d by h whole square whole to the times gamma d. Gamma d is nothing but dry unit weight of the soil above this water table level. So, this way the average gamma of this entire soil layer is calculated based on what is the height of the water table clear. So, this is also available in the book of Kramer 1996 this equation you can see there because it is a pseudo static approach. So, this expression is available. How much is the p static in upstream side? This hydrostatic pressure in the upstream side that will be half gamma w is the water unit weight h w u whole square. And what is the hydrostatic pressure in the downstream side? Half gamma w e h w d whole square. Here why it is not gamma w but gamma w e because see here what is gamma w e is the effective water pressure which is coming here gamma w plus already you have taken this gamma bar effective unit weight of the soil in this earth pressure calculation. So, you are removing this portion gamma bar minus gamma w times r u from this otherwise you will be taking water pressure two times. Why it is deducted? So, this is the reason why this gamma w e is used on the downstream side clear. So, this was published by us that is Choudhury and Ahmed as I said this is the phd thesis work of my second phd student Dr. Syed Muhammad Ahmed. Choudhury and Ahmed 2007 paper in the journal applied ocean research which is an LCVR publication volume 29 page number 37 to 44 you can get all these details in this paper. Now, let us see the results what we have done after using the simple limit equilibrium approach for all the forces involved with respect to sliding as well as overturning mode of failure then the results with respect to the factor of safety against sliding and factor of safety against overturning are shown. So, this figure shows factor of safety value against sliding FSS with respect to various values of horizontal seismic acceleration coefficient k h for a given value of b by h ratio h w by h ratio h w d by h ratio phi value delta value and k v value and r u value for this different curves shows for different height of tsunami wave h t with respect to h w u can you see h w u is nothing but steel water height when it is 0 that means no tsunami. But when it is 1.5 means the tsunami wave height is on top of the steel water level 1.5 times of the height of that steel water level clear. So, for different values we got the different results which is also available as we have proposed in a closed form solution or a complete equation form which can be used by practicing engineers or practical design engineers directly. How? This factor of safety with respect to sliding is calculated mu times mu is nothing but coefficient of friction between the base of the wall material and the soil material against direct sliding and b by h ratio is the property of the wall section which you want to design. So, typically the b by h ratio for design we consider at site between about 0.2 to 0.4 that is a typical range for a rigid retaining wall. So, using those values you can do a iterative study as long as you get a safe factor of safety of more than 1.5 even under this earthquake and tsunami combined condition. So, this k v is the vertical seismic acceleration coefficient. This k is a constant which we have defined as this expression that 0.5 times k p e gamma bar 1 minus k v e times 1 minus r u. How r u is known for a particular soil with known water table and saturation condition you know the pore pressure ratio. So, that pore pressure ratio and given input value you can calculate gamma bar as I have already mentioned knowing the water table level k p e you can use the mononobioca bay or any other pseudo static seismic passive earth pressure coefficients. And these all others are non dimensional term of ratio that h w d by h that is height of the water table with respect to total height of the wall h d by h u is the tsunami wave height over the static steel water table level. Gamma w is nothing but unit weight of water and gamma w e is nothing but equivalent unit weight of the water. So, using this one and this gamma c is nothing but unit weight of the concrete which you are using for the water front retaining wall or the sea wall. So, using all these values or combinations one can easily compute the factor of safety against sliding or they can refer to our this design charts and get for a particular tsunami wave height what should be the factor of safety for this chosen section. Remember this is for a typical chosen data set of value if you change the value this values obviously is going to change as given by this equation. Similarly, for the overturning mode of failure also the factor of safety is represented and can you observe one thing earlier we know for the static condition for the no tsunami condition that factor of safety against sliding is decreasing drastically as the seismicity increases or seismic acceleration increases. Now in addition to that you can see as the tsunami wave height increases the factor of safety is decreasing also drastically can you see that. So, this is for 1.5 next one is 1.125 this is for 0.75 like that. So, as the tsunami wave height is increasing factor of safety for a given earthquake excitation is also decreasing. So, we need to take care of this combined effect of tsunami wave height action as well as the seismic acceleration value for the design with respect to sliding and overturning stability of the wall. So, this gives the details this journal paper applied ocean research as I already said. Now, coming to some more results for factor of safety against sliding this is for the other case of active state earlier we talked about the passive state. Now, we are proposing the design of wall section when the tsunami wave is going back to sea or the receding back. So, for active state the design solutions were proposed by this paper journal paper Choudhury and Ahmed 2007 in the journal Ocean Engineering it is also Elsevier publication this is the volume and page number. You can see here also we have proposed the combined closed form solution which directly the designers can use in practice. In a simplified manner that is factor of safety in this case with respect to sliding there are two equations you can see one is factor of safety sliding with suffix r another is f what are these? These two cases why it is arising let me explain through this basic figure. Now, we are talking about the active state that is wall of movement is in this direction that is tsunami wave is going back. Now, depending on your soil permeability and how you have designed your weep hole for this wall it will depend that whether the water which came from the tsunami and now going back will be restrained on this back side of the wall along with soil or it will be a free flow of water. So, that is why there will be possibility of arising two cases in active state. In passive state there were only one case, but in active state for sliding itself you have two combinations or two conditions based on the permeability of your backfill material. If it is good permeable material that is say sandy material, good sand with high value of permeability with very good designed weep holes or a filter drains etcetera you should use the which equation you should use in that case you should use factor of safety against sliding in free flow condition. But if the permeability is not good and the filter design or weep holes are not designed properly in that case your soil is going to retain back some water. So, that should be used for restrained water conditions. So, factor of safety sliding with respect to restraining. So, for both the cases the closed from equations are given similarly for the overturning mode of failure also there will be two cases of factor of safety one is with respect to restrained water condition another is free flow water condition. You can get the details in this journal paper. These are few results factor of safety against sliding and factor of safety against overturning for both restrained and free water case. You can see here what are the changes in the factor of safety values and with respect to the horizontal seismic acceleration. In that case remember you need not to consider tsunami wave height because now there is no reason to consider that tsunami wave when it is attacking then you are concerned about the height of the tsunami. When it is going back there is no height available. So, that is why there it is not with respect to tsunami wave height but it is with respect to what height of the water it is standing on the downstream side or in the soil side based on that you should do the design clear. Now, let us come to the application of this pseudo dynamic method. So, far we applied the pseudo static method for calculation of this wall inertia, soil inertia etcetera. Now, we are applying the pseudo dynamic approach which we have developed as we have mentioned over here earlier lecture and these are the basic equations of seismic acceleration in horizontal and vertical direction which we have already mentioned. So, first let us talk about the passive case again that is when tsunami is hitting the wall that design criteria is given by Ahmed and Choudhury in 2008. This is the journal paper you can get the details journal of waterway port coastal and ocean engineering published by ASCE USA this is the volume number and page number ASCE publication journal publication and for active state of earth pressure this is for the active state that is wall movement is in this direction considering both earthquake and tsunami that means tsunami wave is receding back or going back to the sea side and these are the results you can see over here that is considering the crater value and Fukui's value that is one is Japanese method to calculate the tsunami wave force another is American way to calculate the tsunami wave force correspondingly the factor of safety for different cases is reported over here considering crater's approach and Fukui's approach. Now, if we want to compare the results for this factor of safety again sliding and overturning with the available results of the pseudo static values of Ebeling and Morison if 1992 which I said used by US Army Corps of Engineers for design of waterfront retaining wall. You can see the pseudo static values are showing not the critical design but the pseudo dynamic method the present study shows the more critical design that means it is giving more precise or more safer design as far as the pseudo dynamic method is concerned compared to the pseudo static method for waterfront retaining wall design. Now, with that we have shown how to design the waterfront retaining wall or sea wall to take care of both the effects of tsunami as well as earthquake. Now, let us move to the seismic design of reinforced soil wall this is our next sub topic that is seismic design of reinforced soil wall we all are aware about the use of this reinforced soil wall it is a very practical common examples which are nowadays people use in other courses I am sure you have the basic knowledge about what is reinforced soil wall where it is used where there is a space constraint etcetera to get the higher stability with the verticality of the wall etcetera we use the reinforced soil wall. So, geosynthetic reinforced soil wall there are several applications worldwide this picture is from Japan you can see a Japan rail is moving which is getting supported by using this geosynthetic reinforced soil wall and in terms of our course of this geotechnical earthquake engineering let us see how this geosynthetic reinforced soil wall performs during an earthquake this is a case study as reported by professor Tatsuoka in 2010 in his paper that remember when the Kobe earthquake in 1995 that most one of the major damaging earthquake in Japan which occurred earlier before the Tohoku earthquake of 2011 you see soon after the earthquake all the building structures everything got collapsed but this geosynthetic reinforced soil wall that survived. So, the wall survived even that high magnitude of Kobe earthquake motion. So, which is used for the rapid transit they call that is the rail system in Japan in Kobe region that shows directly that the geosynthetics reinforced soil wall they performs even better than the conventional gravity type retaining wall under earthquake condition. So, we need to know how to design this reinforced retaining wall. So, there are two basic design criteria for reinforced soil wall one is called internal stability criteria another is called external stability criteria. So, among these two stability criteria let us first discuss internal stability what is internal stability it is meant by that stability of reinforced soil wall section. So, that the reinforcement strength is proper enough to take care of the strength or the stresses coming due to that vertical height etcetera. So, whatever reinforcement you are providing that should have sufficient tensile strength. So, that it can withstand extra large height which we are providing by using this reinforced soil wall that should not break also the anchorage of the or pull out of this reinforcement layers should be ensured these all aspects comes under the criteria of internal stability of the reinforced soil wall. And what comes under external stability external stability is nothing but the entire reinforced soil wall zone should not fail against sliding overturning those are coming under external stability. So, now when we talk about internal stability this is again I am referring to a part of the PHD thesis work of my first PHD student Dr. Sanjay Nimbalkar this is the paper details you can see Nimbalkar Chaudhary and Mandal in the journal Geosynthetics International published by institute of civil engineers London in UK this is the volume and page number in 2006. So, what we have done in the analysis this reinforced soil zone there are several layers of reinforcement as you can see a small infinitesimal element has been chosen at a depth of z with the thickness of layer as d z how this layer was chosen. So, that within the layer at the centre of the layer there is one reinforcement clear that way the element has been chosen why because it should not be chosen at the interface of the reinforcement and soil otherwise there will be shear stresses in this way or additional stresses coming from the this tensile strength of the reinforcement. So, if you look at this infinitesimal small element which is shown over here in this picture what are the forces acting these are shear and normal force on both the sides of the slice this horizontal slice and this t j is nothing but tensile strength of this reinforcement which is at the centre of this small infinitesimal layer and what is w i over here this is nothing but weight of this small infinitesimal section and this q h i and q v i are the seismic inertia force in horizontal and vertical direction. Here we have used pseudo dynamic approach because as I said Dr. Nimbalka developed pseudo dynamic approach and he used that method for all his analysis. So, these are the input seismic acceleration in horizontal and vertical direction the sinusoidal seismic acceleration. Finally, the results are obtained in terms of this parameter k what is this parameter k this is a non dimensional parameter which represents the ratio of the strength of the tensile strength of the reinforcement layer expressed in terms of with ratio of gamma is the unit weight of the soil h j is the thickness of that particular layer at which layer you are using that tensile strength required and d j is the interval between the two reinforcement layers clear. So, using that it is non dimensional once if you get the results in terms of k for different input values of your a h or a v or in terms of k h or k v in terms of coefficient you know how much reinforcement needs to be provided. So, that it can withstand that much seismicity. Another requirement is there for internal stability which is in terms of length of reinforcement how much of this length one is strength how much is the strength of the material you are providing that is which material you have to use which is having that much strength which can withstand that particular design value which you are getting from this proposed method. Another aspect is how much length of that reinforcement you are going to provide. So, that there is no pull out failure or any other internal failure of the reinforcement. So, how to find it out this is the expression this is the effective length of that j th layer l e j with respect to t j that is the strength of the reinforcement tensile strength divided by 2 sigma v j c i times tan phi. These are the few typical results you can see how practicing engineers or design engineers can use our proposed design chart. Let us look at here suppose for your seismic zone you know what is your k h value input k h value let us say it is 0.2 g and let us say the k v value for 0.5 times of k h we need to do the design. So, where we should go we should go to this point. Now, you get the value of this required reinforcement force in terms of that non-dimensional value of k. So, you get the value say it is about 0.6 little less than 0.6 say 0.57 0.58 using that now what you can do let us go back if you know this value you know your gamma of the soil you know your h j and d j how much you are going to provide the interval of the or thickness or interval of the individual strips of the reinforcement. Using that you can get what should be the tensile strength of your reinforcement which you want to provide. Another thing required length that l c how you need to find out same for k h say 0.2 we want to design with k v say 0.5 h. So, this point we have to go length is coming say close to about 1.5 times of this. So, it is expressed again in the non-dimensional form of l c by h let us go back here. So, this is our this value of l it is expressed in terms of this non-dimensional height of ratio of h. So, you know how much height you are going to design or achieve. So, knowing the value from this figure say 1.5 and h you get to know your l e value over here. So, once you know the l e if you go here t j already you have calculated l e you know. So, provide that much length if you want to check whether you are getting that much coefficient with respect to direct sliding or not that is another cross check. If you are not getting that direct sliding you change the value over here. So, this is the coefficient of friction with respect to direct sliding of the material. So, if you have a different material you will have different input values of c i tan phi which is nothing but the interface friction between the reinforcement and the soil clear. This is how the design can be made using our proposed approach as given in this paper. So, these are few comparison. Now, we have also validated our results with the earlier available results, but earlier available results all were using pseudo static approach and ours were pseudo dynamic approach. So, this table shows the comparison with respect to the work done by Shagoli et al in 2001. This is the paper published in geotechnic journal and this is the work done by Lechensky in 1997 and Ling et al in 1997. These are paper published in ASC journal of geotechnical and geo environmental engineering. You can see the comparison for different values of input values of k h what is the required reinforcement strength is required that is in terms of kilo Newton meter the tensile strength. Our present study for different values of soil phi value 20 degree, 25 degree, 30 degree you can see how much our present study gives. In most of the cases our present study at higher values of k h are giving that more critical value of the design. Can you see that that is little higher tensile strength is proposed as per the pseudo dynamic method is concerned where you have a scope to consider all the dynamic factors like frequency of excitation, time duration, then shear wave velocity, primary wave velocity even soil amplification also etcetera. And we have compared our results as you can see over here with respect to Ling and Lechensky's value of 1998 paper with respect to the how much required length is necessary in terms of the non-dimensional value. Remember that L c is nothing but ratio of that Le by h. So, for different values of k v and for input value of k h equals to 0.2 our present study suggest that more length of reinforcement is necessary for better stability of the wall. So, the details can be obtained in this paper. Now, coming to the external stability of this reinforced soil wall we now know how to design in terms of internal stability. When we talk about the external stability of this reinforced soil wall let us see over here as I said there are two cases sliding stability and overturning stability. The details of this analysis can be available in the journal paper by Chaudhary et al 2007 that is Chaudhary Nimbalkar and Mandel 2007 published in the journal Geosynthetics International published by Institute of Civil Engineers London UK. This is the volume number and page number you can see in our analysis we considered a two wage failure mechanism that is there will be a facing material over here and this entire wage a and wage b is the entire zone of reinforcement soil reinforcement. So, these are the seismic inertia forces in wage a which is triangular in nature and wage b which is rectangular in nature and considering the pseudo dynamic approach we calculated the stability in terms of sliding for this picture. Again the stability of the reinforced soil wall was considered in terms of overturning also considering this effective pressure coming from active earth pressure coming from this soil a wage a and then acting on the wage b when it is about to tip about this point clear. So, what are the recommendations or the design values of the results or design charts these are the design charts which gives us what is the requirement length of the reinforcement. So, remember in this case of external stability what we are getting we are getting how much length of the reinforcement to be provided for stability with respect to this sliding and overturning. Earlier we get in terms of internal stability the strength of the reinforcement and length of the reinforcement in terms of pull out failure now we are getting in terms of sliding and overturning failure. So, again designers can use this design chart come to a particular value of k h and corresponding k v get the required length how much is required in terms of sliding failure stability this is the minimum length required. So, that sliding failure will not occur similarly this is the minimum length required. So, that overturning failure is not occurred. So, how much finally when we are designing the reinforced retaining wall to withstand earthquake what should be our final recommendation the strength of the reinforcement material we have to ensure from the value of that t j that proposed design chart of the seismic reinforcement strength from internal stability criteria and how much length of the reinforcement to be provided should be considered from three criteria that is pull out criteria in terms of internal stability then sliding failure criteria of external stability and overturning failure criteria of external stability. So, among these three whichever gives us the maximum value or highest value of the reinforcement length to be provided that has to be adopted at the practice or at the design at the site clear. So, this shows the comparison of the result of this geosynthetic reinforcement length required in terms of direct sliding. So, LDS shows the length required in terms of direct sliding of course it is again represented in terms of non-dimensional parameter for different values of k v and k h with respect to Ling and Lechensky's paper of 1998 which is published in ASC journal, but this is for pseudo static analysis and our present study here mentions the analysis used by pseudo dynamic approach which gives a critical design for the practice to implement which is available in this journal paper. So, with this we have come to the end of today's lecture we will continue further in our next lecture.