 Welcome to the NPTEL lecture on video course geotechnical earthquake engineering. We were going through module number 9 of this course that is seismic analysis and design of various geotechnical structures. So, let us have a quick recap what we have learned in our previous lecture. We started with the subtopic within this module that is seismic design of retaining wall. We have mentioned that what are the various state of earth pressures acting on the retaining wall based on the movement of the wall. Same remains in the case of earthquake condition also. There are various examples worldwide the failures of retaining wall due to the additional destabilizing earthquake forces which comes on the retaining wall during the earthquake process. So, unless we estimate this extra or additional destabilizing earthquake force on the retaining wall and consider those for the design of retaining wall then failure cannot be avoided. So, for the seismic analysis and design of retaining wall we need to compute the combined earth pressure that is static plus seismic whether it is a active state or passive state of earth pressure and to do that we basically follow two different types of approach. One approach is called force based approach where we compute the earth pressure using the force equilibrium etcetera. Another is displacement based approach where in addition to the computation of seismic earth pressure we also compute how much amount of displacement of the wall is going to take place which is necessary for the performance based design. Then we had introduced what is the pseudo static method which was originally proposed by Terzaghi in the year 1950 and this is nothing but a coefficient which is multiplied with respect to the failure soil mass which will give the seismic inertia force. So, this is the additional destabilizing force as we can see from the D'Alembert's principle of mechanics. If there is an acceleration A the corresponding static inertia force or equivalent static inertia force will be A w by g where this small g is acceleration due to gravity. So, for this this book of Tuhata can be referred to for the details. Then we have seen what is the conventional seismic design of retaining wall which is nothing but the pseudo static approach as proposed by Mononobe and Matsu in 1929 and by Okabe in 1926 which is combinedly and commonly known as Mononobe Okabe method of 1929 which is the pioneering work for computation of seismic earth pressure on retaining wall. So, this is how the Mononobe Okabe method was proposed which is nothing but an extension of Coulomb's static earth pressure theory. So, in Coulomb's static earth pressure theory as we all know we have already seen this one for suppose for the active state of earth pressure the wall tends to move in this direction for an assumed failure plane with respect to a failure angle like rho over here. We know what is the value of this failure mass or weight w in this vector totally is known because it is magnitude and direction is known. Then we have the active earth pressure P a acting at an angle delta where delta is the wall friction angle and this soil reaction R which acts at an angle phi which is the soil friction angle. Now, these three forces should be concurrent forces to maintain equilibrium and if we draw the force polygon by knowing the line of action of this P a and R we will get the force polygon or closed triangle like this. So, this is w which is completely known now by drawing this line and line here wherever the intersect we will get the value of P a over here value of soil reaction R over here. So, for design what we do for conventional static analysis of Coulomb earth pressure theory we maximize this value of P a to get the design value of active earth pressure by changing this angle failure angle rho. So, that this maximum value of P a is obtained. Now, in the mononope wocabe method they added this values of K h w in both the direction for analysis and this K v w in both the directions and the critical direction whichever gives the maximum value of P a e that needs to be estimated under the earthquake condition or using the pseudo static approach. Also for the passive state so, this is for the active state as you can see over here. So, similarly for the passive state also we have seen that this is the Coulomb's passive earth pressure closed triangle or force polygon where w is again known R and P p their angle their directions are known, but magnitude was not known, but using this force polygon that is all the forces are concurrent we can get their magnitude also this is P p passive earth pressure this is the soil resistance and in addition to that this K h w and K v w has been introduced by mononope wocabe for pseudo static analysis, but limitations of this pseudo static approach we had already learnt in our previous lecture that is first of all this value of K h and K v what should be used for design is not clear. Also soil amplification we cannot consider in this pseudo static approach variation of the seismic acceleration with respect to depth cannot be considered in this pseudo static approach and effect of the dynamic soil properties are not possible to consider that means suppose if you are designing a retaining wall in a soft soil may be the loose sand and another retaining wall in the dense sand for both of them as far as the chosen soil properties are considered you can only take care of the static soil property that is phi value and wall friction value etcetera, but you cannot take any dynamic soil property which is also important factor as far as earthquake engineering is concerned as we have already learnt. So, what are the various available literature in the literature for the seismic earth pressure estimation using pseudo static approach and the recent pseudo dynamic approach there are many researchers who have proposed this work starting from mononope wocabe as I said 1926 and 29 then Madhavan, Kamesharao 1969, Richard and Elms 1979, Saran and Prakash 1979, Prakash 1981, Nadim and Whitman in 1983, Steedman and Zeng in 1990, Ebeling and Morrison 1992, Das 1993, Kramer 1996, Kumar 2002, Chaudhary and Subbarov 2005, Chaudhary and Nimbalka 2005 and many other researchers. So, we will talk about some of the recent research work on this seismic earth pressure not this pioneering or the older or former research work because that mostly closely relates to the mononope wocabe method the alteration of that method. So, we will more focus on the recent development of this seismic earth pressure theories for both active as well as passive condition. Now when we talk about the pseudo static method of approach the mononope wocabe method was force based method because in that case as we have already seen we consider only the force equilibrium and we estimate the value of P a e or P p e. But as I have mentioned for performance based design we need to look into the displacement based approach where we talk about the displacement of the wall also. So, to the pioneering work in the displacement based approach was proposed by Richard and Ems in 1979 this is the detail of the paper one can easily get you can see the details of the paper Richard and Ems 1979 seismic behavior of gravity retaining walls it was published in journal of geotechnical engineering of ASC volume 105 issue 4 page number is given over here. So, what it proposed in addition to the computation of this active earth pressure P a e under earthquake condition using the same pseudo static approach. But additionally it mentioned that it can be checked with respect to the permissible displacement of the wall and how to estimate the permissible displacement you can see this is the seismic active earth pressure can you see in the picture over here it is point of action also is considered like this. Now, within the wall wall is also subjected to if wall is having weight of w w then wall is subjected to seismic inertia force of k h w w and k v w w. So, these are the wall inertia forces including the wall self weight and for the stability this wall can either move in the sliding pattern that is it can translate or it can rotate or it can have a combination of these two. So, these are nothing but the displacements or movement sliding or the rotational. So, Richard and Ems proposed the sliding mode of movement that is for the amount of movement how to estimate it this is the force of resistance this is the normal reaction this is the reaction of or resultant of this f and n acting at an angle phi b where phi b is nothing but the angle of friction between this base of the wall and the foundation soil. So, displacement should be calculated using the formula and should be checked against the allowable displacement. So, the permissible displacement can be computed with respect to 0.087 v max square a max cube by a y to the power 4 where v max is the peak ground velocity a max is the peak ground acceleration and what is a y a y is known as yield acceleration for the wall backfill system. What is yield acceleration? Let me describe it what is the definition of yield acceleration yield acceleration is that acceleration when the factor of safety with respect to sliding of this wall will be equals to 1. So, a y refers to that seismic acceleration which beyond which if the acceleration increases then wall starts moving because then factor of safety with respect to sliding will be less than 1 right. So, a y gives us the that limiting value or that threshold value of acceleration at which factor of safety is equals to 1 that is known as yield acceleration. So, how to estimate that obviously as we know this will be nothing but factor of safety against sliding will be the stabilizing force with respect to sliding divided by the disturbing force. Disturbing force we know it is the horizontal component of this seismic active earth pressure and this seismic inertia of the wall which are causing the disturbance of the wall or tends the wall to move in this direction whereas what are the stabilizing forces this frictional force which can be computed as tan of this angle times the normal reaction. What will be the normal reaction that is nothing but if you do the vertical equilibrium this normal reaction is nothing but equals to weight of the wall and the vertical component of the pressure which helps to find it out. So, it has to be done algebraically to find out the normal reaction and normal reaction times tan of this base angle base friction angle will give you the stabilizing force right. So, that stabilizing force divided by the disturbing force will give you factor of safety against sliding that has to be equated with respect to 1 to get the a y value. Once you get the a y value you can always get the permissible displacement. Now, this is the permissible displacement under a particular given value of a max and corresponding value of v max. Now, if your allowable displacement for a wall if it is specified at a project then automatically you should keep your permissible displacement within the range of your allowable displacement and accordingly you can design the wall section. So, that your a y value etcetera comes accordingly. So, that this permissible value of displacement is within the allowable range clear. So, that these are the basic steps of the displacement based approach of the pseudo static method of design. Now, for pseudo static method further the method was extended by other researchers as I said the recent research work I will describe Choudhury and Subbarao in 2002 gave the design chart for the estimation of seismic passive earth pressure coefficient for negative wall friction case. What is negative wall friction case? Let me describe it little bit. So, when we are talking about passive condition of earth pressure as we have already mentioned for passive state of earth pressure on a rigid retaining wall. If we have ground like this and this is the failure plane we have weight of this for passive condition this is the R. Now, if we have seen already the P P value will be acting over here at an angle delta. This passive earth pressure when wall tends to move in this direction is nothing but called positive delta, but there can be few cases where this passive force may act in this direction also which is called as negative delta. This passive earth pressure is nothing but an application of this where we can apply this passive earth pressure not only for the design of retaining wall. This passive earth pressure we can apply for bearing capacity of foundation as well as anchor uplift capacity. How? Let me describe you when we are talking about some footing shallow footing problem like this which is let us say embedded at a depth shallow depth like this. As we all know let us consider the Terzaghi's bearing capacity failure theory which is all known to all of us from the basic geotechnical engineering course. So, this is the Terzaghi's theory which says we have one zone over here which is known as this angle is phi for static case. This is a zone one which is named as active zone. This is zone two which is log spiral zone and this zone three is rank in passive zone. This is log spiral zone and it is asymmetrical under the static condition. Now, if we consider this line let us look at here this line as a imaginary wall what we can say there it will be passive earth pressure acting at an angle delta with respect to normal to this. Why it is a passive earth pressure? First of all when we are talking about this imaginary wall let us see how the displacement is occurring when the load is acting on the footing from the column etcetera. This soil tries to punch inside and go inside. So, this surrounding soil moves this direction this wall tends to move in this direction towards the backfill side that is why it is a passive state of earth pressure clear. Why it is a passive state of earth pressure? Because this wall tends to move in this direction. Now, why it is in this direction of passive earth pressure? Because it is an acting at an angle delta and considering this composite failure surface that is log spiral followed by a rank in passive zone. We get the passive earth pressure acting in this direction which will balance this w and this outside load. So, basically 2 pp equals to q plus w this is the application of positive wall friction angle of passive earth pressure where what is the movement of the wall with respect to surrounding soil? Wall moves down this wall moves down compared to surrounding soil surrounding soil moves up that is why it is named as positive wall friction angle. What is negative wall friction angle? So, for positive wall friction angle what happens? Wall moves down surrounding soil moves up that is the relative movement between wall and soil for positive wall friction. For negative wall friction it is reverse in the sense wall moves up and this soil goes down. So, where is the application of that? Its application comes in the form of anchor uplift capacity where we can get that. Let us say this is an anchor plate which is being pulled by this force this is the ground surface we all know from our static theory. So, this will be failure surface let us say. So, if we consider this as imaginary wall imaginary retaining wall what is happening? First of all if we take this normal on this side there will be force in this direction which we will call as negative delta p p. Why? Why it is passive earth pressure? Because this imaginary wall again it tends to move in this direction when we are pulling out the anchor when we are taking out the anchor like this. Obviously, this wall tends to move in this direction towards the backfill. So, when we talk about this backfill and this imaginary wall it is nothing, but passive state and why we call it as negative delta? Because in this case this wall moves up and this surrounding soil moves down because this entire central area goes up this goes down clear. So, that is referred as negative wall friction angle case. So, passive earth pressure is very much utilized in geotechnical engineering for anchor uplift capacity whether it is a negative wall friction angle case or for shallow foundation bearing capacity theory if it is a positive wall friction. So, that is what in this paper if we look here for this negative wall friction means where this anchor uplift capacity is the application as it is mentioned. We had obtained the design chart for this passive earth pressure coefficient under the dynamic condition with respect to the unit weight component. There are two three components if you consider the generalized C phi soil where cohesion is also present and friction angle is also present including the surcharge on the ground surface. So, then the total seismic passive resistance can be computed using this expression and each of the this k values are nothing, but seismic passive earth pressure coefficient with respect to cohesion with respect to surcharge and with respect to unit weight. This chart gives the how this unit weight related seismic passive earth pressure varies with respect to pseudo static seismic acceleration k h and k v. You can see k v is 0, k v is half of k h and k v is 1 k h because these are typical ranges of k v and as per AASHTO code it says about two-third of k h should be considered for design as k v value. So, for different values of phi that is soil friction angle this design chart gives for a particular wall inclination ground inclination and for a particular ratio of wall friction angle to soil friction angle and that value should be negative as I have already mentioned this is the case for negative wall friction angle. So, the details about this concept can be obtained using pseudo static method in this journal paper by this is my work Choudhury with my supervisor Professor Subarao in 2002 seismic passive earth resistance for negative wall friction which is published in Canadian geotechnical journal volume 39 issue 5 these are the page numbers. Now, further that pseudo static method again applied to compute the point of application of the earth pressure because earlier if you remember when we talked about this earth pressure. Let us look at where this total value of p a or in dynamic condition p a e where they will act is not known. Mononobe Okabe had considered that let us say this will act at as one-third of height of the wall from the base as they considered it is a hydrostatic distribution of the pressure. So, they considered one-third from the base, but there is no logical explanation why it should be at one-third it was based on an assumption like is coulombic theory also we assume that it is hydrostatic pressure, but in static condition we know yes it is a hydrostatic condition. So, it has to be one-third from the base, but in this case it is dynamic case also why it should be at one-third that should be questioned. So, it need to be obtained where this point of application of this design value of this p a is acting because for design of this retaining wall we should know not only the value of this force, this value of force will give us to design the section, but to put the reinforcement in the wall suppose if you are going to use the reinforced concrete wall as a retaining wall you need to know how much reinforcement at different levels you are going to put and for that you should know what is the distribution or exact point of application of this p a so that accordingly you can place your reinforcement. So, point of application is also should be known and that also takes care of your sliding analysis, rotation analysis everything because every case the point of application is an important parameter. So, to find out that what has been done you can see in this paper Choudhury et al 2005 we worked this comprehensive review of different methods how to estimate the seismic passive earth pressure using pseudo static approach and their point of application. This is published in the journal current science in 2004 you can see these are the details Choudhury, Sitaram and Subbarao 2004 with my teachers professor Sitaram and professor Subbarao this paper was published in the journal current science volume 87 issue 10 these are the page numbers. So, what we did we considered an inclined rigid wall section. So, this is the face of the wall this is the ground surface horizontal ground surface for passive state of earth pressure we determined that is why displacement of the wall is showing in this direction. The entire failure zone which has to be optimized of course with respect to this failure angle as I have already mentioned, but in addition to that the entire zone we have not taken as a single mass, but we have divided it into number of small horizontal slices infinitesimal small slices like this a b c d of thickness d y at a depth of y from the ground surface. So, for each of this slices we considered what are the forces acting static force as well as the seismic pseudo static forces then considering equilibrium of each slices and doing the analysis and integrating over the entire height we got the expression for point of application for the seismic passive resistance in this form where omega is the excitation frequency v s v p r the shear and primary wave velocities etcetera can be used for estimation of this point of application. Then seismic passive resistance by pseudo static method for a generalized C phi soil was also estimated by us considering composite failure surface why composite we have taken because in the passive state of earth pressure we know planar failure surface seriously overestimates the value for passive case we should get the minimum value, but planar failure surface when the wall friction angle that is delta value exceeds phi by 3 that is one third of the soil friction angle in that case we should not consider for passive case it is always non-linear or curved failure surface. It is no longer a planar failure surface that Tarzaghi himself has mentioned way back in 1943 the same is true for the seismic condition also. So, that is why instead of taking planar rupture surface we considered the curved rupture surface like this which is a composite rupture surface like this portion b d is a part of log spiral and d e is a part of planar failure surface. So, that combination has been taken for this retaining wall phase is a b of height h for the passive earth pressure condition considering the surcharge on the ground uniform surcharge small q this is a c phi soil. So, there are three components of passive earth pressure one is with respect to gamma that is unit weight component which acts at one third from the base then cohesion component p p c d which acts at mid height of the wall and p p q d is the surcharge component of passive earth pressure which also acts at the mid height of the wall. Then summing them up we will get the total seismic passive resistance the details about this study is available in this journal paper seismic passive earth pressure in soil journal of geotechnical and geo environmental engineering of a c published in 2005 volume 131 issue 1 these are the page numbers. So, what we had proposed in this case finally we gave the design chart that is for how to use this coefficient of passive earth pressure for different input values of k h and k v. So, how people can use it very easily for a particular seismic zone in India or if anywhere in the world they should get what should be the input parameter of design value of k h may be based on design basis earthquake or something like that to estimate k h value and k v value based on that input k v and k h value at the site what is the value of phi friction angle of the soil you can go to this chart get the value of this seismic passive earth pressure coefficient and directly using the formula as I had already mentioned this formula you can get the passive earth pressure. So, that way the design of the wall you can do once you know the passive earth pressure they are total value and also their point of application for further design. Now, in pseudo static method other researchers also work as I have already mentioned only the very recent research works I am showing you here like Shukla et al in 2009 they described the derivation of an analytical expression for the estimation of total active force that is static plus seismic active force on a retaining wall with a backfill of c phi that is both cohesion and frictional soil considering both the horizontal and vertical seismic acceleration. So, what is the expression for the computation of total active earth pressure under seismic condition is this one. So, this active earth pressure coefficient k a e gamma is with respect to the unit weight component of the soil whereas, k a e c is the coefficient of active earth pressure with respect to cohesion component and their expressions for k a e gamma and k a e c is given by these expressions where tan theta in this expression this theta tan theta is nothing but k h by 1 minus k v like for mononobe okabe also this is the same expression and alpha c is the critical failure angle with phi is the frictional angle of the soil. So, this is the details about the paper that is Shukla Gupta and Shibakugan this was published in journal of geotechnical and geo environmental engineering of a s c this volume this page numbers in 2009. Now, let us see what are the advantages and disadvantages of this pseudo static method when we are doing the analysis for retaining wall what are the major limitations first let us look at that like representation of the complex transient dynamic effects of earthquake shaking by a single constant unidirectional pseudo static acceleration is very crude. So, that is why it is one of the major limitation of the pseudo static method it neither consider the complex nature of the dynamic load it does not consider the time duration or transient nature of the dynamic load and other dynamic effects like frequency of excitation shear wave velocity primary wave velocity all these seismic waves which are travelling through the soil during this earthquake shaking none of this dynamic properties are considered in this pseudo static method only a single constant value is used. So, that is why it is a very crude estimation of the dynamic nature of a problem in a quasi static manner also the relation between that k and the maximum ground acceleration is also not clear suppose 1.9 g acceleration it does not mean that k value should be 1.9, but what are the advantages why people still use this age old pseudo static method because it is very simple to use as we have seen it is nothing, but an extension of static analysis only it is a quasi static or pseudo static as the name suggests and it is very straight forward. So, there is hardly any complexity involved in the analysis that is why many people many researchers many practitioners they preferred this method because of its simplicity no advanced or complicated analysis is necessary because of the nature of the problem. So, it uses the simple limiting state of equilibrium of the analysis which is routinely conducted by geotechnical engineers mostly the practitioners etcetera when they want to design any retaining wall in the seismically active region to consider the seismic earth pressure on the retaining structures. Now, let us move forward and now let us talk about the development of modern pseudo dynamic approach. This pseudo dynamic approach was originally proposed by Steedman and Zeng in 1991 which has been further modified and the generalized solution for pseudo dynamic approach was given by Choudhury and Nimbalkar in 2005 for the first time. So, what is pseudo dynamic method? Let us look at this problem. So, this is again we are talking about the basic rigid retaining wall analysis this is the line diagram AB is nothing but the vertical face of the retaining wall this is the ground surface. This condition is showing the passive state of earth pressure that means wall it tends to move towards the backfill side that is why this is the seismic passive earth pressure this is the soil reaction this is the weight of the failure mass ABC at a particular failure assumed angle of alpha and these are the seismic inertia forces Q H and Q V Q H is horizontal seismic inertia force and Q V is vertical seismic inertia force. In the pseudo static method how we estimated this Q H and Q V nothing but Q H is coefficient of seismic acceleration that is K H times this W and this Q V that is vertical seismic inertia force was estimated as K V times W where K V is vertical seismic acceleration coefficient. But now we are proposing this pseudo dynamic approach which takes care of all this dynamic effects. So, what are the advantages of this modern pseudo dynamic approach over the conventional pseudo static approach? Let us look at it first like soil amplification which may happen in some of the cases of the soil as we have already studied that in the previous module on site response analysis that can be considered in the design that is the one biggest advantage of this pseudo dynamic approach which the pseudo static approach cannot do. Then frequency of earthquake excitation is also considered. So, this dynamic nature is considered at what frequency the earthquake motion is coming. So, that can be taken care of time duration of the earthquake that is the transient nature of the earthquake can be considered in this method. Phase differences between different waves can be considered amplitude of equivalent peak ground acceleration can be considered and it considers the seismic body wave velocities body waves means S wave and P wave both traveling during the earthquake. So, let us look at this basic picture once again during the earthquake we have this vertically propagating upwards this shear wave velocity V s and primary wave velocity V p. Now, why this is vertical that already we have described that if it comes from a large depth below the ground surface after several layers it will be almost close to vertical. So, it is a good assumption to consider the vertically propagating seismic wave like that. Now, the seismic accelerations which are considered for the analysis are expressed in the form of a sinusoidal motion. So, that is why the name pseudo is still there one can always say why this dynamic approach is still having this component of the name the pseudo because of the reason because earthquake motion will be random in nature it will not follow any sinusoidal or any particular mathematical function it will be fully random in nature. But to have a generalized design approach we cannot do the design with random motion in that case it will be a case specific design that is for each taken random motion you will get different values instead of that the practitioners or designers or the codal provisions will always look for a closed form design solution or generalized design solution. So, to do that what we have proposed we have considered the equivalent sinusoidal motion like once you have the random motion of earthquake take the area under the curve of that acceleration versus time history and find out the equivalent area in the sinusoidal form of the same acceleration versus time history which can be represented in this format. So, horizontal acceleration a h is now no longer a constant, but it varies with respect to depth from the ground surface that is z measured from the ground surface and time t that is up to which this earthquake is occurring which is expressed in this form that is 1 plus h minus z that is at a depth of z a small infinitesimal slice of thickness dz is considered. So, how it varies 1 plus h minus z times f a minus 1 by h that means f a considers the soil amplification factor. Soil amplification factor is nothing but from the bedrock to ground surface how much the PGA value got amplified or the maximum value of earthquake acceleration got amplified. So, that gives us the amplification factor that non-dimensional value of amplification factor is assumed here to follow a linear distribution that obviously people can argue and can take different variation, but when there is no other solution one can fairly estimate the variation of f a as a linear variation with this expression this is nothing but a linear equation with multiplication of a h, a h is nothing but amplitude of the horizontal earthquake acceleration and sin of omega t minus h minus z by v s where this t is the duration of earthquake h minus z by v s it gives nothing but the phase as you can see this is nothing but the phase right. So, that automatically gives us the expression how this sinusoidal horizontal seismic acceleration varies with respect to z because there is a function of z over here and varies with respect to time because it is a function of time and frequency is also involved. So, frequency is also taken care soil amplification is taken care and shear wave velocity is also taken care of because shear wave velocity mostly contribute on the horizontal seismic acceleration as we already know. Similarly, for the vertical seismic acceleration what is the equation the a v is also a function of z and t which is expressed as 1 plus h minus z times f a minus 1 by h times a v this a v is nothing but amplitude of vertical seismic acceleration times sin of omega times t minus h minus z by v p because as we know the primary wave velocity that will contribute to the vertical seismic acceleration. So, that is why this expression is considered for both horizontal as well as vertical seismic acceleration which are now function of depth and function of time. Now, once you know the acceleration what you can do for this infinitesimally small slice what is chosen over here you can find out the mass of it and once you know from the geometry its mass of it you can multiply that mass with respect to corresponding acceleration to get the inertia forces. So, that is what it has been done let us look at here if you do not consider any amplification then the equation boils down to this that is amplification is 1 basically there is no amplification. So, in that case the expression for the seismic horizontal inertia force q h t will be nothing but mass of that infinitesimally small slice that is m of z times the acceleration h z will give you the seismic horizontal inertia force. If you integrate that over the entire height of the wall that will give you the total horizontal seismic inertia force which on simplification can be expressed like this where this value of lambda is nothing but t times v s which is nothing but wavelength of the vertically propagating shear wave and this parameter eta is nothing but t minus h by v s then the total seismic vertical inertia force can also be estimated in the similar fashion that is integrate over the entire height of the wall 0 to h m z times a v times a z t that will give us on simplification this expression where this eta is nothing but t times v p is the wavelength of the vertically propagating primary wave and this psi is nothing but t minus h by v p. After doing this what is done here the details can be obtained in this paper Choudhury and Nimbalkar Dr. Sanjay Nimbalkar was my first PhD student this is a part of his PhD thesis work under my supervision at IIT Bombay in 2005 this paper has been published in the journal Geotechnic published by institute of civil engineers London this is the volume number page number. So, what has been done after finding out the Q H and Q V expression W is already known as I said then the limiting equilibrium of all these forces involved was considered and once you do the limiting equilibrium of all the forces involved in this two dimensional problem the total that is static plus dynamic passive resistance can be obtained like this which needs to be optimized with respect to your this angle of alpha. Remember in static case we need to optimize it with respect to only this angle of chosen failure plane alpha but in the case of dynamic problem it is not only you have to optimize it with respect to alpha but you need to optimize this with respect to duration of earthquake also in terms of the frequency of earthquake how it has been done I will come to that now. Let us see what is the coefficient of seismic passive resistance coefficient of seismic passive resistance which is nothing but our final design value which designers will always look for when they are doing the design of any retaining wall this is the closed form expression for k p e in this case this m 1 and m 2 are expressed by this you can see over here this parameter small t by capital t this capital t is nothing but time period which is related to the omega that is frequency as we all know and this small t is nothing but duration of earthquake. So, we have to optimize this value of k p e what should be the design value of k p e as we know for passive case it should be the minimum value. So, minimum value of k p e needs to be obtained with respect to a combination of this alpha value and this t by t value. So, that optimization has to be carried out it is an optimization problem to obtain the minimum value of k p e or design value of k p e which was not present in the case of static problem in that case we have to minimize only with respect to this alpha value. Now how to get the seismic passive earth pressure distribution over the entire depth this is one advantage of this pseudo dynamic method like in pseudo static method earlier as we have mentioned in most of the cases we were getting only the total value of the earth pressure except for one method where we have mentioned about that method of slices right otherwise we were getting the total value we were not getting the distribution, but in this case of pseudo dynamic method as we have already taken this infinitesimally small slices. So, at each level of the depth you are getting the design value of this p p e. So, automatically that gives you the distribution or it can be mathematically expressed in this form this is the variation of seismic earth pressure distribution with respect to depth. So, once you know the distribution automatically you know the point of application of the total passive resistance as well which will help you to put the reinforcement in your wall section when you are designing the wall fine. So, all these details are available in this journal paper of Geotechnic by Choudhury and Nimbalkar 2005. Now, let us see how the results are varying as far as the proposed pseudo dynamic method is concerned. So, for the seismic passive resistance you can clearly see now the seismic earth pressure distribution is fully non-linear earlier in all the pseudo static method we were getting linear earth pressure distribution which should not be the case because we know due to seismicity the earth pressure cannot be hydrostatic or the triangular distribution it cannot be a linear variation it will be a non-linear process. So, that non-linearity has been also captured in this pseudo dynamic method. So, these are all the advantages of pseudo dynamic method in addition to that what I have mentioned in the first slide of this pseudo dynamic approach. So, you can see the results this red line shows the value when k h equals to 0 that means under the static condition because k v we have considered for 0.5 times k h. So, if k h is 0 k v is also 0. So, this red line shows the passive earth pressure under static condition and as the seismicity increases as we know the direction of the seismic acceleration will act in both the directions whether it is horizontal or vertical, but we have to take the critical value for the design and what will be the critical value for passive case it will be the minimum one and minimum one is nothing but it should reduce with increase in the seismicity. So, that is what from the static case of this red line as you can see as the seismicity increases from k h value 0 to 0.1, 0.2, 0.3 g this value of passive earth pressure distribution under earthquake condition is also reducing. So, those are design values and as the seismicity increases the non-linearity of this variation is also increasing. So, that automatically shows that point of application which was in the static case at one third from the base of the wall is no longer at one third, but it is below one third for the case of passive earth pressure fine. Some more results when we are talking about the effect of soil amplification how this amplification effects the design value of this earth pressure. Let us see the seismic passive earth pressure coefficient when there is no amplification that means f a equals to 1, f a equals to 1 means no amplification factor. That time depending on this value of h by T v s you can see this is the variation of k p e for no amplification case, but if there is amplification in the soil if suppose your soil gets amplified 1.4 times or 2 times then in that case the design value of k p e is still decreasing what does it mean that means the design value should be much lower than what it is the case for the non-amplified soil clear. So, this criticality also can be captured in the case of pseudo dynamic approach which is not possible in the case of pseudo static approach. So, all these are advantages of pseudo dynamic method. This is another comparison you can see for this case of k h equals to 0.2 and k v equals to half of k h with this value of phi and delta. Mononobio-Cabe method which is pseudo static method that will give a linear variation or hydrostatic distribution straight line variation like this the red color whereas, the pseudo dynamic method in the passive case it is non-linear and this is the non-linear variation. You can always say the total value of the passive earth pressure there is that can be said almost similar for this particular chosen set of data value, but the passive earth pressure coefficient values may vary which is necessary for the design. Let us look at this values which is also available in this journal of geotechnic by Chaudhary and Nimbalkar 2005. This passive earth pressure coefficient the design value this dotted line is for Mononobio-Cabe with this rectangular boxes then this dotted line we can see with circles are proposed by Chaudhary in 2004 and the triangular one is the present study means this pseudo dynamic approach. You can see the design value k p e is minimum what we are obtaining in the pseudo dynamic method compared to the conventional pseudo static method. So, what we can say the critical value at the most desirable design value is proposed by pseudo dynamic model not the pseudo static model for the passive state of earth pressure which is clear from this picture also. Now, let us come to the next case of seismic active earth pressure and applying this new pseudo dynamic method. So, for the case of seismic active earth pressure on the retaining wall this is the line diagram this is the retaining wall planar failure surface is assumed again this is the active state of earth pressure. So, that is why this is the position of the active earth pressure seismic active earth pressure p a e at an angle delta with respect to normal to the wall and w is the weight of this failure wage q h is the seismic horizontal inertia force q v is the seismic vertical inertia force remember they act in both the directions. So, and we have to find out the critical direction of it by doing an optimization of analysis and f is the soil reaction. So, here also this infinitesimal small element is been considered at a depth of z of thickness d z z is measured from the ground surface and v s and v p are the shear wave velocity and primary wave velocity of the earthquake excitation. The details are available in the publication of Dr. Sanjay Nimbalkar's phd thesis work that is Choudhury and Nimbalkar 2006 in this geotechnical and geological engineering an international journal published by Springer this is the volume number page number etcetera. So, what we can see the similar approach has been adopted in this case it is the active state. So, the seismic inertia force in both horizontal and vertical direction is obtained in the similar manner, but when you are doing the equilibrium and finding out the p a e you have to do the proper analysis of all the forces involved and their corresponding directions etcetera to get the design value of this and once you get the expression for p a e next step is to do the optimization once again and this is the expression closed form expression or the design expression which designers will like to use always this is the k a e expression with m 1 and m 2 values over here. You can see here also the optimization requires with respect to the chosen alpha angle and this t by t ratio and here again we are getting the advantage of seismic active earth pressure distribution the variation of it with respect to wall which gives us the what the point of application of the total seismic active earth pressure will act that also will give us the idea. So, this is the result you can see over here this red line is the result for the static case and as the seismicity increases as we know for the active state that design value or critical value should increase right because in active case we maximize the design value. So, that is why as the seismicity increases the design value increases compared to the static one and also as the seismicity increases the variation of the active earth pressure distribution with respect to depth it is highly non-linear as you can see over here what it can be said the point of application of the total seismic active thrust which is one third from the base of the height of the wall and static case is no longer at one third, but it shifts upwards than one third in passive case it moves downwards in active case it moves upward. So, that gives us the exact position where and what amount of reinforcement in the soil reinforced soil wall when we are trying to construct and design a reinforced soil wall when we are trying to construct RCC rigid retaining wall where the reinforcement has to be provided that is clear from these results which is not available in the pseudo static approach. Again the variation of this amplification factor can be seen over here like when there is no amplification this black line this is the value of k a e the design value without amplification, but when the soil gets amplified during the earthquake process that material property also can be taken in this pseudo dynamic approach and you can see the design value of k a e is increasing can you see that automatically says that we need to take the higher value of seismic active earth pressure when we are designing our wall in a soil which is going to get amplified subjected to some earthquake motion. So, these details are available in the publication the effect of amplification etcetera on the seismic active earth pressure publication of nimble current Chaudhary in 2008 in the journal of earthquake and tsunami this is the volume number and page number. So, with this we have come to the end of today's lecture we will continue further in our next lecture.