 Last class, we have discussed this about the slope stability analysis different methods and this factor of safety. If factor of safety is equal to 1, then it is said to be the slope is failure. If factor of safety is greater than 1, then we can say that slope is safe. Then there are two types of slopes as we have earlier discussed. One is your infinite slope analysis or there is your finite slope analysis. We have finished for this infinite slope analysis. It is a function of factor of safety for C, phi, gamma, beta, angle of this slope D and U. Then there are water forces on the soil considered. So, for this there are two methods. One is equivalent methods for water forces. Case one is boundary water forces, side water forces or at the boundary it has been considered plus your total unit weight has been considered. Case two it is a seepage force plus the submerged unit weight has been considered. We have discussed this boundary water force case as well as the seepage force cases. Then come back to your finite slopes. We have discussed this planar failure surface and in this planar failure surface where are the cases the planar failure surface or block slides occurs. That also we discussed. Then your finite slope of circular failure surface by means of rotational slides, method of slices and particularly cohesive soils and ordinary method of slices by means of a linear method. This is a simplified method and Spencer's methods we have discussed. And after that now we are going to start this finite slopes of non-circular failure surface, finite slope of non-circular failure surfaces. So, case one in this case of if it is not circular failure surface, case one is your wage method. Second is given by Jambu. This is called Jambu simplified method. And third one is your Morgenstein price method. So, it is called simplified that by means of price method. Let come back to wage method. In case of wage method failure surface consist of two or more planes and applicable to slope containing several planes of interfaces and weak layers. That means failure surface consist of two or more planes and applicable to slope containing several planes, several planes of interfaces and weak layers. Generally force equilibrium has to be satisfied in case of wage method. And the assumption is that resultant of side forces on each slice either arc horizontally or at varying angle from the horizontal typically up to 15 degree, typically up to 15 degree. Then in this case if you look at this wage method failure surface consist of two or more planes and applicable to slope containing several planes of interfaces and weak layers. So, failure surface consist of two or more planes in case of wage method that if you look at the layer one and layer two or I can say layer A and layer B. That means if you look at here this is one wage and other wage third like I name it one, two, three, four. There are four wages. If you take out this number of wage let us consider any of the wage. So, this resultant of the side forces would locked at an angle theta. And below this normal and tangential force normal forces at arc as an angle theta m. Now in wage analysis equilibrium of force in each slice is considered to adjacent consider to adjust the inter slice forces and balance them resulting in a correct solution. If you look at here a typical example has been considered. So, for layer one this is a four feet height layer one of soil is a four feet height layer two layer two soil is a ten feet height with this the properties has been given gamma c and phi prime for layer two c is zero. That means it is a cohesion less soil for layer one phi is equal to zero this is a cohesive soil. So, considering one, two, three, four the four wages. Let us start with this start with this part one number one wage. So, this is your number one. So, it is weight component means resultant of normal and shear forces on a b this plane resultant of normal and shear forces on plane a b. If you look at here resultant and normal forces will locked in the plane a b. Now go back to your slide two if you go back to your wage two this is the wage two the forces coming here wage two will be if you go back to your normal force slide will locked in the plane in the plane b c and this is your plane b c and this this one if I take it out if I look at here this one is your side force between slice one and slice two if this is my side force between slice one and slice two. So, this is slice one and this is slice two in this case this is your side forces whatever the side forces coming in slice one and slice two and this part is your cohesion force this part is your cohesion force this is your cohesion force acted on the acted on the on plane b c along the bc what is the cohesion force because here phi is equal to 0 and c is that that means this is a purely cohesive soil in this case this force equilibrium that polygon has been drawn so this is a trial solution for trial solution for assumed factor of safety of 1.5 1.5 now if I come back to slice third slice look at here third slice in case of third slice if you look at here this part is your this part is your side force between slice three and four slice force side force between slice three and four this is my three and this is the four this is your side force these are all your side force acted on this and this will be there then this is your w three if I complete this equilibrium condition has to be satisfied the force polygon has to be completed so this is your weight w three acted I put it here vertically and here it is your side force acted up and this part this force if you look at here from here to here it says side forces between slice two and three slice two and three if you look at here there are two side forces come into picture for slice three one is your side force for slice two and three acting in this plane similarly for slice three and four side forces acting in this plane so particular this part this is your representing side forces acting between the slice two and three between the slice two and three and this is representing this is representing particularly this this this part if you look at here this part this is representing your side force between three and fourth between three and four and this is your weight weight of your slice three and this straight line forces normal force on c d normal force on the c d it acted on this this normal force has been brought it and this part this part is your cohesion force cohesion force on c d because here if you look at this layer two there is a cohesion force c is equal to five hundred pound per feet square five is equal to zero so this force polygon has to once this is a in equilibrium condition that means this force polygon has to be completed from there you can find it out what is your factor of safety or to get your factor of safety if you look at here side force between three and four then the for four polygon per case one slice case two slice case three slice case four this is a case of case four means slice four slice four if I take this slice four slice number four and this is your weight this component is your weight component weight on of this slice and this is your cohesion component on the four on the face d e in this face d e this is your cohesion component this is your normal force normal force in d e this is your normal force and this part is your as if you look at this once I start from slice one two three four if the moment I go to the slice four and this part of here this is the because this is not a complete force polygon so whatever this we are getting this is your unbalance force of on slice four this unbalance force on slice force that means slice four that means whatever factor of safety you have assumed that is not correct means the methodology how we are going for wage analysis if you look at here once once again I am explaining for example you consider this is two layer soil and this is your slopes and take it into wage number of wages or number of slices one two three four this slices has been assumed this slices has been assumed you can take it also more number of slices in between also so then what will happen you start with one by one slice slice one slice two slice number three slice number four and each slice this equilibrium condition has to be satisfied to satisfy the equilibrium conditions look at the forces acted on this considering the side forces in this method wage method they have considered the forces between this slices that means slice one and slice two side forces has been considered not like earlier the side forces has been cancelled so each slice this force polygon has been drawn and a trial assume you take a trial assumption of your factor of safety you start with a factor of safety assumed value of factor of safety let us say assumed value of factor of safety let us we have trial trial with this factor of safety is equal to one point five and start with this slice one two three four once you start taking this assume factor of safety one point five and we have started from the top to bottom we find that this is our on balance force that means this force polygon is not completed that means equilibrium condition has not been satisfied then what will happen because you assume the factor of safety of one point five depending upon this what is your on balance force accordingly whether it is a positive or negative side accordingly or larger or smaller accordingly you can take your factor of safety either you can take one point three one point one or may be one point six depending upon that you assume this trial factor of safety till this force polygon will complete till this force polygon will complete that means equilibrium condition satisfied so this is the way how we solve this particularly wage analysis if it is a non circular non circular case then how we are going to solve this case then second method is jambu simplified method so jambu has given a method of slices applicable to circular as well as non circular failure surfaces jambu has given both for circular as well as non circular failure surfaces and factor of safety is a function of c b weight pore water pressure and f zero is a correction factor if you look at this factor of safety is a function of it is written in terms of f zero f zero is a correction factor that varies with depth to length ratio depth of the slope and length of the slope ratio that varies as depth to length ratio of sliding mass and type of soil so it is based on what jambu has simplified method given a simply a correction factor has been given correction factor has been given based on your length of the sliding mass as well as depth of the sliding mass if you look at here this is my depth of the sliding mass as well as this is the length of the sliding mass length and depth of the sliding mass of each has been taken into consideration based on that because it has to satisfy your equilibrium conditions a correction factor has been applied and this method can be applicable for both circular as well as non circular failure surfaces so jambu has given particularly in this case if you look at here as I said if this is my non circular failure surfaces this is the depth this is the depth and from here to here because these are all non circular these are all non circular failure surfaces or then this is the depth and from here to here top to bottom from here to here this is a length based on the length and depth they have given your correction factor f 0 for different kind of soil phi for case one this is a case of ratio of d by l depth to length ratio it is varying this is the case of your factor safety not factor safety it is your correction factor with this for phi is equal to 0 this is your case for c phi soil for c is equal to 0 in this case once phi is equal to 0 that means this case is your purely cohesive soil and this is for c and phi cohesive as well as frictional cohesion less soil this is your purely cohesion less soils this correction factor chart has been given then third one is your price method in this case no assumption is made the advantage of this both this methods over the this price method over the last two method no assumption is made regarding inclination or point of application of resultants if you go back here if you go back to your wage method what is your assumption if you look at this assumptions resultant of resultant of side force on each slice either earth horizontally either earth horizontally or at a varying angle from the horizontal typically of two fifteen degree that means if this is my slice the resultant of side force the resultant of side force it will act either the resultant of the side force it will act either horizontally look at here this way it will act or at a varying angle with horizontal if this is my horizontal at a varying angle maximum varying angle will be fifteen degree that means if I make this angle if this make this angle more than fifteen degree it may possible that the resultant may may act at an angle more than fifteen degree this is the limitation this is the limitation in this case of this wage method as well as jambus method so to overcome that morgan's morgan's turn and price they have not done any assumption they have not made any assumption particularly for non circular cases of the failure surfaces so no assumption is made regarding inclination or point of applications that means there is no assumption whatever this inclination is there accordingly the point application point of application of resultant will be acted and these are determined as the part of the solutions and these are from determined from the part of the solutions there is no assumption what is the actual resultant forces where it is acted with respect to horizontal this has to be determined as a part of the solutions and because other methods are very simplified it requires computer programming to solve the basic equations this basic basic equations it is somehow or somewhat you can say that it is exact but not practical it is more exact but it is not practical so these these are all basic summaries of this slope stability analysis what are the methods and references has been given references has been taken from this dunkan et al 1987 and your thus bm thus from this book and your soil mechanics design manual by means of your departmental of navy in united states us in 1982 one more thing I want to show it now if I go back to there are other cases also if you go back to previous one what I have discussed about rotational slip analysis or rotational analysis for undrained frictional friction less failure total stress generally for a fixed undrained friction less failure that means if it is a undrained friction less failure generally what happened we follow total stress analysis for cohesive and frictional failure method of slices as well as bishops conventional method has been used we have discussed this method of slices also bishops conventional method also we have discussed now if I go back to this for rotational slip for total stress analysis if you look at this total stress analysis for total stress analysis it is assumed that phi u phi is your frictional resistance in undrained conditions is equal to 0. So strength parameter are those in undrained soil particularly strength parameter are those in undrained soil this is your strength parameter for those in undrained soil if you look at this total stress analysis this is your circular slip surface in this circular slip surface you identify the center where the failure surface has been made and the center with radius r and find it out the weight find it out weight of this weight of this entire soil mass within this failure surface that is your weight w and it will acted in the center of gravity c g how far this weight component is far away from your center of rotation from 0.0 this distance is your point e if I take considering the equilibrium condition taking in taking movement at the point o your factor of safety is coming about to be c r square c r square theta by w e if you look at here factor of safety is your resisting or restraining movement divided by disturbing movement divided by your disturbing movement if you come back to your restraining movement in this case c r square theta if you look at here this cohesive force c this is acted on by c with this this total length we can get it from because this cohesive force from where it will act this unit cohesive force it will act along the very very of your failure surface. So if you look at here the failure surface what we have assumed is your rotational or circular slip surface this is acted along a and b. So unit cohesion into length l has been found out by r into theta this is your radius it will acted on this angle is your theta. So this restraining movement will come out to be c r theta it will be acted over if I take this it will multiply into r about point o. So this will be your c r square theta that means resisting movement. Then which one is your disturbing movement this is because of your weight component of the soil mass along the slope along the failure surface that will be acted at a distance this is your weight component at a distance e from your center of rotation o point o. So this will be w into e in this case your w is your weight component this is your w is your weight component it acted at the point of center of gravity c g here it is your c g c g then this c g is distance is at a distance e from point o. So if I put it w into e w into e this is your clockwise and c r theta into r c r theta into r that means c r square theta it will be anticlockwise. So factor of safety is your resisting movement divided by disturbing movement which is equal to c r square theta by w e where c is your cohesive strength either you can it is generally Pascal or kilo Pascal then r is your as I said slip circle radius this is your slip circle radius r is equal to slip circle radius in meter theta is equal to slip sector in radius this is your theta this is in terms of radius w is your weight of sliding sector w is equal to weight of sliding sector e is equal to eccentricity e is equal to this weight component how far from there e is equal to eccentricity this is your in movement this particularly this has been a total stress analysis in this case the assumption is in case of total stress analysis phi u is equal to 0 phi u is equal to 0. Now in this method of slices there are two cases one is your by means of slice thus that has been given by swedish circle method. So it is used for cohesive soil as well as and also frictional soil c as well as c phi soil if you look at here this is my circular failure surface circular failure surface assume circular failure surface r is your radius and theta this failure surface make at angle theta then it has been made into finite number of slices. So as I said last class n number of slices you can go up to 1 2 3 4 5 up to n number of slices depending upon that how many number of more slices you will take it more number means means your result correctness of the result will be good. So it has been taken 1 2 3 4 5 6 for example 6 slices has been consider any slice you take into take into consideration for any of this slice find it out is your force equilibrium then find it out what are the forces going to act if you look at this forces going to act in this cases lateral forces l 1 and l 2 your weight component this is your weight component acting vertical downward normal component and as well as your tangential component if I take into consideration that means what is your factor safety as I said this is your resisting movement by disturbing movement. So in this case resisting movement is your cohesion force it will be anticloquize c r theta then plus your n n is your normal component n n n into tan phi and is your disturbing movement for each slice if I take into t n tangential component is coming that is your t n that is your disturbing movement. So summation of that means if I have n n number of slices 1 2 3 4 5 6 7 8 or may be say for example in this case 7 number of slices that means for each slice you have to find it out normal component as well as tangential component. So summation of normal n tangential component you make it. So you can find it out your factor safety so that means here summation 1 to n and here 1 to n. So factor safety is equal to for method of slices c r theta this is for c phi soil n tan phi divided by tangential component t n then there is another chance it may happen that there will be a tension crack because for particularly cohesive soil if cohesion component is there there is a possibility that there is a tension crack what do you mean by tension crack that there will be gap in the soil there will be gap in the soil in this case if you look at here this is the tension crack this tension crack you can find it out up to at what height up to what height tension crack can develop. So what will happen because of the tension crack what will happen it will reduce this angle of sliding sector it will reduce this angle of sliding sector this theta will be reduced to theta prime sliding sector this angle will be reduced because of your tension crack then you will have to find it out your height of tension crack for friction less soil for friction less soil you will have to find it out for cohesive soil you have to find it out your height of tension crack you can get it 2 c by rho rho is equal to unit weight of soil or 2 c by gamma you can put it c is equal to cohesive strength either it is a pascal or kilo pascal 2 times of c by gamma or rho. So you can find it out then from there for this is for friction less soil and for cohesive as well as frictional soil that means friction as well as friction less soil that means particularly c phi soil typically if I say c phi soil this is for c soil this is for c phi soil the height of tension crack h c you can find it out 2 c by rho or gamma 2 c by rho or gamma tan 45 degree plus phi by 2 from there you can find it out height of tension crack then once you get the height of tension crack then next part is your as I discussed what is the effect of the tension crack because of the effect of the tension crack this angle will be reduced why this angle will be reduced because once you draw the failure surface the failure surface will go from here to here it will end here earlier cases what happen once there is no tension crack the failure surface will go propagate and it will end up to the surface look at here. So in this case what will happen this because there is a tension crack there is a tension crack the failure surface will go up to the tension crack that means what you have to find it out you have to identify what is your height of your tension crack once you get your height of tension crack then you can assume failure surface you can you can draw your failure surface then your theta will be modified to theta prime because of tension crack this theta prime will be reduced once this theta prime will be reduced. So your weight as well as c component also that resisting movement also will be reduced then second part second second second concern is your first concern is your in this case tension crack second concern is your location of slip circle center this is my slip circle this is my slip circle how do I locate where this slip circle center is lying where exactly it is lying I do not know where is the center where is the radius so that I can draw the slip circles. So this is there is no simple way this generally determined by means of trial and error by means of trial error. So f is in this case f is more sensitive to horizontal movement than vertical movement factor of safety is more sensitive horizontal movement than your vertical movement horizontal movement than your vertical movement. So it has been this center has been found out by means of a trial and error method. So by trial and error you will get your slip circle of the center with your critical failure surface that means every you draw the failure surface 1 2 3 4 and find it out your center then by trial and error you will identify for critical failure surface where exactly your center will be located this is your second concern first concern is your tension crack second is your location of slip circle then we will discuss little bit more about your effective stress analysis may be next class I will stop it here.