 Last class, we have started this stability analysis. If I quick redraw, what has been done in the last class? A gravity retaining wall has been taken into consideration, simple retaining wall without any backfill is horizontal and there is what are the forces acted on this, this has been taken into consideration, this is toe, this is your heel and total half of the part is your b by 2, this is my b by 2, then distance from here to here is x, distance from here to here is your e. So, we have considered this last time stability analysis, if you take a simple retaining wall of a kind of gravity retaining wall, what are the different forces acting? One is your force because of soil retain, this is your active earth pressure and retaining wall self weight, this is w and this is weight w, this is a distance a from the toe of the retaining wall and total resultant force r will be acted at a distance of x from the toe and what are the different stability you are going to find it out? First part you will find it out x bar, what is your distance, what is your distance? This you can say that x prime, what is the distance of resultant forces acted at this point from the toe, if you take moment about this point toe, you will find it out x prime that I have already said, then once you get x prime, then you find it out all these factor of safety first one is your nose sliding, for case of nose sliding, so mu v, mu v should be greater than is your horizontal forces, so factor of safety is equal to mu v by horizontal forces, mu is your coefficient friction between a wall and soil, h is your total horizontal forces reaction forces horizontal forces, v is your vertical forces, this factor of safety should be greater than 1.5, if there is no sliding, no sliding means this wall is suppose to if it is a less than 1.5, what will happen entire wall will slide along the base and it will go somewhere else, so for nose sliding factor of safety should be greater than 1.5, if it is less than equal to 1.5, so may be shear key may be provided at the regular interval, we will see in the example problem, then after nose sliding the second factor of safety you will have to consider that is your no overturning, in no overturning factor of safety is equal to movement of resistance summation of movement of resistance by summation of movement of overturning and it generally varies between 1.5 to 2.0, 1.5 to 2.0, if you look at no overturning what do you mean by this retaining wall because of earth pressure P a, it may possible that it will overturn above the 2, so that means what is your movement of resistance by movement of overturning, movement of resistance is by self weight, it will not allow to overturn, self weight will self weight times of a, this is your movement of resistance, that means it will not allow to overturn, what is your overturning movement, this active earth pressure acted it will take a distance from here to here, it will your overturning movement, so check this for no overturning the factor of safety should be 1.5 to 2.0, then third is your no bearing capacity failure, no bearing capacity failure means this retaining wall should not fail by means of bearing capacity, because this retaining wall rest on the soil it should not fail by bearing capacity, so find it out the pressure below the maximum pressure below the retaining wall, this is your b by b, b is your summation of vertical forces, b is your width into 1 plus 6 e by b, if I write q maximum by q minimum, so it will be plus minus 1 plus minus 6 e by e by b, so for factor of safety against no bearing capacity failure it should be q n a by q maximum, q n a is your allowable bearing pressure of soil, so it should be for no bearing capacity failure it should be greater than factor of safety should be greater than equal to 3 for no bearing capacity failure, then fourth part is your fourth part is your no tension for no tension for no tension crack e should eccentricity when is then e should be less than equal to b by 6, that means this eccentricity e this eccentricity e it should be less than equal to b by 6, for that you will have to find it out your means this is for your no tension, so what will happen these are the stability analysis case one for say gravity wall all walls, but I am writing it right now gravity walls a simplified case what are the forces the disturbing force is your active earth pressure because of your soil and the stabilizing force is your because of your self weight, so you find it out what is your resultant force and find it out what is the distance x bar once you get distance x bar you can find it out e eccentricity you can find it out and check all the factor of safety first case is your no sliding if there is a sliding then what will happen the factor of safety should be less than equal to 1.5 in that case what will happen this retaining wall will be entirely move at the base of the wall, so if it is if it is if it is less than equal to 1.5 that means this design is not ok, so then remedy should be taken either shear key will be provided here at the toe or may be at intermediate position below the base of the wall and for no overturning that means movement of resistance movement of resistance force should be greater than movement of overturning and this factor of safety should be greater than 1.5 to 2 if it is less than 1.5 or 2 then what will happen the retaining wall will topple the retaining wall will topple along the toe that means as a body as a whole body it will rotate along the toe, so there is a failure of chances is there because of overturning then for bearing capacity once it is resting on the soil once it is resting on the soil you find it out what is the maximum pressure at the base of the wall maximum pressure at the base of the wall you can find it out maximum and minimum the movement suppose this is your retaining wall let us say this is my base of the retaining wall, so what are the possibilities what are the possibility for the two condition let us start with no tension no tension mean e should be less than equal to b by 6 in that case what will happen the pressure will be in positive side what is the below this below this wall the pressure will be in positive side if the pressure is not in positive side suppose this kind of pressure distribution you are getting that means this is positive and this is negative what does it mean it if you say it is a negative pressure that means this part of the this part of the base of the base of the wall this part is not going to in contact with in contact with your soil, so what will happen your pressure distribution will be done by only this part, so redesign has to be done, so as far as possible as far as possible you always satisfy no tension criteria, so once you satisfy the no tension criteria that means your q maximum q maximum and q minimum there should be in positive side positive side below the retaining wall, so how do you find it out q maximum and q minimum q maximum and q minimum it should be b by b b is your summation of vertical forces b is your width 1 plus minus 6 e by b if it is a plus then you are getting q maximum if it is a minus you are getting q minimum, so once you get q maximum, so you check your factor of safety against bearing capacity failure of retaining wall, so q n a is your net allowable bearing pressure of soil from where you will get it you can get it from Terzaghi's bearing capacity theory c n c gamma d f n q plus 0.5 gamma b n gamma once you find it out divided by net allowable bearing pressure that means net means minus gamma d f minus gamma d f will be there allowable means a factor of safety will be there, so factor of safety complete factor of safety is your net allowable bearing pressure divided by your q maximum what is the q maximum you are getting from your pressure distribution below the wall and it should be greater than 3. Once the stability analysis has been satisfied then you will go for a structural design there are two parts of retaining wall two parts of retaining wall design one is your geotechnical other is your structural in geotechnical part you first have to satisfy all the stability criteria four stability criteria once it has been satisfied then you go for your structural part for gravity retaining wall no need for structural wall structural design but for reinforced means kind of counter for retaining wall and other retaining walls you need to have your structural design we will see this other two parts of your stability analysis second case is your cantilever retaining wall let us consider a cantilever retaining wall this is b width b then your suppose say resultant forces r is equal to w s plus w c plus p v now this is your w c weight of the concrete and this is your p so it will be coming p v as well as p h then this is your f r and if I put it in this way so your active part pressure will act upon at an angle beta with p a is equal to half gamma h 1 prime into k a so this is my h 1 prime so w s is your weight of a b c d weight of a b c d if I write this is my a this is my b this is c this is your d so w s is equal to weight of a b c d w c is equal to weight of concrete and p h is equal to p h is equal to p a cos beta p v is equal to p a sin beta now if I write it factor of safety against sliding factor of safety against sliding first one then it will be factor of safety is equal to f r by f d f t is your horizontal driving forces f r is your horizontal resisting forces and f t is your horizontal driving forces so factor of safety if I write it this will come r tan phi prime plus c dash b plus p p by p h p h then it should be greater than it should be greater than 1.5 it should be greater than 1.5 as I said earlier then other checks are other checks are as usual there are what are the different other other factor of safety this is your sliding second is your overturning third is your bearing capacity fourth is your no tension if you look at here I have consider a typical cantilever retaining wall and this cantilever retaining wall has been constructed by means of r c c concrete reinforced concrete material with along with the steels along with the steels these are the steel bars has been provided along with the steels so typically I have consider also a surcharge will be there means this soil it is with an angle it will retain the soil mass so first you identify what is your stabilizing forces what are your stabilizing forces so if you take a typical cantilever retaining wall this part is your retaining soil mass retaining soil mass may be with this level of the ground may be inclined I have consider a typical inclined face of this soil mass retaining soil mass so active earth pressure will be acted upon by at an angle beta so it will be it has two component p v and p h first you identify what is your means resisting forces what are the resisting forces weight of the concrete acted on these these are vertical downward weight of the concrete because this has been constructed by means of r c c reinforced concrete so this is your stabilizing forces similarly the weight of these weight of these part of the soil retained it has a self weight it has a self weight it is say w s w s is your weight of soil weight of soil soil in a b c d a b c d weight of soil so this is your also stabilizing forces what is the other stabilizing forces p v this p a has two component p v and p h so it is again a stabilizing force so what is your destabilizing forces or may be you can say that driving forces this is your horizontal horizontal forces p h this will try to try to push this wall or overturn this wall this will be your driving or this is this p h is your horizontal forces if i consider all this p h p v and all this so there are as i as we have discussed there are four stability criteria factor of safety against sliding factor of safety against overturning factor of safety against bearing capacity and there should be no tension if you look at the factor of safety against sliding it is your resisting forces by driving forces so resisting forces what is coming whatever the vertical forces coming w c w s p v you find it out its resultant r so each action has equal and opposite reaction so r has been taken into consideration r is equal to w s plus w c plus p v so it will be r tan phi because phi is a angle of angle between this soil angle of the this is your phi angle internal friction angle of your soil r tan phi c dash b what is c dash b if the soil has two parameter c and as well as phi c and phi c prime and phi prime this is your strength parameter c is strength parameter this unit cohesion will act along the base of the wall so that is why it will be your c prime b prime this is your horizontal resisting forces as well as r tan phi and p p what is your p p once you retaining wall you place once you place the retaining wall for retaining the soil mass you cannot place all along above the ground surface you have to take the retaining wall some depth below the ground surface so that it can be achieved some stability so this force because of the soil mass here it will be acted by passive r pressure p p so it will be r tan phi prime c prime b and p p by your what is your horizontal driving forces as I said earlier your horizontal driving forces will be p h horizontal driving forces will be p h so this factor of safety should be greater than 1.5 it should be greater than 1.5 so then all other things overturning and bearing capacity and known tension for overturning movement of resistance by movement of overturning means driving so with this help of this forces you can find it out what is your resisting movement so resisting movement will become all along your weight of the soil weight of the concrete and p v about the toe if this is my toe and this is your heel the resisting movement will be because of your weight of concrete weight of soil and p v along the toe this will give resisting movement and what is your driving movement because of p h it will try to drive so you can find it out both the movement and check this factor of safety overturning movement it should be varying from 1.5 to 2 then bearing capacity check then once c and phi is there you can find it out q n a bearing capacity will be your net allowable capacity of a soil by your q maximum as I said q maximum is equal to b by b into 1 plus 6 c by b then it should be greater than 3 then you will know no tension it should be less than equal to b by 6 this is your no tension so what are the steps then this is part of your geotechnical this is part of your geotechnical engineering that means you are first for making this stability analysis this kind of wall this kind of wall you follow this procedure so there will be less chance of committing mistakes again and again so if I follow this