 Now we will solve a example problem about this design of counter fort retaining wall. Now look at this example, it is given design a counter fort retaining wall for this following given particulars. First one is given height of wall above ground level which is equal to 5.5 meter, then safe bearing capacity of soil safe bearing capacity of soil is given 160 kilo Newton per meter square, phi value is given phi is equal to 30 degree and unit weight of soil is given 16 kilo Newton per meter cube spacing of counter fort spacing of counter fort counter fort about three meter three meter and R C C unit weight unit weight of R C C or unit weight of R C C which is equal to 25 kilo Newton per meter cube and use M 15 grade concrete and F e 250 steel. As I said earlier now this is a typical example problem we will have to design for counter fort retaining wall. No dimensions has been given dimensions has to be taken derived then your stability analysis then your structural design it is given height of wall above ground level is 5.5 meter that means from ground level to above is 5.5 meter and the soil below the ground level it is bearing capacity is given 160 kilo Newton per meter square phi and gamma of the retaining soil retaining soil is given and spacing of counter fort walls is about three meter and unit weight of R C C reinforced concrete 25 kilo Newton per meter cube and it has been the plan to use M 15 grade of concrete and 250 steel mile steel 250 for designing this counter fort retaining wall. Now first step is to find it out this wall has to be if I take this retaining wall like this if I go for this counter fort retaining wall like this what will happen you will have to consider some depth as a depth of the foundation as a depth of the foundations. This depth of the foundation generally provided about 1 to 2 meter either you can assume or you can this depth of foundation you can find it out from this depth of foundation you can get it p by w into 1 minus sin phi by 1 plus sin phi whole square where is your p is your say bearing capacity of soil pressure in this case and it is coming about 160 thousand divided by your w is your unit weight of soil based on that it is coming about 16 thousand into one third whole square which is coming about 1.11 meter. So, let us take about foundation depth is equal to 1.2 meter foundation depth then once you are taking this foundation depth is equal to 1.5 meter then overall height below the above the ground surface overall height of wall overall height of wall overall height of the wall is equal to 5.5 plus 1.2 which is equal to 6.7 meter if you look at here it is given height of wall above ground level the ground level is here because retaining wall cannot stand above the ground level you have to make sure that there is some foundation depth retaining wall has to be put it below the ground level. So, this height is given about 5.5. So, this depth of the foundation is we have taken 1.2. So, overall height of wall is coming about 6.7 meter now thickness of the base of the wall now the dimension one this overall height and the depth of the foundation is over part one is your dimension of the retaining wall dimensions of the retaining wall in this dimension of the retaining wall first find it out your thickness of base slab thickness of base slab you can get it as I have said earlier for counter forward retaining wall 41.7 root over of h it should not be lower than that and 20 l root over of h and it is coming about to be 1.5 meter. 324 mm and this coming about to be 402 mm. So, this thickness of base slab should not be lower than this. So, now we can take a higher value let us provide thickness of the base slab say 400 mm 400 mm. So, now the actual height of upright slab upright slab is your this is your upright slab this is your applied slab. So, thickness of base slab we have found out it is to be 400 mm that means 0.4 meter. Now this upright slab for upright slab actual height height of upright slab height of upright slab is 0.4 meter. Is equal to 6.7 minus 0.4 which is equal to 6.3 meter 3 meter. So, this height is about coming 6.3 meter. Now depth is coming about depth of the foundation it is coming about is equal to 1.2 meter. Now we will have to find it out this particularly design for your upright slabs in case of upright slab. What is your pressure intensity in upright slab for upright slab what is your pressure intensity. Pressure intensity is equal to gamma k a h which is coming about 16000 into 6.3 into 1.3 this is coming about 33000 into 1.3 600 kilo Newton per meter square. Now maximum bending moment is coming about to be maximum bending moment which is equal to W L square by 12 or P L square by 12 P is your pressure intensity P is equal to your pressure intensity maximum bending moment is equal to P L square by 12 from there we can find it out P is equal to 336000 into 3 square by 12 and this comes out to be 25200 Newton meter. So, once you get maximum bending moment then what you are what we can suppose to get it we can find it out your area of steel this mild steel for m 15 grade concrete the permissible stress in concrete c is equal to as I said earlier it is equal to 5 Newton per m m square for for for m 15 grade concrete. Then for mild steel F e 250 steel permissible stress in steel t is equal to 140 Newton per m m square based on this if I equate maximum bending moment to area of steel this should be come out to be you can either equate maximum bending moment to area of steel or maximum bending moment to your q b d square because you have to find it out this thickness also we got it base slab thickness now we will have to find it out thickness of your operate slab at the base. So, from this q b d square which is equal to maximum bending moment. So, q for m 15 and 250 steel it is 0.87 b into 1000 into d square which is equal to 252 into 100 into 1000. So, d is equal to 170 m m now if I am using effective cover of 40 m m this is your effective cover then it comes out to be 210 m m now let us say thickness of operate slab let us say 210 m m 210 m m you can keep this thickness of this operate slab either you can keep it this way or you can keep it continuous throughout from top to bottom this thickness will be same there is no varying in thickness. So, in this case we are considering this thickness uniformly throughout that means uniform throughout this thickness we are considering this this about 210 m m then after getting the thickness means width of your operate slab at the base it comes out to be 210 m m then thickness of your base slab is 0.4 meter or 400 m m. Now base width this base width has to be come out and this width has to be come out then we will go for your stability analysis next part is your base width. So, base width is generally varying from 0.6 h to 0.7 h. So, if I take total height h is equal to 5.5 it is this height operate slab is your this height is there. So, if I am taking into this is coming about to be 6.7 into 0.6 6.7 into 0.6 and this is your 6.7 into 0.7 it comes out to be 4.02 and 4.69 meter let us provide let us provide this base width is 4.5 meter. So, base width is comes out to be 4.5 meter. If you look at here this base weight base width is generally 0.6 to 0.7 times of total height of this retaining wall. So, total height if I am taking it 6.7 5.5 plus 1.2 this is coming about to be 6.7 6.7 into 0.6 6.7 into 0.7 this base width you can take it between 4.0 to 2 4.6 nanometer either you can take it 4.03 4.3 4.5 4.6 just I put it any value between this 4.5 meter base width. Now next part is your toe projection toe projection means how much it will be projected if this is my retaining wall and this is your toe and this is heel and this part is called heel slab heel slab and this is your upright this is your upright slab this part is your this entire part moving towards this this is upright slab. Now how much is your toe has to projected beyond this retaining of soil mass. So, this toe projection comes out to be it says that one fourth one fourth of b one fourth of b and this value is your b. So, this comes out to be 1.1 meter. So, you can provide either 1.1 or you can provide a toe projection of 1 meter. Let us provide this toe projection of 1 meter toe projection of 1 meter. Now if I provide this toe projection of 1 meter that means this dimension is almost over tentative dimension is almost over. Let us make a sketch how the dimensions of this it looks completely retaining wall you can show in a sketch of your tentative dimension for your stability analysis 1.1 meter and this is about to be this is about your 1 meter and this is your 210 mm and this is your 0.4 meter and this is your 0.4 meter and this is your 3.29 meter. So, with this with this this is your tentative dimensions and now with this tentative dimensions will check this stability analysis if this stability analysis satisfies four factor of safety one is your factor of safety against overturning other is your factor of safety against sliding and e should be your eccentricity should be less than equal to b by 6 and bearing capacity that means the dimensions are ok. If this stability analysis if it is not satisfy either you modify the dimension or you can go for some remedy. So, we can check this now this stability analysis calculation of this forces we can check how it is how it is coming about means total resisting force and driving forces all has to be calculated then once you calculate then you can go for your bending movement and pressure at the base all these things. Now if I name it this is w 2 this is w 1 w 1 w 2 you can name it w 3 also this part is your 6.3 this part is your w 3 now start 1 by 1. So, w 1 is your what is your w 1 w 1 is your weight of this upright slab from here to here. So, that means unit weight what is your width 0.21 into 6.3 6.3 6.3 this to this height is your 6.3 into unit weight of concrete is 25000 which comes out to be which comes out to be 33075 kilo newton. So, this is a vertical force because this is vertical downward by self weight this is vertical downward similarly you can find it out w 2 w 2 is your weight of base slab w 2 is your weight of base slab which is about to this is this width is your 4.5 meter this width is your 4.5 meter. So, w 2 is equal to 4.5 into 0.4 into 25000 which is coming about to be 45000 and this is again vertical if I am putting it this symbols and this symbols this means force is acting vertical this means force is acting lateral in that direction. Then third part is your w 3 w 3 is equal to 3.29 into 6.3 into 16000. If you look at here 3.29 is this width and this height is your 6.3 with this counter 4 soil has to retain. So, unit weight of soil is again this is vertical 16000 and which is coming about to be 3 3 1 6 3 2 again this is vertical. Then fourth one is your earth pressure earth pressure that is your k a gamma h square by 2 which is equal to one third 16000 into 6.7 whole square by 2 then. So, earth pressure is there then moment you can find it out from this moment this is your pressure. Then moment you can find it out k a gamma h cube by 6 which is equal to one third into 16000 into 6.7 whole cube by 6 now this is the vertical force and this is your lateral forces or horizontal forces acted by this directions. So, these are about your force now you can find it out lever arm lever arm lever means means this force acted how far from toe or how far from your heel. So, in this case from this toe this w 1 is coming about to be 1.105. So, because it is 1 meter and 0.21 by 2. So, this is coming about 1.105 and w 2 is your it is straight forward is half it is coming about your 2.25 and w 3 is coming about 2.5. So, 855 and earth pressure will act at a distance 6.7 by 3. So, then after that you find it out your clockwise moment and anti clockwise moment you find it out your 2 moments clockwise moment and anti clockwise moment about your toe. If you look at this how this clockwise moment all vertical forces this will give a clockwise moment lateral forces earth pressure this will give your anti clockwise moment. That means if I write is clockwise moment and anti clockwise anti clockwise moment and from this you can find it out this clockwise moment all the term for w 1 it is coming about 36547.8 and this is coming about 1.01250.0 and w 3 is coming about 946809.36. So, and anti clockwise moment is coming about 267345. So, this comes out to be 108467.2. So, now net moment you can find it out net moment is equal to clockwise moment minus anti clockwise moment which is coming about to be 817262 0.24 Newton meter. Now we can check you can find it out eccentricity you can find it out this is the resultant force are acted. So, how far from your c g how far from your c g you can find it out e. So, first find it out your x bar distance of your eccentricity x bar from the toe which is equal to m by v total moment resisting as well as clockwise as well as anti clockwise and total vertical forces it is coming about to be 817262.24 by 4097.07 which is equal to 1.99 and e is equal to e is equal to b by 2 minus x bar which is equal to 0.26. So, b by 6 is equal to 0.75 that means e is equal to less than b by 6 which means there is no tension crack. One stability criteria has been satisfied we find it out what is your eccentricity eccentricity should be less than b by 6 that means there is no tension crack that means there is no tension crack below the base of the slab there is what do you mean by tension crack once there is a tension crack there is a gap between soil and the wall. So, there is no gap between soil and wall. So, similarly factor of safety against sliding you can take from this factor of safety against sliding it is coming about 4.0 which is I just leave it for you then sorry factor of safety against sliding is coming about 1.711 you can assume mu is equal to friction between concrete and soil is equal to 0.5 you can assume this friction between concrete and soil is equal to 0.5. So, 1.771 now which is ok which is greater than greater than your 1.5. So, and your factor of safety against bearing capacity also it is ok it satisfy now we can find it out what is the value of p maximum at the base and p minimum it will comes it will be calculated with v by b 1 plus 6 e by b and it will be calculated v by b total vertical force v by b 1 minus 6 e by b and this comes out to be 1 to 1 394 Newton per m m square and this is equal to 0.5. This comes out to be 6 0 6 9 7 Newton per m m square. So, now if I draw it if I draw it this is my pressure distribution is coming. So, this pressure distribution this part is coming about 1, 2, 1, 3, 9, 4 then 6 0 6 e by b 1 minus 6 e by b 1 minus 6 e by b 9 7 this is your pressure distribution around this below the base of the slab. What I did after the after the dimension has been fixed based on the calculation or based on the assumption value then you can do it systematically by taking into your forces which your vertical forces lateral forces then clockwise movement and anticlockwise movement. So, I have made it into very small small part. So, that there will not be any mistake I make it upright slab as w 1 base slab is your w 2 soil retaining in the counter for part this is your w 3 then is your how much is the earth pressure coming all I have taken into small small part and total movement has been calculated. Once you calculate your total movement and total vertical force also you can calculate from here from this these these these what is your total vertical forces. So, x bar you can find it out from there you can find it out eccentricity and you check it whether e is less than b by 6 or not if e is less than b by 6 that means there is no tension crack 1 factor of safety satisfied. Second factor of safety again sliding has been checked I leave it for the student to calculate it is about 1.711 which is greater than 1.5 mu has been assumed as 0.5. So, and also factor of safety against overturning as well as factor of safety against bearing capacity has also been checked it is within this limit then once this stability analysis has been done then what will happen this dimensions whatever we have taken that is ok now that is ok now. So, step one dimension is ok step two stability analysis has been done step three is your design structural design of your RCC counter for retaining walls. So, from this stability analysis we calculated P maximum maximum pressure below the base of the wall minimum pressure below the base of the wall and now in the next class we will go for your structural design of your counter for retaining wall.