 As we discussed last class about the design consideration of breast excavation, first one is your bottom heave formation or bottom heave. It occurs generally in cohesive soil. Let us say this is a cohesive soil and this part is your ground surface and this is your height h of the excavation height h and this is your over burden gamma h and this you can say that bearing capacity failure, bearing capacity failure, then this comes as a heave. For homogeneous soil clay q ultimate or clay q ultimate is equal to c u n c, c u is your undrained cohesion and for critical depth for critical depth gamma h c is equal to c u n c, this is your critical depth that means h c is equal to c u n c by gamma and you can write it down like this gamma h c by c u which is equal to n c, generally the value of n c is 5.726. So, gamma h c by c u it is called stability number, stability number, if stability number or stability factor, if stability number or stability factor is greater than n c, greater than n c, then failure occurs. For a factor of safety of say, for a factor of safety of 1.5, factor of safety of 1.5 you can write out gamma h by c u which is equal to n c by 1.5 which is equal to 4. Now, what does it mean? If you look at there, if you remember well last class we discussed about the one of the design criteria or design consideration is about your bottom heave. This bottom heave generally occurs in what kind of soil, generally it occurs in case of cohesive soil. How the bottom heave has been formed? As you go more the deep, more depth you cut vertical cut means below the ground surface, then what happen? This overburden pressure will increase. Because of this increase in overburden pressure below the cut, what will happen? This soil will fail by means of bearing capacity. Once this soil will fail, then nearby soil will be pushed so that there will be heave formation, there will be heave formation in the excavation. So, this is your excavation. This shaded area is your completely excavation. And in this excavation once there is a heave formation, heave means the soil from the bottom. The soil from the bottom will come inside your underground excavation. Once this thing has been occurred, what will happen? Whatever the structure constructed here like metro rail underground, so it will also fail, it will also fail. So, to start with this, understand this phenomena. Let us say for this cohesive soil, this is a cohesive soil. What is your bearing capacity? For cohesive soil bearing capacity is purely cohesive soil. There is no phi c n c or c u n c. c is your undrained cohesion. Then this c u n c is nothing you have to equate it to critical height, critical height, critical height. So, that means it is not in a kind of critical height. If you equate it into this will be your gamma s c, critical depth or critical height. So, this c u n c depends upon your critical depth. Now, from their critical height is coming s c is equal to c u n c by gamma. If I take this all the terms gamma s c by c u, this is called stability factor. Gamma s c by c u, this is called stability factor which is equal to n c. That means this n c value is generally for bearing capacity factor for cohesion value for purely cohesive soil it is 5.7 to 6. That means gamma s c by c u, this is your stability number or stability factor which is if it is this stability number or stability factor, if it is greater than n c. If it is greater than n c, that means failure occur for cohesive soil. Failure occur means what will happen? There will be a bottom heap formation. That means this is the condition this stability number or factor it should be less than equal to n c. It should not greater than your bearing capacity factor for cohesive soil. Now, for example in Kolkata soil in Kolkata soil c u is equal to in Kolkata soil c u is equal to 2.5 ton per meter square and h generally cut typical example I am taking in Kolkata metro h is equal to 10 meter say height of cut exhibition. So, gamma h by c u which is equal to gamma is equal to 1.8 into h is equal to 10 by 2.5 which is equal to 7.2 greater than n c. That means if there is a underground construction underground construction in Kolkata soil there is a chance that bottom heap will be formed bottom heap formation will be there. So, this is your first criteria first design consideration that is your bottom heap. Then another example this is you can say that this is a uniform soil. Suppose this soil is purely uniform there is no layered soil that means a constant value of c u is there this is your calculation. Now, if I go for second example. So, how do you prevent before I solve the second example how do you prevent the prevent will be you have to dewater you have to dewater from the cohesive soil. So, that water table will go below your far below this there. So, that it will become harder slightly harder. So, that bearing capacity failure may be may be avoided. So, example 2 there are 6 layers of soil as I said this is not a uniform soil layer. So, layer 1 is your firm clay layer 2 is your soft clay c u is equal to 2.5 ton per meter square. Layer 3 again it is firm clay c u is equal to 6 ton per meter square. Then layer 4 is your sandy sand sand sand sand. And layer 5 is your stiff clay c u is equal to 10 ton per meter square. And layer 6 is equal to your dense sand. Now, there is a chance that there is a cut off up to sandy silt. That means vertical cut off you do the vertical ground excavation you go for if there these are all your soft soil at what depth you can go for your vertical excavation. Where you are getting a kind of hard stratum this is sandy silt or dense sand you can go here. So, if I try with these there is a vertical cut is excavation up to sandy silt. Let us say with these sandy silt say this is your this is your d f this is your d f this is your d f. And this is your kind of you can write h. And let us put it layer by layer layer 1 layer 2 layer 3 layer 1 is equal to gamma 1 c u 1 gamma 2 c u 2 gamma 3 c u 3. Then from there it starts like this then it starts like this. Then it starts let us say this is my case of d 1 then this is gamma 4 and c u 4. Then it if it is coming in this way up to the middle up to the middle the bearing capacity failure I am showing up to this middle like this. So, let us say the failure envelope from here you can extend up to b 1. And here it will be completely b now factor of safety is equal to factor of safety is equal to c u n c plus gamma d f plus c u h by b 1 by gamma h by b 1 by gamma h plus d f. So, that means this is your all stabilizing forces divided by all disturbing forces all disturbing forces. Then what will happen the factor of safety greater than 2.0 it will be ok. If the factor of safety is coming out to be for against your bottom if it is greater than 2.0 then it is ok. So, what we are basically doing how what we are basically doing we are taking a influence line influence of failure envelope here up to a distance b 1. And this is if this is b here a distance b 1 how to calculate b 1. So, b 1 is equal to b by root 2 and d 1 from there it is your lower value. If you look at here first we check your stability number and factor it should be greater than n c. So, that if it is greater than n c then failure will occur based on that you can calculate your factor of safety for finding out factor of safety. Let us say this is an exhibition there are 1, 2, 3, 4, 5, 6 number of layers and at 2 positions or 4th number of layer this is a sandy silt and 6 position is a dense sand. So, that means your exhibition you can extend from here to here or exhibition you extend from here to here. That means the base of your exhibition till exhibition where you are doing up to that depth it should be ensure that up to that depth at that depth there is a hard stratum line. That means this is your hard stratum line sandy silt or dense silt. So, what happened I am taking a this is my involvement d 1 is your d 1 is your influence area based on your bearing capacity. So, once you are getting d 1 then there will be depth of foundation this will be also given height also given b also given only you have to find it out b 1 and d 1. So, d 1 will become it will come based on your bearing capacity calculation then once you get d 1 then you can check your factor of safety your factor of safety is equal to there are 5 layer 1, 2, 3, 4, 4 layers that means c u 1 n c 1 c u 2 n c 2 all for you calculate plus gamma d f gamma into depth of foundation depth of this depth gamma d f plus c u h by b 1 this c u n c this c u n c is your this one is your c u n c and c u h by b 1 this summation of c u h by b 1 by gamma h plus d f divided by gamma h plus d f if you look at here all stabilizing forces divided by all disturbing forces all stabilizing forces divided by all disturbing forces from there from there you have to find it out your factor of safety is equal to 2.0 and b 1 you are going to calculate b 1 is equal to b by root 2 or d 1 b 1 is equal to as it is given b and d. So, b by root 2 or d 1 that means which is lower value of b 1 you are supposed to take it this is for design concentration of bottom heave then next part is your clay bursting how to calculate factor of safety for your bottom heave this has been done then for clay bursting what is the condition for clay bursting clay bursting means clay soil sample will come out along with the water if there is a clay layer of shallow depth then there is a water table then water table is a ground surface then below the shallow depth of clay if there is a kind of sandy soil sand or sandy soil or sand then there is a chance of clay bursting in this case the factor of safety how come this clay bursting will come this sand will give upward movement the sand will give upward movement if you look at here this is your clay of a shallow depth of the clay layer is there then here there is a sand layer. So, once you start excavation that means you completely excavate excavate and small part above your sand you are there you are stopping there what will happen this sand along with this water it will make upward force. So, what will happen mixture of sand and clay it will come up in the excavation. So, you have to stop the clay bursting clay brushed means it will brushed out and puncture and the clay will come out from this excavation that is the meaning. So, what is the factor of safety you should take it the factor of safety factor of safety is equal to gamma d plus 2 c u d by b divided by gamma w h plus d which is equal to greater than 1.3 gamma d gamma times into d gamma times into depth of d plus 2 c u d 2 times c u d into b it is nothing, but c u d by b by 2 c u d by b by 2. So, this b by 2 this 2 because this half of the part will be taken into consideration. So, once this half of part has taken into consideration. So, b by 2 that means the 2 has gone there. So, 2 c u d by b then for upward thrust this is your upward thrust upward thrust will be gamma w into h plus d gamma w is your unit weight of water and h plus d is your this height up to this height it is applying your upward thrust. So, this factor of safety should be greater than 1.3 for design consideration of clay bursting that means each case we have to check this whether this factor of safety is satisfying or not. If it is not satisfying then this design has to be changed then that means the depth has to be changed or other criteria this struts has to be provided very close to each other.