 So, I will start the second lecture of this module 2 that is on shallow foundation. Now, as I have already discussed that the bearing capacity and the settlement are the two main design criteria of a shallow foundation. So, that means the soil the bearing capacity of the soil should be such that that it is adequate to carry the load that is coming from the super structure as well as the settlement should be within permissible limit. Now, today in this lecture I will discuss about the bearing capacity how to calculate the bearing capacity of a shallow foundation. Now, first before I go to this calculation or the analysis part I should discuss some terminology of this bearing capacity. So, first I will first one is the gross pressure. Suppose, this is the foundation and this is the ground level and D f is the depth of foundation and B is the width of the foundation and then this gross pressure is the pressure that is coming at the gross pressure that is coming at this base level of this foundation. That means the pressure that includes load that is coming from the super structure, the self weight of this foundation and the weight of the soil. Now, next one is the net pressure as the settlement the soil will settle after because of this net pressure because this soil pressure is already existing in this soil because then you have to remove this soil then we place this soil. So, that means at the base of this foundation the pressure that is coming from due to this soil is already there. So, the settlement will start or this after due to this net pressure that means net pressure is the gross pressure minus the pressure that is coming due to this soil. That means the if q g is the gross pressure then the net pressure will be q g minus depth of the foundation into the unit weight of this soil. Net the ultimate bearing capacity of the soil the ultimate bearing capacity of the soil that is the ultimate load that this soil can carry before it fails. So, that means this is the ultimate load carrying capacity of the soil. So, if we re-notice the q u which is the ultimate bearing capacity of the soil next will be the net ultimate bearing capacity of the soil. So, that means the same again the net ultimate bearing capacity would be the ultimate bearing capacity of the soil minus d f into gamma where gamma is the unit weight of the soil. Then when we will get the ultimate bearing capacity and the net ultimate bearing capacity of the soil then you have to go for the net safe bearing capacity and all gross safe bearing capacity. For this net safe bearing capacity you have to apply a factor of safety here f is the factor of safety that f varies either f will be either 2.5 or 3. So, if we divide this net ultimate bearing capacity by this factor of safety then we will we will can calculate the net safe bearing capacity of the foundation or the soil. Now, the gross safe bearing capacity similar as the if the net safe bearing capacity is n s then the gross safe bearing capacity will be n s plus d f into gamma. Now, again this will be n u divided by f plus gamma u. Now, the next one is the allowable bearing pressure. Now, as I have discussed that from the bearing capacity consideration basically we will get the two pressure. One is the bearing capacity consideration that the the net load that maximum load that soil can carry before it fails. So, that means, that term we will get the net safe bearing capacity and you have to apply the factor of safety there. So, we will get the get the net safe bearing capacity and from the settlement consideration we will get another pressure that means that pressure is. So, there is within the permissible settlement that means, if we apply that that pressure within the soil then settlement of that foundation will be within that permissible limit. So, that if there is a permissible limit for the settlement that is corresponding that pressure is called the pressure for or the allow that is the maximum pressure the soil can carry for that settlement consideration. So, there is basically two criteria one is settlement consideration another is bearing capacity consideration. So, that means bearing capacity consideration that net safe bearing capacity that means it is the net safe bearing capacity of the soil and then another is the pressure that cause the maximum permissible limit. So, minimum of these two will give us the allowable bearing pressure that is q allowable net. So, one is from the net safe bearing capacity then bearing capacity consideration another is the within the that settlement consideration that the minimum of these two bearing and the settlement that will give us the allowable bearing capacity bearing pressure. So, now so these are the different terms that we will use for this analysis. Now, there are now we have to calculate the bearing capacity of the foundation the first expression that I will the analysis that I will explain that is given by the Terzaghi. So, that means this is the Terzaghi's bearing capacity theory. Now, Terzaghi proposed this bearing capacity theory. So, this is the this bearing capacity theory. So, there is the some assumptions are there. So, before we start this analysis part of the bearing capacity analysis part we will go for this assumptions or the that means the footing is loaded is footing is the long strip or continuous footing resting on a homogeneous soil having shear parameters c and phi. So, that means the soil is homogeneous with a shear parameter c and phi and it is a strip footing or continuous footing. Now, as this is the strip footing or continuous footing that means the analysis is 2 d analysis. Now, soil fails in general shear failure. So, last class I have discussed the different types of failure that the one is general shear failure then the local shear failure and the punching shear failure. Now, this particular analysis of Terzaghi bearing capacity theory is valid for the general shear failure for this one. Now, the load is vertical and concentric. Now, ground surface is horizontal the base of footing is laid at a shallow depth. Now, this is for the shallow foundation. So, that means this is the base of the footing laid at the shallow depth. Now, shearing resistance of the soil between the surface and the depth of footing df is neglected. So, the footing is considered as a surface footing with uniform search as equal to depth of footing into unit weight of the soil at a level of the base of footing. So, the resistance coming from the surf the soil that is within between the surface ground surface and the base of the footing is basically neglected in place of that it is considered that one surcharge that is equal to the pressure that is coming due to that soil part is acting at that base level. So, now, if we go for this Terzaghi bearing capacity part. Now, if we draw the failure surface for this suppose this is the ground surface and this is the foundation width say. So, this is ground level. Now, where width is b, b is the width of the footing and this is the base level of the foundation. Now, this depth of foundation is df, df is the depth of foundation, b is the width of foundation and say we are talk calculating this pressure that is acting of the base of the foundation that is this pressure is q u that means ultimate bearing capacity of this soil. So, q e that is the ultimate load that this footing can carry. So, now, as in the first assumption this is a footing is the strip footing. So, that means this is a 2 d analysis now soil fail in general shear failure the load is vertical. So, that load that is applying is vertical and concentric that means is centrally loaded. Now, suppose this is the center line. Now, base of the footing is laid is the shallow depth. So, df is a shallow depth and the shearing resistance. So, as I mentioned the shearing resistance of the soil between this zone that means the ground this is also ground level, ground level and the base of the footing is neglected. So, in place of that it is assume that is the load that means the total pressure that is acting in this point or the base of the foundation that will be equal to q that will be gamma df into gamma where d is the depth of foundation and gamma is the unit weight of the soil. So, that is the pressure that is acting at the foundation base level. If we draw the failure surface that is assume for this analysis that is general shear failure. So, this is a triangular portion then there is a logarithm spiral and then is a straight portion. Similarly, here the triangular portion thing logarithm spiral portion and the straight portion. So, this is the failure surface that is considered for this analysis. Now, basically in this this is a symmetric. So, this part so we can say this is a three zone. So, this is the first zone or we can say this is a three zone this is the first zone. Now, this is second zone this is also second zone and this one is the third zone. So, three zones are there. So, one is this triangular portion this is the first zone. So, zone one this is called the triangular zone. Now, similarly zone two this is called zone of radial shear. Now, the zone three this is Rankine passive zone. So, these are three zones one a first zone is triangular zone this is this triangular part. The second zone is zone of radial shear I mean this zone and this zone and third zone is the Rankine passive zone. So, now the angle is considered for this analysis that means this angle is 45 degree minus phi by 2 where phi is the second zone. Fictional angle of the soil this is also 45 degree minus phi by 2. Similarly, this angle is also 45 degree minus phi by 2 and this angle is also 45 degree minus phi by 2. Now, this angle is considered phi and this angle is also considered as phi. So, this angle now here this portion this part is logarithm of spiral part is considered this is logarithm spiral then the straight portion. So, triangular part then the logarithm spiral then the straight portion. So, this straight portion this passive Rankine passive zone this angle is 45 minus phi by 2 and this angle is also 45 minus phi by 2. Similarly, these two angles are 45 minus phi by 2 in the triangular part this angle is phi this angle is also phi. Now, if I draw this free body diagram of this triangular portion. So, if I draw this free body diagram of the triangular portion. So, here we can say this is a b and d. So, and this is the center line and this is the width of the footing that is b. So, this triangular portion we are drawing here and this load that is acting that will be q u ultimate bearing capacity or load carrying capacity. So, as this consider this angle is phi and this angle is also. Now, here a d and b d act as a rough back of the rigid wall. Suppose, this consider here a d and b d act as a rough back of rigid wall. So, where the c and the phi this two parameter c and phi are the equivalent of the wall addition. Now, c is equivalent to the wall addition and phi is equivalent the angle of wall friction. So, now for this mean that suppose this is the free body diagram and this is the perpendicular line perpendicular to this a d and perpendicular to b d. So, these dotted lines of the perpendicular lines perpendicular to a d and perpendicular to b d. Now, here that is the p p. P p is the passive resistance that is coming from this zone 2 and 3. So, that means this portion this resistance in this triangular portion is coming from this zone 2 and these two other zones. So, this is the passive resistance that is coming from this zone that is a p p. P p is the passive resistance. So, which is acting at an angle phi with this vertical line. Similarly, here also p p is acting at a angle phi and this c a. So, because as now this phi is equivalent to this wall friction angle. So, now p p is acting as a angle of phi with this perpendicular line of this wall. Similarly, here also p p is acting as a angle phi with the perpendicular line of this wall. Now, this c a is the addition that is acting for a d and b d line. So, where c a is the addition is acting this two line. Now, we can say this c a that is equal to c into a d or c into b d, where c is the addition. So, similarly c will be the cohesion and p p is equal to the total passive resistance. So, p p is the total passive resistance that is acting here. So, this is the passive resistance c is the cohesion. Now, in this figure if we calculate the weight of this triangular portion that mean weight of. So, weight of which this triangular portion a b d if you want to calculate the weight. So, that weight will be this is the half the area this means this is the b and then this height. Now, this height is basically b by 2 into tan phi. So, with this base of this triangle is b height is b by 2 into tan phi and then the unit weight of the soil. So, this will be the weight. So, finally, we can write 1 by 4 comma b square into tan phi. So, this is the weight of this triangular wedge. Now, next we have to calculate the different components of this triangular wedge. So, if we take the different components then we can write. So, if we take this figure again we can see. So, there is a in the vertical direction if we take the different components. So, one is p p that is acting in the upward direction then q u that is acting in the downward direction then adhesion it has one component in the upward direction one component in the horizontal direction. Similarly, here also one component in the upward direction and one component in the horizontal direction. So, if we take this vertical components of all the forces then we can write that q u into b. So, q u is the total load that is acting in the downward direction that will be equal to it is the load bearing capacity of the soil that means 2 p p this passive resistance both are acting in the upward direction plus C A this adhesion in the vertical component this C A into sin phi as it is acting in the both the sides. So, this will be 2 into C A into sin phi sin phi we multiply with this C A it will get the vertical components. Then this weight of this wedge that is acting so this is the passive resistance component this is the ultimate load carrying capacity of the bearing capacity of the soil and this is the adhesion components and this weight of the wedge that is acting in the downward direction. So, this this this will be minus 1 by 4 gamma p square into tan phi. So, these are the total forces here this 2 this one and this one acting in the downward direction and this p p and this C A components you acting in the upward direction. So, now we can see now here as we have mentioned that C A is equal to C into D A or C into B D. Now, from this figure we can write this D A is equal to B D that will be equal to B by 2 divided by cos phi. So, here in this figure we can write this B A as this height is B by 2 into tan phi. Similarly, this side ad or B D is B by 2 divided by cos phi. So, we can replace this value here in the C A. So, we will get C A is equal to C into B by 2 divided by cos phi. Now, if we put this C A value here in this equation. So, this will be Q U into B equal to 2 p p plus 2 into C into B divided by 2 divided by cos phi minus 1 by 4 gamma into B square into tan phi. So, this will be the tan phi. So, further simplify you can get this 2 p p plus B into C. There is another term here that is sin phi because this 2 C A is replaced by C into B by 2 cos phi and sin phi is there. So, this 2 to cancel. So, this will be this B C into sin phi by cos phi this will be tan phi. So, minus 1 4th gamma B square into tan phi. So, now this consider this p p this total passive resistance it has 3 components basically. So, this p p is a summation of p p gamma plus p p c plus p p q. So, this resistance are coming from 3 parts one is for the weight of the soil this gamma c is due to the cohesion and q is due to the surcharge. So, that surcharge we have consider here for this part. So, this q is the surcharge that we have consider here. So, that means this q that is replaced the surcharge that means gamma d f d f into gamma. So, now here p p will get this p p gamma that will get. So, first components is p p gamma. So, this is produced by the weight of the soil in which this assume the soil is cohesion less and negligible surcharge. So, this is the contribution because of the weight of the soil. This is produced by the weight of the soil in shear zone and it is as per the assumed that soil is cohesion less that means c is equal to 0 and negligible surcharge. So, q is also equal to 0. So, q and c both are 0 in this condition the resistance that is produced by the weight of the soil is called as p p gamma. Second one this second resistance this is p p c it is the produced by the soil cohesion. This is produced by the soil cohesion assume that the soil is weight less and negligible surcharge. So, assuming that weight less soil and negligible surcharge. Similarly, the third component that is p p q it is also produced by the surcharge assuming that the soil is weight less and also cohesion less. So, in here in this third condition soil is weight less and cohesion less that contribution due to the surcharge is called p p gamma. So, now if we put this value in the final expression. So, our expression was q u b is equal to 2 p p plus b c into tan phi minus 2 p p plus minus 1 by 4th gamma b square into tan phi. So, if p p is the summation of 3 parts. So, if we replace this equation by this is 2 p p gamma plus p p c plus p p q plus b c into tan phi minus 1 by 4 gamma b square into tan phi. Now, let we consider that 2 p p gamma minus 1 by 4 gamma b square tan phi. Now, let we consider that 2 p p gamma minus 1 by 4 gamma b square tan phi is equal to b into half gamma b square tan phi. So, we replace this term 2 p p gamma minus 1 by 4 gamma b square into tan phi by this expression b half gamma into b into n gamma. And 2 p p c plus b c into tan phi is replaced by b into c n c. But c is the equation and again 2 p p q is replaced by b into q into n q. Now, where this n c n q n gamma these are toss the keys bearing capacity factor. So, this is the toss the keys bearing capacity factor. So, finally, if we write this replace this expression by this terms then the final expression will be q u. So, if we put this in this expression so, this will be q u is equal to c n c plus q n q plus half gamma b n c plus half gamma n gamma. So, where c is the cohesion of the soil, n c is the bearing capacity factor of toss the keys bearing capacity factor q is we can write that q is equal to gamma d f is d f into gamma. So, gamma is the unit weight of the soil and this is half b is the width of the foundation. So, this is the 3 components of 3 part. This is the contribution due to the cohesion, this contribution due to the surcharge and this is the contribution due to the weight of the soil. So, ultimately this is the final expression where we will get the ultimate load carrying capacity of the foundation. So, this q u is the ultimate load carrying capacity of the foundation that is q u equal to c n c plus q n q plus half gamma b n gamma. Now, where we can write this terms this bearing capacity terms that we can write that n c toss the keys given this bearing capacity factors value this is cot phi into a square divided by 2 cos square divided by 2 cos square divided by 2 pi by 4 plus pi by 2 minus 1. This is n c similarly, n q will be a square divided by 2 cos square pi by 4 plus pi by 2 and n gamma we can write this is half gamma b n tan phi k p gamma minus k p gamma divided by cos square phi is total minus 1. So, where this a is equal to e to the power 3 pi by 4 minus pi by 2 into tan phi and k p gamma divided by 2 is equal to passive earth pressure coefficient. So, this is passive earth pressure coefficient. Now, if phi is equal to 0 then n c will be 5.7 n q will be 1 and n gamma will be 0. So, from this expression we can write q u will be n c 5.7 into c plus n q part is 1. So, into q and n gamma part is 0. So, this is the ultimate load carrying capacity of the soil and then q net ultimate that will be q ultimate minus q. So, this will be 5.7 into c plus n q part is 1. So, into q and n gamma part is 0. So, this is the ultimate load carrying capacity of the soil and then q net ultimate that will be q ultimate minus q. So, this will be 5.7 into c or 5.7 into c undrained cohesion of the soil. So, now by using this expression we can determine what will be the this load bearing capacity of this soil or this foundation. Now, this is the basic expression this expression now where c q gamma b this we can this gamma is the unit weight of the soil. So, b is the width of the if we know this. So, from this expression we can see this bearing capacity of the soil this depends on the soil property that is c and phi of the soil. Then this surcharge of the I mean the depth of the soil depth of the foundation and width of the foundation that means the geometry of the foundation then the soil properties also. So, now this is the basic expression which is derived for the vertical and concentric loading condition it is for homogeneous soil and so this is simplified expression. So, now in the next lecture I will discuss the different condition the general type. So, for inclined loading or if there is a moment is there then how will determine this bearing capacity. So, this is the simple expression that we can use. So, here one thing that this n c n q n gamma expressions are given because these are the n c n q n gamma expressions. So, by using this expression we can calculate the this bearing capacity Tazaki's bearing capacity factors for different see different phi values. Now, Tazaki has also proposed or given this bearing capacity factor in a tabular form for different phi values. So, that expression values are given here. So, here we can see this is the Tazaki's bearing capacity factors this is the table for different phi is starting from 0 this is for as I have mentioned for the 0 degree phi value this n c is 5.7 n q is 1 and n gamma is 0. So, it starts from 0 to 50 degree. So, corresponding n c value n q value n gamma value. So, these values are given. So, by using these charts we can determine. So, if we know the geometry of this foundation or the width of the foundation then the depth of the foundation and we know the if we know the soil properties then by using this table we can determine this bearing capacity factor. Then we can use this bearing capacity factor in this our general expression and then we can calculate the ultimate load bearing capacity of this foundation. Now, another thing is given that this expression that is derived is valid for the different general shear failure. So, that I have mentioned now if the soil fails in local shear failure then how to incorporate those effect in this expression. Now, that is also possible in this expressions. So, suppose this is for the previous one was for the general shear failure and this is for local shear failure. Now, local shear failure Tazaki's as recommended that the shear strength parameter this parameter this is C is converted to C m and phi is also converted to phi m. Now, how this for this is this C is for cohesion for the general shear failure and C m is the cohesion for local shear failure and phi is the cohesion angle for general shear failure and phi m is the friction angle for local shear failure. Now, how to convert this thing now C m will be two-third of C and tan phi m that will be equal to two-third of tan phi. So, phi m value it will be tan inverse of tan phi two-third of tan. So, now in place of phi and C we have to use this phi m and C m when we calculate the load bearing capacity of the soil in place of local shear failure. So, now we have to use this expression in case of local shear failure that is q ultimate this will be two-third of C n c dash plus q n q dash plus half gamma p n gamma dash. So, where this n c dash n q dash n gamma dash are the bearing capacity factors in case of local shear failure. Whereas, n c n q n gamma these were the bearing capacity factors for general shear failure. So, this n c dash n q dash n gamma dash are determined using the same expression but we have to replace phi by phi m and then we can determine this n c dash n q dash n q dash n gamma dash by using the same expression. Here also, Tazdaq has given this n c dash n gamma dash n q dash values in a tabular form. So, here so this is the Tazdaq bearing capacity factors under this local shear failure where this is the phi value from 0 to 50 degree. So corresponding phi value what will be the value of n c dash n q dash and n gamma dash. So, these values are given here. So, if you know the phi value and if the soil fails in local shear failure then we have to use n c dash n q dash n gamma dash instead of n c n q and n gamma. So, here we will get the different value of n c dash n q dash and n gamma dash. Now generally it is observed that if phi is greater than equal to 36 degrees then it indicates that this is a general shear failure and if phi less than equal to 29 degrees then it is local shear failure. So, if soil fails in general shear failure or if it is a dense soil or stiff clay then we can use this previous expression for the general shear failure and if soil fails in local shear failure then this medium dense or moderate sand then we have to use this expression we have to convert this c to c n and phi to phi m then we have to use this expression. So, this final expression is for the local shear failure. Now the next thing is that these expressions are derived for strip footing. So, these expressions are derived for the strip footing. Now if soil this footing is circular and square footing then also we can use this expression with some modification. So, Tazaki has recommended that how we will use because this expression general expression is valid for the strip footing. So, how we will use this expression for the circular footing or square footing. Now if the so that the general expression for any type of footing so we can write this is alpha 1 c n c and plus q n q plus alpha 2 gamma p n gamma. Now as I have already mentioned that for the strip footing this alpha 1 is equal to 1 and alpha 2 is equal to 0.5. Now for the circular footing this alpha 1 is equal to 1.3 and alpha 2 is equal to 0.3. So, alpha 1 is 1.3 and alpha 2 is 0.3. Now for the square footing alpha 1 is also 1.3 and alpha 2 is equal to 0.4. So, for the square footing the expression will be 1.3 c n c plus q n q plus 0.4 gamma b n gamma. So, n q n c n gamma values that we will get from this table if we know the phi value. So, here also for different other type of footing the circular square we can use this expression also. Now one thing that in this expression that we have derived. So, in this expression we have not considered the effect of water table. So, that effect we have to consider here. Now here see if we take the effect of water table suppose this is the footing whose width is b and depth is d f. Now these are the points that we have position of water table. Suppose this is the so water table can be above the base of footing or below the base of footing. So, now we have taken the different we will consider the different cases where we put this water table. So, that can be above the base of footing. So, this is the position of the water table or that can be below the base of footing. So, this is the base of the footing that can be below the base of the footing. So, this is another position of the ground level. So, here this ground level we can locate this ground level at distance of d w from the ground level and if it is below the base of the footing then you can locate this ground level position by d w dash is measured from the base of the footing. Now, we will consider the different cases. Now, for the first cases or first case now here we consider that above this soil level. So, suppose if we take this figure. So, this is the ground level this is the base of the footing where b is the width and d f is the depth of the foundation. Now, suppose this is the position of the ground level here this is d w and this is the another position of the ground level ground water level. So, this is d w dash now first case that if case 1 the first case if d w dash greater than equal to b. So, first case is that the ground level that means this position of this ground water table below the base is equal to or greater than width of the foundation. So, now in this general expression we can write our expression is q u c n c plus q n q plus half b gamma n gamma. In this n q term this is d f is gamma. So, we can see that here this d f into gamma is there. So, this gamma is basically the unit weight of the soil above the foundation base and this gamma because these two gammas in two parts. So, this gamma is the unit weight of the soil below the foundation base. So, this is the second first term second term and third term. So, first term is unit weight that means gamma less and then the second term there is one gamma that gamma is basically above the foundation base and the next one is the below the foundation base second one. Now, in the case 1 if d w dash is greater than equal to b then gamma will be gamma t or gamma bulk. So, gamma will be gamma bulk as this is natural unit weight condition. So, both the case that means the water table will not affect this bearing capacity expression. So, if the natural gamma is natural unit weight of the soil is gamma t or gamma bulk then the both the cases. So, this expression also you have to use the gamma bulk and this expression also you have to use the gamma bulk. Now, if case 2 you will consider this d w dash is equal to 0. So, this is the case 2. So, if d w dash is equal to 0 this case 2 that means the water ground water table level is at the base of the footing then we have to consider the gamma sub that is gamma sat minus gamma w. So, this is gamma sub merge gamma sat. So, this gamma we have to used for second term for half b gamma n gamma portion and gamma t or gamma bulk for q n q part. So, here if first case d w dash greater than equal to b then you will have to use gamma t for both the cases this part and this part. Now, if d w dash is equal to 0 then you have to consider gamma sub for this part. This part you have to use the gamma sub and this part you have to use gamma bulk. Now, for the case 3 that d w dash is less than equal to d w dash b or less than equal to b or you can say this is greater than b. Then we have to use gamma is equal to gamma sub plus d w dash divided by b into gamma t minus gamma sub for half b gamma n gamma and gamma t for q n q. So, if d w dash is less than equal to b then you have to use this gamma value in this part and gamma t for this part. Now, case 4 that if water table is at g l. So, if water table is at g l then both the parts you have to use the gamma sub. Here also you have to use the gamma sub here also you have to use the gamma sub. So, here also for both the parts. Now, for the case 5 that d w is less than equal to 0 and less than equal to d f then gamma will be gamma sub plus d w by d f into gamma t minus gamma sub for q n q and for the second part for this part you have to use the gamma sub and gamma sub for half b gamma n gamma. So, if the water table is greater than 0 and less than d f then you have to use this gamma for this part and gamma sub for this part. So, these are the 5 cases where we can include the water table effect putting this water table position at different location and we can incorporate the water table effect with a different unit weight that is used in the expression. In the next lecture I will explain the other effect this is the loading inclination effect then the shape of the foundation effect depth of the foundation effect in this expression. So, in general time those things I will explain in the next lecture. Thank you.