 Last lecture we have finished this basic introduction to this slope stability analysis that means different modes of slope failures and what is the reason and in this part we are going to discuss about this slope stability analysis in this lecture outline. We will discuss this common features of slope stability analysis method, water forces on soil, infinite slope analysis and finite slopes as we have discussed also last class plain circular and non circular failure surfaces. Now common features of slope stability analysis method if I say the factor safety s f is equal to s by s m where s is equal to shear strength s m is equal to your mobilized shear strength shear resistance. So if this factor safety is equal to one for particularly slope stability analysis if this factor safety is equal to one then you can say that it is slope is in slope is in failure it is in failure if it is one then you can say that it is fail that means slope has been failed factor safety should be greater than one for stability analysis point of view this is the basic requirement then safe and location of failure surface is not known a priority but assume generally we do not know the safe and this location of the failure surface generally it has been assumed by trial and error by trial and error method this what I what I mean this safe and location of the failure surface is not known to us. So let us start with a simple slope with a slope you start with this with this slope what we are supposed to do you assume this failure surface different failure surfaces you assume with respect to different failure surfaces by trial and error method find it out the factor safety suppose for example case one and case two case three these are the three assumed failure surface for one two three if the factor for one if the factor safety 1 point 1 for two if the factor safety is 1 point 2 let us say for three failure surface factor ST T is 1 point three that means out of these three the minimum factor ST minimum factor ST comes out to be one point one that means this failure surface generally consider your limiting failure surface. Then your static equilibrium static equilibrium is a equilibrium forces and movements on the sliding masses that means equilibrium forces and whatever the movements on the sliding masses that has to be considered then your two dimensional analysis this is the outline of this slope stability analysis methods or features now in case of infinite slope analysis as I said earlier this shape it will be in infinite so translational failure along a single plane failure surface parallel to your slope surface in case of infinite slope analysis a translational failure along a single plane failure surface parallel to slope slope surface the ratio of depth to failure surface to length of failure zone is relatively small the ratio of depth to failure surface the ratio look at this the ratio of depth to the failure surface to length of failure zone depth to length of failure zone is relatively small in case of infinite slope analysis applies to surface revealing in granular material or slab sliding in cohesive material if you look at this apply to the surface that means in granular material or slab slides slab slides in case of cohesive material in case of cohesive material particularly slab slides in case of cohesive material equilibrium of forces on a slice of the sliding mass along the failure surface is considered equilibrium look at here equilibrium of forces on a slice of the sliding mass on a slice of this sliding mass on a slice of a sliding mass along the failure surface is considered along the failure surface means whatever along the failure surface this equilibrium condition has to be considered. Now, take this case of infinite slope in this infinite slope let consider a small part in the infinite slope let us say gamma is unit weight above this above this water table there is a water table below this water table soil property is gamma saturated c and phi and w is your weight of your mass and normal force as well as tangential force d is your depth of your infinite slope now force is a function of force is a function of if you look at your c prime phi prime gamma beta d and u. So, if I come back to here beta is your angle look at here beta is your angle at what angle this infinite slope makes to respect to your horizontal. So, now this is a function of c prime phi prime gamma this is your properties of this soil beta is your angle d is your depth u is your pore water pressure. So, with respect to that where you can find it out r u r u is your different r u for c page parallel to the slope surface c page emerging and c page downward d c for granular soil factor particularly this factor of safety this f granular soil in sorry in case of infinite slope analysis this factor of safety is a function of if you look at here it is not force this is a factor of safety it is a function of function of c phi gamma b beta d and u. So, for granular soil f is a function of tan phi prime by tan beta 1 minus r u 6 square beta for dry granular soil where there is no water table r u is equal to 0 f is equal to tan phi prime by tan beta factor of safety is equal to tan phi prime by tan beta tan phi is your it is your property of soil in angle of this soil. Then beta is your angle with this slope has made. So, for cohesive soil factor of safety decreases with increasing depth of failure plane factor of safety decreases with increasing depth of your failure plane that is that means if c is sufficiently large small c is sufficiently large that is your unit cohesion is sufficiently large and d c then for factor of safety 1 may be large and infinite slope failure may not apply in this case infinite slope may not be apply if c is sufficiently large. Now, next part is your water forces on soil water fills the voids water fills the voids and increase weight which increases your driving forces particularly what happen in inside the void void space water fills once water fills inside the void space if this is my if this is the case of this this is my this is your infinite slope if there is a crack or may be there are the voids inside the voids inside the voids. So, what will happen water fills the voids that means earlier this is a soil mass this mass means c prime and phi prime and your gamma gamma is your unit weight of soil and if water pour inside the voids what will happen it increase the weight of it increase the weight means it will increase this driving force water also exerts pore water pressure as you know water also exerts pore water pressure because of water which decreases your effective stress therefore the strength if I take into effective stress like sigma prime is equal to sigma minus u. So, u is your u is your pore water pressure once water fills this voids the pore water pressure increases that means it decreases your effective stress therefore the strength also it will have affect your strength there are mathematically two equivalent ways of taking water forces into account in stability analysis there are two ways to take consideration of water forces first one is your equivalent methods for water forces first one is your boundary water force plus total unit weight boundary water force plus total unit weight u is equal to h p into gamma w gamma saturated gamma w or gamma saturated consider soil element particle and water filled pores as a single solid mass considering soil element considering this soil element considering soil element as well as that means particle soil element means particles of the soil along with your water it consider as a one single soil mass one single solid mass then case two second condition seepage force plus submerged unit weight submerged unit weight that means your factor shift coming i gamma w v where r gamma consider soil element as particle skeleton soil element as particle skeleton with water external to it with water external to it now if you look at here if you look at here this is an infinite slope at an angle beta if you look at here these are this is my equipotential lines and these are my flow lines if I make it into flow net if I make this flow net in the infinite slope and this are my this is your flow lines this is your flow lines and this is your equipotential line this is your flow line and this is your equipotential line case one we are discussing that is your boundary water forces if I consider a small element of this part if I consider a small element of this part if this is a width is b and this is your weight w acted if you look at here both the sides both the sides how the water pressure generated and at the bottom also your pore water pressure at the base as well as sm and normal force will be acted on this. So what is the case one in this case one boundary water force plus total unit weight in the case one we are considering a small element with the small element of soil we are taking the boundary at the boundary what is the water forces acting and plus your total unit weight we have consider in this case what will happen soil element soil element means inside soil element if I if I consider if you look at here first case this is a boundary if this is my if this is the this is the soil element I am going to consider this is your soil element let us say one one two two in this soil element what will happen if I go to back boundary water forces that means these are my boundary water forces these are my boundary water forces here one boundary here other boundary here other boundary at the bottom. So these are the boundary water forces at the left hand side at the right hand side at the bottom the boundary water forces consider and another assumption is that water inside the soil if this this part I have taken into consideration that means within this part there is a soil mass within the soil mass what will happen water also there this has to be this has been consider as a one solid element. So this is your this is your one solid element of your weight w with b and these are all of your boundary water forces this is your lateral water forces lateral at the bottom. So this is your case one considering your water forces now if you look at this beta hydraulic gradient case two if I take into consideration of case two what is the case two case two is your case two there are two methods you can find it out one is your considering your boundary water forces second is your seepage force either of the two method you can solve this seepage force plus some much unit weight consider soil element as particle skeleton with water external to it particle skeleton with water external to it particle skeleton with water external to it particle skeleton with water external to it. Now if this is your infinite slope in case two that means seepage force seepage force plus some much unit weight to be consider now here hydraulic gradient I is equal to sin beta with respect to that seepage force if I take the soil element here this is my seepage force. So this seepage force this seepage force will be acting this way. So this seepage force is I w I into gamma w I is your hydraulic gradient gamma w is your unit weight of water in terms of volume then effective weight effective weight is your nothing but is your w is your effective weight which is equal to gamma into volume effective weight w is equal to gamma is equal to gamma is equal to unit weight of the soil mass for volume unit weight of the soil mass for volume means mass into or weight into the volume from there you can find it out effective weight. Then gamma prime some much unit weight is equal to gamma minus gamma w or gamma substituent minus gamma w from there you can find it out your by means of second method you can also find it out. Now this is the case we have discussed particularly if I review it again this is the case we have discussed particularly taking into consideration of infinite slope I have also discussed this infinite slope by taking this infinite slope what first you consider a small element and take the equilibrium into consideration that means I have taken this element of this then equilibrium consideration point of view this is the weight then this has been acted at an angle beta then tangential force and normal forces acted on this say with this infinite slope there are two analysis has come into picture as far as water forces are considered case one we have consider as a water field void case one we consider one is your boundary water forces with your total unit weight other other there are two equivalent method one is your boundary water force plus total unit weight second case is your shipage force plus some of unit weight we have discussed also now proceed to finite slope this is the case infinite slope has been over now proceed to finite slope in this finite slope first case is your plane failure surface so where the plane failure surface occur that means it is a translational block slides along single plane of weakness or geological interface translational block slides along single planes of weakness or geological interface so factor of t f is equal to c prime into l length w cos theta minus u is equal to pore water pressure into length into tan phi divided by w sin theta plus f w now if you look at this block slide analysis block slide analysis that means a planner or plane failure surface in this case in case of if I take into consideration if I take into consideration this slope as like this this slope as like this if I take into consideration in which case is generally occur suppose there is a rock mass above the rock mass there is a soil slope so what will happen it will fail in a plane failure surface in a plane failure surface it will fail so that means what will happen it makes an angle with this theta this is your plane failure surface with this if this is my length l so there is a weight as well as normal force with respect to your theta this is called block slides now another case of this sim of weak soil there is a hard soil above this then below this there is a weak soil in that case also plane failure surface suppose to be occur so there is a block in this block there is a unit weight that means your weight of the soil mass as well as weight of the soil mass plus normal component as well as tangential component with respect to that we will have to find it out this factor of safety so case two sim of weak soil we are discussing right now where this block slides occurs case one if there is a rock mass rock mass above the rock mass there is a soil is there what will happen there is a block slide that means plane failure surface will occur now in this case case two if there is a weak soil above the weak soil there is a soil mass what will happen again there is a block slide will occur this is a case we are discussing now for your plane failure surface where it occurs, second is your finite slope that is your circular failure surface arc, circular failure surface arc generally we do by means of rotational slides, rotational slides method of slice method, apply to slopes containing cohesive soil, apply to the soil the condition here the slopes particularly the slope where it has where it has cohesive soil, it has been applied to your cohesive soil ordinary method of slices, it has been solved by ordinary method of slices this has been given by your failure method, then also there are other methods this is called Bissoff simplified method or Spenser's method these are also other methods available, now if you look at this force involved in effective slip circle analysis what are the forces involved if you look at here there is a circular slip particularly in case of cohesive soil, so we will have to find it out where is your center of rotation, where is your center of rotation O with this center of rotation O these are my radius with that with respect to that that your circular surface is there, so then if this is O say then if this is my C g if this is my C g how far if where the C g w will be acted how far from your center that is your r sin alpha, if I consider a slice now you take into consideration of a slice of any n slice if I name into 1 2 3 4 5 6 it may be n number of slices I can do it also, so if you take it into say let us consider a 1 slice let us consider 1 slice that is your n slice n n th slice, so in this case n th slice if you look at this slice I have taken into in the bigger picture, so what will happen in this slice what are the forces supposed to act if you take it here this slice will be acted by a lateral force what what happen one it has been pushed, so one one way it will be these are all your frictional resistance around the two sides weight of your soil lateral resistance then bottom bottom is your normal as well as pore water pressure pore water pressure, so what will happen, so you take all this make it into a equilibrium conditions equilibrium conditions means all the forces once it will be in equilibrium conditions, so all the forces should meet if I draw it by head and tail it should be much it should satisfy this equilibrium condition from there you can find it out your factor of safety, now go to this ordinary method of slices as proposed by felinus in this case assumption that the resultant of slide forces on each slice are collinear look at here this assumptions resultant of slide forces resultant of slide forces on each slice on each slice are collinear resultant of slide forces that means resultant of slide forces on each slice are collinear act parallel to your failure surface and therefore, cancelled each other resultant of side forces if you look at your what is your side forces this is your side forces this side forces so what is it mean resultant of side forces on each slice are collinear and at parallel to the failure surface at parallel to your failure surface and therefore, cancel each other one is up other is down so therefore, it cancel each other now once it has been cancelled each other so what are the forces remaining now in simple method of ordinary method of slices in this case if you look at here if this is my slice one slice has been considered only weight w n normal forces as well as tangential forces as well as pore water pressure so what will happen this is my normal forces this is my normal forces this is your shear stress or tangential forces then along the normal forces also pore water pressure act into picture so this makes angle at an angle alpha n or theta n it depends on that so other forces is your weight of this slice and this side forces cancel to each other in case of ordinary method of slices so here it is written that resultant of all side forces assume to be act in this direction so n n normal force found out by summing forces in this direction particularly particularly in this direction what is your normal forces summing up where they are meeting to your c g now this is your ordinary slice method who has given this ordinary slice method it has been given by means of a felinus felinus is the person who has given this solution it is called ordinary method of slices now come come back to second one is your b sharp simplified method b sharp simplified methods in case of b sharp simplified method assumes that the resultant of slide side force forces on each slice act in horizontal direction assume that resultant of side forces assume that resultant of side forces on each slice resultant of slice side forces on each slice act in horizontal direction and therefore vertical side forces vertical side force component can components cancel each other so this is their assumption the assumption they have given with that resultant of side forces on each slice act in horizontal direction and therefore vertical side force component cancel each other so factor of c p comes out to be c n b c is your unit cohesion b is your width plus w n minus u n n b n u is your pore water pressure tan phi n 1 by m alpha divided by summation of w n sin alpha so m alpha n alpha is equal to cos alpha n plus sin alpha n tan alpha n divided by u for undrained analysis factor of c p is this is for undrained analysis c n l n by w n sin alpha n so b sharp they have given chart to find it out m alpha alpha is positive when slope failure arc is same quadrant as ground slope you see alpha is positive when slope failure arc slope failure arc if this is a slope alpha is positive when slope failure arc when slope failure arc is same quadrant is same quadrant as ground slope in the same quadrant as ground slope if suppose this ground slope is this in the same quadrant quadrant 1 quadrant 2 quadrant 3 quadrant 4 then it is considered to be positive so based on that based on that positive and negative values of alpha they have given the charts they have considered the value of alpha from 0 degree to plus 60 degree and 0 degree to minus 30 degree and with respect to that they have given this values of m alpha varying from 0.6 to 1.4 if you know this value of m alpha you can calculate from this cos alpha sin alpha tan phi prime by factor of safety so alpha is your slope angle means slope is at what angle it makes with your ground surface cos alpha plus sin alpha tan phi prime by factor of safety so alpha is degree is given you take this alpha what degree it is there then with respect to that you calculate your m alpha once suppose for example alpha is equal to 10 degree I am taking with this suppose value of m alpha is 0.8 0.8 now where it intersect if you look at it intersects somewhere else from where you can find it out tan phi prime by factor of safety so you know the phi so you can find it out your factor of safety this is a chart for your m alpha has been given then side forces in besoft method side forces in besoft method as I said earlier resultant of all side forces assume to act in this directions resultant of all side forces side forces resultant of all side forces resultant of all side forces are assumed to act in this directions and nn found by summing forces in this direction nn is found by summing forces in this direction your un is your pore water pressure in normal direction sm is your shear forces then bn is your width of that nth slide if you look at here b is equal to width n is equal to nth slide if I consider number one slide so this will be b1 if I consider number two slide this will be b2 if I consider number i slide it would be bi so alpha n here alpha n if you look at here it makes an angle the slope makes an angle with your horizontal or the ground surface now last one is your that is your spencer's method in this spencer's method assumes that the point of application of resultant of side forces point of application of resultant of side forces on each slide is at mid height of each slide look at here this assumption it is a modified of that earlier two method first one is your by simply slides method second is your b's of method third is your this your spencer's method in this case the assumption is that point of application the point of application of resultant of side forces the point of application of if this is my slides point of application of resultant side forces on each slides resultant side forces on each slides is at mid height of each slides that means if this is my height h at mid height it will act at mid height of this slides but no assumption is made regarding inclination of resultants look at the look at these drawbacks particularly in case of each method in case of spencer's method there is no assumptions there is no assumptions made regarding inclination of resultant inclination of resultant means the moment I inclined the slope is inclined the moment the slope will be inclined obviously the resultant of your side forces will be inclined so there is no assumptions whatever the slope angle they say with respect to that slope angle they have considered resultant forces of each slides is at mid height of each slides resultant forces of each slides is at mid height that means whatever the angle resultant of this forces will act as mid height and this method if I compare with this method with your b's of method this method is more or less you can say that more exact more appropriate than your b's of method we will discuss more in the next class about this finite slopes non circular failure surface in the next class thank you.