Welcome to lecture number 2 of module 5 stability analysis of slopes in the course advanced geotechnical engineering. So the title of this lecture is lecture 2 which is the part of module 5 stability of slopes. So in the previous lecture we understood about different types of slope failures and causative factors for slope failures and we also discussed about infinite slope stability analysis methods and we have also discussed that there are two types of slopes predominantly they are infinite slopes and finite slopes. Then the finite slopes are the one which are man made slopes to the maximum extent and some hill slopes also can be called as a finite slope. Now when we look into the analysis of finite slopes there are different possible failure surfaces. The failure surfaces can be planar in nature or if it is a homogeneous soil then there can be possibility that you can get a circular failure surface and if you are having a different stratification of soils there can be possibility that we can get non-circular failure surfaces. So in this slide three different failure surfaces are shown one is defined as planar failure surface this occurs along a specific plane or weakness. So where the potential weak plane will exist and this failure surface occurs along a specific plane or a weak plane of weakness. This basically excavations into stratified deposits where strata are dipping towards the excavation that is the layers of soil dipping towards the excavation and in earthen dams along the sloping course of weak material not likely to occur in homogeneous soils. So these planar failure surfaces they occur along a specific plane of weakness and the circular failure surface this is for soils exhibiting cohesion C or cohesion and friction angle and no specific planes of weakness are great strength. So this actually takes in the form of a circular failure surface and non-circular failure surface when the distribution of shearing resistance within an earthen dam is non-uniform and failure can occur along surfaces more complex than a circle. So this is said that non-circular failure surface mostly occurs in layered soils. So in the typical cross section of a finite slope with one vertical and horizontal where the small n is the number if n is equal to 2 it indicates that the slope is having one vertical to horizontal sloping inclination that is about 26 degrees with horizontal. And if n is equal to 1 that indicates that the slope is actually having a 45 degree sloping inclination with the horizontal and the horizontal surface indicates that the back slope which is 0 degrees but in practice there can be a possibility that the slopes can also have a particularly especially for natural slopes in nature there can be with natural sloping inclination of back sloping inclination ranging from 10 to 20 degrees or so. So in this for a cross section of a finite slope a typical planar failure surface is depicted here and then a two-bedge type of failure surface like as it has been told in earthen dam suppose a sloping core and if this happens to the potential weakness plane then the two-bedge failure mechanism can come and this is the typical circular failure surface and this is also a circular failure surface where the majority of the failure surface extending into the soft soil which is underneath the say for example beneath this level and this is a tow slope failure where it can be seen that the entry and the exit point is from the tow of the slope. So this is called the entry point commencement of the entry point of the failure surface and this is called the exit failure surface. So in a given slope there can be number of failure surfaces but one need to determine the failure surface which actually gives the critical factor of safety or the least factor of safety and that particular failure surface is called as the potential failure surface. Now in this particular slide there can be typical rotational slides which are shown one is a cylindrical shaped failure surface you can see that this is a circular arc but the failure surface which actually shown is mostly this is common for plane strain structures there can be a settlement at the certain point away from the crest of the slope and a failure surface which actually occurs with the soil movement which is actually shown here and this is a typical rotational slide and which is called as a spoon shaped failure surface. So you can see that this failure mode or a slide which is actually called as a spoon shaped slide. Now we actually yesterday we introduced ourselves to that there is a effective or total stress parameters. We know that the total stress parameters may be can be adopted for the short term conditions and effective stress parameters can be adopted for the long term conditions or getting the long term stability of a slope. Total stress parameters can be obtained for the short term stability assessing the short term stability of a slope. So the short term basically low permeable soil basically in clays at the end of the construction the soil is almost still undrained. So here the adoption of total stress analysis making use of undrained shear strength U is adopted. So at the end of the construction of the soil which is clay is almost still undrained hence the total stress analysis making use of undrained shear strength C U is adopted. Some free draining materials sands and gravels if you look into it. The drainage takes place immediately hence effective stress parameters C dash and phi dash are used. Suppose if you have got a free draining materials sands and gravels when the drainage takes place immediately so there is a need for adopting effective stress parameters that is C dash and phi dash. Long term relatively after a long period of time let us say that an embankment is constructed on a soft soil after having waited for a certain duration the fully drained condition might have been attained. So the fully drained stage will have been reached hence the effective stress parameters C dash and phi dash can be used. So what we broadly say is that the effective stress parameters are used for long term conditions are also used for you know the assessing the stability of a slope with free draining materials like sands and gravels. In case if there is a short term condition with low permeability then the adoption of total stress parameters is more relevant. So the total stress strength is used for short term conditions in clay soils whereas the effective stress strength is used for long term conditions in all kinds of soils or any condition where the pore pressure is known. So this is after John Boo 1973 stated that the total stress strength is used for short term conditions in clay soils whereas the effective stress strength is used in long term conditions in all kinds of soils or any conditions where the pore water pressure is known. So this pore water pressure may result due to ingress of rain water into the slope that we have discussed one of the causative factors of the slope instability is rainfall. So because of this once the ingress of the rain water is captured in the form of periodic surfaces we can actually get the pore water pressure having known then we can actually estimate the total stress parameters and then the long term conditions can be used. The analysis of the finite slope particularly with the factor of safety the factor of safety for finite slopes depends basically on the assumed location of the center of rotation of the slip surface. So as it was shown in the typical failure surfaces particularly for circular arc failure surface wherein it all depends upon the location of center of rotation in suppose if a slope analysis which is done in two dimensions that is basically for a plane strain structure in x and y direction then assumed location of the center of rotation for the slip surface and the radius of the failure surface basically what radius it actually exists and type of failure that is toe failure base failure or slope failure. So before attempting the analysis methods before attempting the analysis methods let us look into the various definitions of factor of safety here the definitions are given with respect to limit equilibrium point of view force equilibrium point of view and moment equilibrium point of view the topmost figure if you see there is a failure surface which is indicated and there is a shear strength which is actually shown and there is a shear stress which is actually shown the shear stress is mobilized due to the disturbing forces and the shear strength is mobilized to as a resisting force. So as a limit equilibrium condition for total stress we can write factor of safety as shear strength available divided by the shear stress in case of effective stress which can be written as factor of safety is equal to C plus sigma dash tan phi dash by the shear stress which is mobilized. As far as the force equilibrium is concerned for a typical wedge failure which is having a planar failure surface extending a unit length perpendicular to the plane of this figure wherein we can actually write the force equilibrium as a factor of safety is equal to sum of resisting forces divided by sum of driving forces or disturbing forces. So here if you look into it the resisting forces are nothing but the resistance offered by the soil that is nothing but the shear strength and driving forces is nothing but the because of the weight of the slice which is actually acting vertically downwards and by resolving this along the potential plane of failure that is the planar failure where which is actually taking place and that component will work out to be W sin alpha. So by writing factor of safety is equal to Su by W sin alpha. So we can write this as Cl, Cl is nothing but cohesion which is mobilized along the, it is assumed that the cohesion is mobilized uniformly and over a length L, L is nothing but the length of the plane of surface that is measured from this point to this point into plus n that is n is nothing but the normal force tan phi divided by W sin alpha where total length of the sliding plane which is actually shown here which is nothing but this is the length of the sliding plane. Then similarly for moment equilibrium the factor of safety is defined as resisting moment by driving moment. So here the summation of resisting moments divided by summation of driving moments. So when we try to get the resisting moment by taking the shear strength along this particular arc, let us assume that we are having Su1, Su2, Su3 and Su4. Then we can say that Su1 into dl1, Su2 into dl2, Su3 into dl3 plus Su4 into dl4 into R that is the, this is the force which is acting over this particular length. We have taken each arc length as dl1 into one unit that is the unit perpendicular to the plane of this figure into R we will get the force into moment that is the resisting moment divided by W which is nothing but the entire mass is actually assumed as the CG of this area is assumed to act here and the at the center of gravity here this W into this horizontal distance from the center of rotation that is x. So that is actually called as Wx. So factor of safety is equal to in the integration form if you show that R is equal to 0 to L Su into dl. So this indicates that you know the factor of safety can be defined from the force point of view or limit equilibrium point of view or moment equilibrium point of view. Now let us try to review the different stability analysis methods. Basically all limit equilibrium methods utilize the more column expression to determine the shear strength tau f along the sliding surface. So prime of AC the more column expression is used to determine the shear strength tau f along this sliding surface and then as we defined in the previous slide the factor of safety is given by Su by tau which is for total stress conditions and for effective stress conditions F is equal to or Fs is equal to C dash plus sigma dash tan phi dash by tau. See basically the available shear strength tau f depends on the type of soil. So if it is say a particular type of soil and the effective normal stress whereas the mobilized shear stress tau depends upon the external forces acting on the soil mass that is the self weight of the soil which are called as geostatic conditions and as well as any external loading if it is there on the crest of the slope that also adds to the destabilizing or disturbing force. So here different methods are summarized here but we are going to discuss about some selected methods. We have many methods as you can see from this slide. The first method is ordinary method of slices and then it is followed by Bishop's simplified method is there. So in all these methods the zone or area within the failure surface is assumed to be divided into the number of slices basically vertical slices and then the slice equilibrium is considered either from the for the four circular point of view or moment equilibrium point of view and based on that the deductions for the factor of safety have been obtained. So for example in the case of ordinary method of slices the circular failure surface is assumed and moment equilibrium is considered and in this method the both normal forces acting inter-slice forces that means that normal forces acting on the perpendicular to the vertical face of the slide as well as the tangential forces are neglected. In the case of Bishop's simplified method it is for circular failure surfaces and whereas moment equilibrium is satisfied but it considers E that is the normal force perpendicular to the vertical surface of the slide but on both sides of the slice but neglects the tangential forces, the tangential forces are assumed to be 0. In the case of John Bush method it is mainly predominantly for non-circular failure surfaces and this also can be used for circular and non-circular failure surfaces but here this method is based on predominantly on the force equilibrium. So the moment equilibrium is not satisfied the force equilibrium is satisfied. So it considers the normal forces perpendicular to the inter-slice forces but it again like Bishop's method it neglects the tangential forces that is one of the components of the inter-slice forces. Then we have methods by Spencer method wherein it considers the T and U with some constant inclination where T is equal to a relationship between tangential force and the normal force E which was given as with a constant inclination T is equal to tan theta into E and then Morgan Stern's method wherein it can be used for both circular and non-circular failure surfaces and it satisfies both moment and force equilibriums and the assumption for T and E which is defined by a function X where T is equal to function X into a constant delta into E. So if you look into this the Sermas method and Morgan Stern's price method it satisfies both moment equilibrium and force equilibrium methods but the majority of the applications the Bishop's simplified method is used or to some extent the John Booth's method is also used. So this is the typical section of the unit width assumed for the analysis as the slope is assumed to be like a plane strain structure. So a unit width is considered for the analysis like a per meter width of the analysis is considered for doing a two dimensional stability analysis. Of course now there are the methods which are actually available for performing the three dimensional slope stability analysis. Now before discussing about the ordinary method of slices let us look into the evolution of different types of methods for undrained conditions and drained conditions and then we will introduce for the ordinary method of slices and Bishop's simplified method and the other method like Morgan Stern price method. In the case of circular arc analysis for undrained condition or it is also called as pi suffix u that is angle of internal friction undrained condition is equal to zero analysis. Basically this is for a analysis basically performed in terms of total stress analysis and applies to short term condition only for a cutting or embankment assuming the soil profile to come price of fully saturated clay. And the on the right hand side a cross section of a slope or an embankment which is actually shown here and this is the slope inclination which is required to be determined. Here what slope inclination you need to be provided so that the adequate factor of safety need to be ensured that is the point of the importance. Now here it is assumed that the potential failure surface which is indicated with this yellow broken line can be seen here and this is the undrained coefficient is assumed to be mobilized along the failure arc and W is the CG of this weight of the entire area which is subscribed in this zone A, B, C, D in this zone. So this requires determination of the, determination of area of this portion and then by knowing the area into the one meter length perpendicular to plane of the spigar we can calculate by knowing also the unit weight of the soil for use for embankment or unit weight of soil in the cutting we can determine what is the weight and with respect to the moment of rotation which is considered we can also determine what is the horizontal distance from the center of rotation, horizontal distance that is D is nothing but the distance from the CG of the area where the weight W is acting to the center of rotation. So for this by taking moment equilibrium where by taking moment about the moment of all resisting forces about the center of rotation to moment of the all the driving forces about the center of rotation. So here there is no this W is because of the self weight of the soil so factor of safety is equal to MR by MD where MR is nothing but Cu into LA, LA is nothing but the length of the entire arc into that R, Cu LA into R is nothing but the resisting moment divided by WD, WD is nothing but the driving moment or the disturbing moment. So here one of the disadvantages of this method is that accuracy in which actually you will determine the weight W or area and the determination of D which is actually involved. So the calculation of factor of safety if you are having let us say an external load on the within the failure zone. If the external load which is either due to distributed load or due to if there is a distributed load then one need to consider the you know the if this is identified as the potential failure surface then in this zone the load is distributed over this length into the length which is perpendicular and at the CG of that load is actually located. But in this case let us assume that there is a boundary wall which is located at a certain distance where when we consider about this intensity of the load we can actually now take the disturbing moment is nothing but W2 into D2, W1 that is due to the self-heated soil into D1. So the resisting moment is nothing but still the Cu into L into R. So the factor of safety in this case is nothing but Cu LR divided by within brackets W1 D1 plus W2 D2. Now from the undrained analysis Taylor 1948 has developed a method wherein the potential failure surface is given so that the least factor of safety can be determined. So this development is sourced from the undrained analysis which is which we have discussed just now wherein here it is actually considered that this entire the soil portion below this depth is considered as D and then this height is say H and here this is the slope inclination which is beta and beta is equal to 90 degrees means it is a vertical cut and this inclination is 2 theta and this inclination with horizontal is alpha and this is the cohesion which is actually mobilized and this is the weight and then this is the D from the which is horizontal distance measured from the CG of the weight from the weight of this entire portion from the center of rotation. So D is nothing but the depth factor what we call which is the nomenclature usual for developing stability charts wherein here the factor of safety we know that we have just discussed in the previous slide where factor of safety is equal to C L R W by W D where factor of safety is the lowest factor of safety obtained from the circular arc analysis and the factor of safety or the self-weight of the area or portion involved in the failure which is the active zone what it is called is function of gamma H and geometry of the failure surface. The geometry of failure surface can be characterized by three angles alpha beta and theta and so what it has been taken is that the by rearranging the terms here C by factor of safety is written and which is written as C R is equal to gamma H into function of alpha beta and theta the CR is nothing but the required cohesion of a soil just to maintain a stable slope and function of alpha beta and theta is a pure number and basically that is actually designated as a stability number. So the Taylor's stability number is actually given by CR is equal to CR by gamma H. So CR is nothing but required cohesion to just maintain a stable slope otherwise it can also be written as C by gamma H by FS into CR when you substitute for C by FS we can write by C by FS into gamma H. So for a factor of safety is equal to 1 it actually induces that NS is equal to C by gamma H. So the required cohesion CR is nothing but the required cohesion to just maintain a stable slope. So here with reference to angles alpha beta theta what has been done is that the weight portion which is actually considered that is the weight which is subscribed as the function of gamma H and geometry of failure surface are considered in the form here in the form as a to represent the geometry of the failure surface. So based on that deliberations the Taylor actually has given stability charts which are known as popularly known as Taylor's curves wherein on the x axis we have sloping inclination it ranges from 0 degrees to 90 degrees and on the y axis it is the stability number that is NS is equal to C by gamma H. So here it can be seen that one up to beta is equal to less than 53 degrees it is actually depend upon the the D which is nothing but this failure surface is assumed to pass through below the base but when we when the beta greater than 53 it is found to independent of D. So it is only depend upon the sloping inclination which is when it is more than 53 degrees. So it can be seen therefore more than this it it is the constant the stability number will be constant. So so here this is used for u is equal to 0 and mostly for undrained conditions this is used. Now the undrained analysis of the by using this Taylor's method basically for beta less than 53 degrees as we discussed in the previous slide the stability number is a function of beta and D by H. So that means that here there can be a possibility of the base failure for gentle slopes the critical failure surface goes below the toe and always restricted above the strong layer hence this depends upon the its location. For beta greater than 53 degrees when the sloping inclination is greater than 53 degrees the stability number is only function of sloping inclination that is beta all critical slip surfaces are passed through the toe. So this is basically because the for such steep slopes the critical failure surface passes to the toe of the slope and does not go below the toe. Let us assume that say for vertical cut beta is equal to 90 degrees NS is equal to 0.26 from the short term condition. So it can be seen here for beta is equal to 90 degrees the NS value is about 0.26. So when we calculate back when by substituting this particular expression wherein the critical height is nothing but where the factor of safety is equal to 1 the critical height is defined as a height at which the factor of safety tending to 1 by doing this we can actually get Hc as 3.85 C by gamma this is actually obtained from the Taylor stability number whereas one performs the conventional analysis by using at pressure 3d it is can be obtained as C is equal to 4 C by gamma which is nothing but by considering the earth pressure equation like sigma a is equal to k a gamma H minus 2 C root k a and wherein if you consider both negative pressures that is the depth of the tension crack that is where the negative pressure is existing and equivalent portion below the this particular zone and when you take the equilibrium of that we can actually get the 4 C by gamma that is nothing but the critical height or the critical height of a vertical cut which is actually called and this is the height at which by attaining this height the soil supposed to fail. So undrained analysis stability charts given by Taylor method the position of the critical surface may be limited by two factors one is the depth of the start term in which the sliding can occur and the possible distance from the toe of the rupture surface of the toe of the slope. So the possible distance from the toe of the failure surface from the toe of the slope that is nothing but from the Taylor actually has given charts for determining this particular value n, nH in terms of nH it has been given like this is you know the distance in terms of n multiplied by H, H is nothing but the slope height and this is the failure surface which is obtained for slopes which are actually less than 53 degrees also. So the stability numbers for homogeneous slopes for phi is equal to 0 say for example by knowing d by H that is the depth factor and beta the nS and n can be obtained from this chart. So here n is nothing but by knowing this we can see that where it actually cuts and that is the n and then by taking this horizontal projection out onto the stability number we can get the stability number nS. So for d by H is equal to 1 that is d by H is equal to 1 beta greater than 53 degrees the n is equal to 0 that means that the slope actually the failure surface passes through the toe of the slope. So and some more important points as far as the Taylor's method is concerned they are actually summarized in this slide it is necessary to ignore the possibility of the tension cracks otherwise geometrically similar failure surface do not occur on slopes having different heights. So the possibility of the tension cracks is ignored here and Taylor's stability numbers were determined from the analysis of total stress conditions only that is basically for undrained or short-term conditions only and total Taylor's method is practically restricted to problems involving undrained saturated glaze or too much common cases where the pore pressures is everywhere 0 that means that the pore water pressures are everywhere 0. In case if you are actually getting a tension crack let us assume that the tension crack which is can be determined let us say from the by using the earth pressure fundamentals where sigma a is equal to k a gamma h minus k a gamma z minus 2 c root k a when for a clay soil when phi is equal to 0 we can say that c is equal to c u and k is equal to 1 and at point where the pressure tends to become 0 we can say that z naught is equal to 2 c by 2 c u by gamma that is nothing but the depth of the tension crack. Now when we consider the tension crack and let us assume that if the tension crack is filled with water and it can actually cause a destabilizing force and the portion within this tension crack zone that is the circular arc which is actually passing beyond this point is not considered in the analysis that means that in the factor of safety determined by this particular method is nothing but c u into L ac from here to here that is the length of this arc into the radius divided by w into d in plus of gamma w z naught square that is this force into the lever arm that is say L the L is nothing but from this distance to the vertical distance from this from the horizontal vertical distance L is nothing but the vertical distance measured from the center of rotation to the location of the horizontal force Pw. So like this when we have the tension crack this has need to be accounted. Now let us look into the ordinary method of slices in this method the potential failure surface is assumed to be circular arc with center o and radius r. The soil mass basically the soil mass abcd above a trail surface ac is divided by vertical planes into a series of slices as we have discussed that one of the limitations of the undrained slope stability analysis with phi is equal to 0 analysis is that determination of area and then you know determination of the weight and then d. Here what has been done is that the portion within the soil mass abc which is undergoing failure surface is divided into number of slices. The number of slices depends upon the by convenience it is actually divided and these slices are divided such a way that they have some uniform horizontal distance not necessarily uniform but mostly the uniform horizontal distance and the base of each slice assume to be straight line and so the circular arc is assumed to be as a straight line and factor of safety is defined as the ratio of the available shear strength to the shear strength tow m which is which must be mobilized to maintain the condition of limiting equilibrium. So here what actually we can say is that the ordinary method of slices satisfies both the moment equilibrium for circular surface but neglects the inter slice normal that is here the normal forces on these slices and tangential forces so this is a typical slice so the free body diagram of a slice in case of ordinary method of slices where there is a vertical force which is shown that is this is the W is the weight of that a particular slice if there are n number of slices and if this is the ith number of slice then the weight of ith slice is Wi and this is the resisting force that is s and this is the normal force that is n dash so it satisfies moment equilibrium condition and neglects inter slice normal and shear forces normal and shear forces shear forces in this direction one acting downwards when acting upwards and gives the most most conservative factor of safety so this gives the most conservative factor of safety basically useful for demonstration. So ordinary method of slices the moment equilibrium for slip surface but neglects both inter slice and normal shear forces the advantage of this method if you look into it is simple in solving the factor of safety since the equation doesn't require any iteration process. So this is described in the form of a figure which is actually shown here where this is a typical slope and this is ABC this is the failure surface which is assumed to be a one of the potential failure surfaces there can be innumerable number of failure surfaces and but ABCD in this particular case is assumed as a ABC is assumed as a failure surface and D is the point of the crest and O is the center of rotation and R is the radius of rotation and if you divide the slices of certain horizontal length let us say is B and if you consider the free body diagram of this particular this thing and it actually has the normal forces e1 and e2 both are different and x1 and x2 the shear forces when we consider when you don't consider these forces then e1 minus e2 is equal to 0 x1 minus x2 is equal to 0 and then here this e1 is equal to 0 and e2 is equal to 0 x1 is equal to 0 and x2 is equal to 0 and this is the you know from the when the vertical this is the weight is actually assumed to be weight of each slice is assumed to act at the center of this slice of width B horizontal distance and the arc is assumed to be as a straight line here and this is the tangential force and n dash is acting on the base of the slice and UL is actually acting a pore product pressure acting on the base of the slice so n this is n dash UL which actually gives this normal force and this angle which is subtended from the this n dash directly extends to the center of rotation that means you can see here and this angle is called as the alpha so as we traverse from this side to this side the angle alpha changes that can be noted from here. So the method of slices from the limited equilibrium condition point of view it can be given as tau m tau factor safety as tau f by tau m the factor safety is taken to be the same for each slice implying that there must be mutual support between the slides and force must act between the slices and the total weight of the slice W is equal to gamma bh so if you consider the slice of width B and the height is actually measured at the center that is this height which is actually regarded as the height of the slice this is the height of the slice at the center where weight is actually acting and then the weight is equal to gamma into B into H and if so let us assume that in determination of this we are having in a given slice there are three layers of soils then we have to take gamma 1 H 1 gamma 2 H 2 gamma 2 gamma 3 H 3 into B the V is nothing but the width of the slice the total normal force n is equal to sigma into L sigma is the normal stress acting over the length L into perpendicular to plane of that figure what we consider is 1 meter so it is the force is nothing but sigma into L into 1 so this includes n dash is equal to sigma dash L and U is equal to U L and U is the pore water pressure at the center of the base and L is the length of the base and the shear force on the base is nothing but T is equal to tau m into L the tau m into L the L is the length along the the failure surface in a given slice of having width B the total normal forces on slice even in E 2 and the shear forces on the slice x1 and x2 so the method of slices considering the moment about O the sum of the moments of the shear force T on the failure arc AC must be equivalent to the moment of the weight of the soil mass ABCD so this we can write as sigma T into R is equal to sigma WR sin alpha so by writing by using T is equal to tau m into L is equal to tau m is equal to tau f by factor of safety in L and we can write sigma TR as sigma tau f by factor of safety in L is equal to sigma W sin alpha so factor safety is given by by rearranging these terms we can write factor safety is equal to tau f into L divided by which are sigma that is nothing but the resisting resisting portion divided by the the driving that is disturbing one W sin alpha so factor safety is nothing but sigma tau f into L divided by sigma W sin alpha this is from the limited equilibrium condition from for an analysis in terms of effective stress we can write factor safety is equal to sigma C plus sigma dash tan phi into L so by writing sigma dash into L as N dash we can write C dash into LA plus tan phi dash into sigma N dash divided by W sin alpha so equation one is exact but approximations are introduced in determining the forces N dash so in factor safety is equal to C dash into LA plus tan phi dash in case if you are having a cohesion less soil slope then we get to factor safety is nothing but when C dash is equal to 0 the first term will get cancelled and where we have tan phi dash into sigma dash N dash divided by W sin alpha so this can be used for both undrained the short term stability as well as for the long term stability by substituting the relevant characteristic strength parameters. The felilius or Swedish solution is actually given it is assumed that for each slice the resultant of the interslice forces is 0 and the solution involves resolving the forces on each slice normal to the base where here we give N dash is equal to W cos alpha minus U alpha and this is actually given by rewriting the equation one which is shown in the previous slide as this particular one C dash LA plus tan phi dash into sigma dash N dash where N which is actually rewritten this equation in terms of so this is nothing but C dash LA plus tan phi dash into sigma W cos alpha minus UL divided by W sin alpha sigma. So this expression is actually or this with this modification this is called as the felilius or Swedish method of slices both are ordinary method of slices and felilius method of sizes one and the same but this is the minor difference which is actually there C dash into LA plus tan phi dash into sigma of W cos alpha minus UL divided by W sin alpha which is summation. So the how we can actually do the analysis by using the felilius method of slices means in a given portion we need to assume the potential failure surface and then divide the slices of the horizontal having horizontal width and then number the slices like the numbering is done from one order 1 2 3 4 5 6 and with a given center of rotation and for a given potential surface this is the entry point and this is the exit point and where in from the each center of the slices is actually identified and the angles alpha 1 alpha 2 alpha 3 alpha 4 alpha 5 alpha 7 are determined. So for an analysis in terms of total stress the parameters C U and phi U can be used and the expression will be like this and for a phi U is equal to 0 analysis which is nothing but factor safety nothing but C U LA divided by W sin alpha. So this particular Swedish method of slices was extended by Bishop in this solution basically assumed that the resultant forces on the slides of slices are horizontal where x1 minus x2 is equal to 0 is considered. So the resultant forces on the sides of the slices are horizontal and assumed that these x1 minus x2 is equal to 0 by with this assumption the equilibrium of the shear force on the base of any slice is given as T is equal to 1 by factor safety into C dash into L plus n dash tan phi dash. So resolving the forces in vertical direction we can get W is equal to n dash cos alpha plus UL cos alpha plus C dash into L sin alpha by factor safety plus n dash tan phi dash sin alpha by factor safety after some rearrangement and using L is equal to b secant alpha because that inclination is alpha and horizontal distance is b L is the length along the straight line length along the slice. So by simplifying we get the factor safety expression as 1 by sigma of W sin alpha in a summation C dash b plus W minus UB within brackets tan phi dash into secant alpha divided by 1 plus tan alpha tan phi dash by factor safety. So this expression wherein you can notice that the factor of safety is existent in the both the methods. So the generally how it is done is that this is done by iteration methods and the factor safety which is obtained by performing some couple of iterations will yield to a factor safety. So for this here what it is done is that the factor safety is determined first primarily by Swedish method of slices and that is used as the initial value and then the number of iterations are specified or performed based on the logic which is actually set in the software which is actually used for determining this or manually with a couple of iterations it can be determined by using this expression. In Bishop 1955 it is simplified method of slices he showed that how non-zero values of the resultant x1 minus x2 could be introduced into analysis but refinement is only a marginal effect on the factor safety. So the pore water pressure also can be related to total field pressure where it is given as the pore water pressure coefficient which is nothing but U by gamma H, U is nothing but the pore water pressure at any point divided by gamma H, the H is the let us say the height of the slice. So it is pore water pressure is defined in terms of a total field pressure at any point by means of dimensionless pore water pressure ratio which is called as RU. When RU is equal to 0.5 it is said that the slope is completely saturated and for any slice RU is equal to U by W by B by rewriting this one in the previous equation what we have given we can write or express factor safety as 1 by sigma W sin alpha into sigma C dash B plus W into 1 minus RU tan phi dash into secant alpha 1 plus tan alpha into tan phi by factor safety. So in the Bishop simplified method as is actually shown here it satisfies the equilibrium of the factor of safety and satisfy the vertical force equilibrium for n and moment equilibrium for factor of safety. By determining factor of safety the moment equilibrium is satisfied and vertical force equilibrium is satisfied with respect to n and consider the inter slice forces even in E2 and more common in practice applied mostly for circular failure surfaces. The salient features are actually given here and this is the typical free body diagram of a slice in case of Bishop simplified method where the weight of the slice and the shear strength which is actually mobilized along the value surface and dash is the normal force acting on the failure surface normal to the failure surface and even in E2 are the inter slice forces on a particular slice force. So the Bishop simplified method consider the inter slice normal forces but neglects the inter slice forces it further satisfies the vertical force equilibrium to determine the effective base normal force n dash and further we also said that the John Booce simplified method. In John Booce simplified method is basically based on a complete slip surface and the factor of safety is determined by the horizontal force equilibrium. Basically here both horizontal and vertical force equilibrium are satisfied and it does not satisfy the moment equilibrium. John Booce simplified method does not satisfy the moment equilibrium and considers the inter slice forces even in E2 like Bishop only and it is commonly used for composite shear surfaces. So basically for layered soils or stratified soils or when you are having a non-homogeneous soils the John Booce method is advocated for its use and basically it is used for composite slip surfaces or non-circular slip surfaces and the factor of safety is determined by horizontal force equilibrium only and moment equilibrium is not satisfied and in this slide the free body diagram of a slice which is actually considered in the John Booce simplified method is shown wherein which is actually very similar to simplified method wherein really the difference is that it does not satisfy moment equilibrium only the force equilibrium that is vertical and horizontal are satisfied. So in this lecture we try to introduce ourselves to different failure surfaces and we will also look into different types of methods which are actually used for analysis. We have seen the phi is equal to u0 method and Taylor's stability chart method which is the deduction of from the extension of unrained slope analysis method and from there we also discussed about the ordinary method of slices and felonious method of slices and then Bishop's simplified method which is mostly used for commonly used for circular failure surfaces and for composite slip surfaces or non-circular slip surfaces in stratified soils the John Booce simplified method is also introduced. So in the next lecture what we do is that we will try to look into some examples wherein we can actually see how the problems can be solved by using typical calculations with the manual calculations as well as the in this particularly we will actually try to see some demonstrate some problems by using some relevant packages for academic purposes.


Hello everyone, I am Prasanthesmalgay, Assistant Professor, Department of Electronics Engineering Vulture Institute of Technology, Solapur . Today, we will discuss some of the techniques for testing the logic circuits, learning outcome. At the end of this session, student will be able to derive the test set for the given circuit for detecting faults, contents, introduction, fault model, stuck cat model, single and multiple faults and complexity of a test set. Once the circuit is designed, we need to verify that whether the circuit performs as required or not, how one will verify this. So, today we will discuss some of the techniques for verifying the functionality of a circuit. The basis of all the testing methods is to apply a set of inputs known as a test to a circuit and to compare the output results with the responses that the correct circuit is supposed to produce. The challenge here is to derive a small number of test set to ensure that the circuit is functioning correctly. Now, we will see a fault model. The circuit behaves incorrectly if something goes wrong with it, may be a transistor fault or may be an interconnection fault . See for example, a transistor switch can break so that it is permanently either closed or open. A wire in the circuit can be shorted to either VDD or to ground or it can be simply broken. Unwanted connection between two wires or maybe a logic gate generates a wrong output signal because of some fault in the circuitry which implements that particular gate. Only to deal with all the varieties of these faults is a difficult, but it is possible to restrict these faults to some simple faults and obtain satisfactory results. Now, stuck at model, most of the circuits which we deal with consist of the logic gates as their basic building blocks . So, a good model for representing the fault in such circuits is to assume that all the faults appear as if a wire may be input or output of these gates being permanently stuck at a logic 0 or logic 1. Say for example, if a wire W has an undesirable signal that always corresponds to the logic value 0, we say that the wire W is stuck at 0 which is denoted as W slash 0. On same lines, if W has an undesirable signal that is always equal to logic 1, then we say that W is stuck at 1 which is denoted as W stuck at 1. So, the stuck at model is also useful for dealing with the faults of other types which often cause same problems as if a wire is stuck at 0 or stuck at 1. So, we restrict our discussion to stuck at faults and discuss the testing process assuming that these are the only faults available in a circuit . Now, single versus multiple faults. Now, the circuit may be faulty because of either a single fault or possibly be possibly many faults, but dealing with multiple faults is difficult because each fault can occur in many different ways. So, a practical way is to consider single faults only. Practice has shown that the test set that detects a single faults in a circuit also detects majority of the multiple faults in a circuit. So, actually a fault is detected if the output value produced by the faulty circuit is different from the value produced by the good circuit when an appropriate test is applied. Each test is supposed to be able to detect the occurrence of one or more faults. So, a complete set of tests used for given circuit is referred to as a test set. So, our aim is to derive a minimal test set required for testing a circuit. So, now, we will see the complexity of a test set. So, one possible method for testing is to apply a test set that comprises all the possible input valuations and check if the output values produced by the circuit are same as a specified by the truth table. But this particular method is effective if the circuits are small because where the test set is not too large, but it becomes impractical for a larger circuits with many input variables. Fortunately, it is not necessary to apply all two raise to n valuations as test for a n input circuit. A complete test set capable of detecting all single faults usually consists of smaller number of tests. Now, consider the circuit shown in figure. Here we have three inputs W1, W2, W3. There are four wires A, B, C, D and fifth is an output. Now, think about which faults will be detected by applying inputs W1, W2, W3 equal to 000. Pause the video for a minute and write down your answer. Okay. Now, I will see what happens when the inputs are 000. When W1, W2, W3 are 000 the correct output is 0, but a faulty circuit will give you an output 1 and this fault may be due to either this wire A stuck at 1 or D stuck at 1 or F stuck at 1. So, in case when we are applying 000 and the output generated is 1, in that case the circuit may be having either A stuck at 1, D stuck at 1 and F stuck at 1. So, these three faults will be detected by a test vector 000. Similarly, we can obtain the faults detected by each of the different test vectors. Say for example, 001 in case of 001 once again the correct output is 0, but faulty circuit will produce 1. This may be due to once again A stuck at 1, F stuck at 1, D stuck at 1 or this B stuck at 1. So, additionally here this B stuck at 1 is also detected. The same way we can do it for others and we get with 010 as the faults detected. Similar procedure we can follow and we can see that different vectors which detects the different faults. Say for example, in case of 111 the last one correct output is 1, but it may be 0 and this particular 0 output may be possible only with F stuck at 0. So, this is how for all the different test vectors we have detected the faults. Now to derive a minimal test set what we can do is we will select the input test vectors so that all the faults will be detected. Now for example, B stuck at 0 is detected by only 011. So, this must be taken as a test vector in a test set. Similarly, B stuck at 1 with only 001, C stuck at 0 with only 011 and D stuck at sorry. So, this 001, 010 and 011 will be taken and this 100 will detect A stuck at 0 as well as F stuck at 0. So, therefore, a minimal test set that covers all faults in the circuit can be derived from the table as 001, 010, 011 and 100. So, in this case as we have seen instead of 8 test vectors only these 4 test vectors will detect all the possible faults in a circuit. So, this is how a test set can be generated. References. Thank you.


Five seconds. Never Refuse Gold Given. Dire's bottom tower is under attack. Given. Dire's bottom tower is under attack. Five seconds, Rima- Use gold given. Dire's bottom tower is under attack. Seconds remain. Never Refuse Gold Given. Dire's bottom tower is under attack.


What's up everybody. This is Chris from the rewired soul where we talk about the problem but focus on the solution And welcome back to another video where we're gonna be breaking down an episode of black mirror season 4 in this video We're gonna be talking about episode 2 arc angel and the overall theme of this being Overly controlling your children as well as the false idea of control So if you haven't watched this episode yet go ahead turn this video off go watch it It's amazing come back and let's talk about it. All right, but anyways quick recap of episode 2 arc angel What it's about is a mother named Marie father's out of the picture. We don't really know why irrelevant But anyways, you can see in the beginning of the episode how anxious she is as a parent She has a c-section and my son was born as a c-section I was losing my mind too so I can relate and you can see her freaking out like it's my baby Okay, it's my baby. Okay is my baby. Okay, and her daughter was okay fast forward a few years where her daughter is at the park with Marie her mom and The daughter ends up chasing a cat and she gets lost and Marie loses her mind as any good parent would So as a result of this because of her fear and if you haven't checked out my other videos on this channel Check right here. I have a video about how fear destroys mental health makes us make very bad decisions Anyways, her fear of something happened to her daughter made her go out to this Experimental program called arc angel and what that is is it put an implant into Sarah's head and then the mom can monitor Her daughter from this tablet. She can see when her daughter is stressed. She can see what her daughter sees She can see through her eyes. It also has a GPS locator in it. So if her daughter ever gets lost again, so In order to have more control over her daughter and not experience this anxiety She did this experimental program of shooting something in her to her daughter's brain Now one of the other aspects of this is that the the arc angel what it can do is it can filter out anything Negative anything the mom deems negative she can filter it out so the daughter can't see it can't hear it So the the mom starts doing this more and more and more she starts filtering out just about anything her daughter can see like Let's say for example like scary movies violent movies there's a dog that barks at Sarah every day she walks to school and her mom filtered that dog out and Sarah ends up meeting some kids at school who watch a bunch of crazy stuff online, which kids do I hope my son doesn't do it. Anyways, and The daughter can't even see it and she's known as the weirdo at school But what what ends up happening at this young age because her daughter is weird and different She ends up having this violent outburst and she starts stabbing herself because she can't even see blood She can't even see when she's coloring blood and making drawings like that's how well this filter works So Marie ends up taking her daughter to a psychologist and she starts Unfiltering everything for her daughter and she gets rid of this whole archangel thing Fast-forward to Marie being a 15 year old teenager Now's the time when Marie starts going out hanging out with boys and one night She doesn't come home when she said she would she lied to her mom about where she was going and Mom bust out the old archangel after maybe you know five six seven years She busts out archangel and she sees everything and I mean everything like when she spied on her daughter Her daughter was in the middle of losing her virginity Kind of awkward right but then she finds that her daughter was trying cocaine for the first time and She goes out and she threatens her daughter's boyfriend.  She finds out that her mom actually put in a Like a plan B pill into a shake that she made her daughter. So Sarah was actually pregnant. So as you can see Marie is very very very controlling of her daughter out of this fear out of this pure fear and this fear Makes her try to be so controlling and see every little thing her child does and this This whole thing it touches on the social commentary of This idea that we need to shield our children from anything negative in life We need to protect them from everything and what we got to think about is Is this what is this what's best for our child? Or is this what we're doing to try to subdue our own fears and our own anxiety for example I've been meaning to make another video on this But the idea that we should take technology away from children is ludicrous It's ludicrous if you think that in the next 10 20 years when your child is You know an adult and looking for a job that technology is gonna be less than what it is now You're out of your mind. You're completely out of your mind So what we need to do is rather than trying to shield them from everything We need to just have open minds of communications when I was sitting back watching this episode arc angel I was sitting there and just wondering like what if the mom just had conversations with her daughter What if she just talked to her daughter?  What if Marie actually had a conversation with her daughter about drugs, right? Because what happens is Overly parenting trying to control this illusion of control. We have it actually Turns kids into worse people than they would have been if we would have just kept the open lines of communication I grew up here in Las Vegas around a ton of Mormons right and Mormons are often shielded in the same way don't have sex until you're married Some of them can't even have caffeine and what I saw as a result of this around the age of high school, right? Where Sarah was in black mirror they started experimenting with everything they started having sex They started drinking they started doing drugs all these things because the parents are trying to control every behavior They have so I love how black mirror like brought this to the forefront like this is this was an amazing episode And hopefully a lot of parents are watching it. I know this series is kind of targeted towards like millennials and stuff So people around my age like my son is actually, you know, nine years old right now. So People around my age younger, you know, when we have kids, we got to start thinking about what are we doing? Are we trying to overly protect our children? And is it for their best interest or is it for ours? Are we blocking things from them because we think it's good for them or do we think it's good for us?   Like why are we gonna say that these people can't be in love because you don't know how to talk to your child That's the same way we got to think about this Like we can't just keep thinking that we can maneuver our children through every single bad situation Just because we don't know how to have a conversation with them Like I hate to break it to you But it's part of what being a parent is is having these discussions Some of you who follow my channel, you know that I am an addict in recovery One of the biggest fears I have is that my son is gonna become a drug addict or an alcoholic But I know that it is completely insane of me that illusion of control If I think that my son is never going to try a drink or a drug Studies have proven that the best way to talk to kids about this is to make sure that they know that there is an open line of Communications the more that your child is able to come to you and ask you about sex about drugs about bullying about their mental health studies have shown that you have a better chance of helping them out and then They can make the proper decision, but when we're shielding them from these things That's when they want to go experiment and try things on their own and most of the time they're not doing it in a safe way and Basically at the end of archangel what happens is Sarah finds out mommy's been spying on her Sarah finds the little tablet and she basically beats her mom unconscious with it and then runs away So that's what happened when Marie tried to control Sarah too much But anyways, I don't I would love to know if you've checked out Episode two of black mirror season four So please leave your thoughts in your comments down below But if you like this video give it a thumbs up and if you're new here be sure to click that little subscribe button I'm always making videos to help you out with your mental health If you want to see some more videos I've done about season four of black mirror click or tap on the thumbnail up there Or if you just want some other videos to help you out with your mental health Click or tap on the video down there. All right. Thanks so much for watching. I'll see you next time


How's it going everybody, Dato Doi here with another Dragon Ball Fighters video, and today I thought we'd take a break from the ranked matches to talk a little bit more about the future of Dragon Ball Fighters, and more specifically Season 2 when I think we're going to get it and what I think they are looking to include with it. Before I go any farther though I should mention that Season 2 has not been officially teased or announced or anything like that, I just think it's something that's extremely likely because of how well Dragon Ball Fighters has done and the fact that the last thing we got was seasonal outfits for Halloween, I have never once in my life seen a game put out seasonal outfits for Halloween and then never do them again. But with all of that out of the way, let's go ahead and discuss the when when it comes to Dragon Ball Fighters' second season. Now when it comes to this section of the video I want you to keep in mind three important dates, the first one being the 17th of November which is coming up here pretty shortly, now this is usually when the month's edition of V Jump starts leaking but this date can fluctuate from like the 17th to the 20th depending on how fast the leaks come out. I don't think this date is too likely to receive any big news for fighters though as ever since the release of Android 17 and Cooler's last bit of information, fighters news on here has been pretty dry as there simply hasn't been anything ready to be announced. Regardless though I did want to mention it in this video just in case something does come up, you never really know. The second and third date I want you to keep in mind are more likely candidates for me for big announcements regarding the game, the second date being Jump Fest of which occurs on Friday December 21st all the way through Sunday December 23rd. Those of you that have been following Dragon Ball Fighters for a long time you'll remember that the last year's Jump Festa is where we got the announcement for Hit, Goku Black and Beerus so that was a big event for fighters, we got three very fan favorite characters from Super so obviously this is a big event for anime and last year it was a big event for fighters too so I wouldn't be too shocked if we see anything here. If Jump Festa is a bust for news as well though the last date being January 26th 2019 that is the finals of the Saga World Tour event for Dragon Ball Fighters that has been going on now for a while. For those of you out of the know this is Dragon Ball Fighters World Tour where the best of the best go to tournaments all around the world trying to gain Dragon Balls to prove who the best of the game really is. The finals of course being the climax of this tournament series it's bound to have some sort of announcement something hype going on if no news is here I'm sorry guys but even I'll admit that Dragon Ball Fighters might be done or they might be moving on to Dragon Ball Fighters too or something like that uh this just seems like the final date for something to be announced like I said though I think it's pretty likely that something will be announced either here or along the way so uh don't even get too worried about nothing being announced there by the way just a little bit of extra trivia the finals take place on January 26th 2019 and Dragon Ball Fighters originally released in January 26th 2018 not really necessary for any reveals but it would be kind of poetic to see Dragon Ball Fighters get a rebirth on its one-year anniversary of release and that all pretty much covers the when when it comes to Dragon Ball Fighters second season but I also did want to cover some things that I personally think we're going to see along with a second season of support for this game and again just a reminder none of this has been officially confirmed or even teased for that matter these are just some things that I think are likely to be expanded upon in the future for this game so for starters I think alternative costumes might be an area they look to expand in the only reason I really bring this up at all is because there is an old datamined icon that does appear to be for switching costumes for characters or something along those lines doesn't really inherently mean all too much because this is something they could have put in the game at a very early stage just in case and then decided to scrap all together or for literally any other reason it's hard to speculate on this I just think it's something that might happen so I thought I'd bring it up the other thing I think we'll see along with the new support is new maps we did get one new map for free at the end of season one that being galactic arena this doesn't inherently mean they're going to add more maps for free and I say this because it is a more esports type of arena it shows the fighters their team in the background all that so they might have just wanted to throw that in for the world tour but I do think it does serve as even a slight indicator that they might want to be adding more maps in the future to accompany dlc characters and speaking of characters that brings us to the last thing that we can obviously expect to see that being new characters to expand the roster even further now I know that's pretty obvious but I just wanted to bring this up because there has been a what I want to call a renaissance of dragon ball recently along with the dragon ball super broly film where it just feels like almost everything surrounding dragon ball has been put to a halt uh or just slowed down dramatically to build up hype for this film uh you know if you play any dragon ball mobile games you know those are kind of slowing down right now new content isn't that fast zenoverse has been leaked some dlc uh so even that game as old as it is uh is building up hype for some new dlc characters specific to the dragon ball film that I don't want to get into because that's kind of spoiler territory and dragon ball fighters has also been on hold with in terms of news so I think all of this just culminates to the fact that you know dragon ball super broly I think they just want to make this a splash and this is all speculation on my part but if you've been writing the broly hype train for all along you've probably been noticing these things uh and it just feels like we're definitely going to get some characters and speaking on the last stage far I would say even stages from from the new broly movie and characters from that movie and that's really as deep as I want to go into that to avoid but actually you know what okay spoilers in five seconds so please click off the video uh seriously just leave right now if you don't want any spoilers thank you so much for watching that's really all you need to know anyway okay going into spoiler talk so for those of you that have been writing the hype train for dragon ball super broly you know that gojita is more than likely going to make an appearance it's been confirmed by a lot of very credible sources I think it has been confirmed to actually be gojita blue uh he's also coming to zenoverse and I just think it's very very likely that if that is true which it looks like it 100% is uh that we would definitely see that come to fighters I don't know fighters does feel like an outlier in terms of dragon ball games but the hype just feels so surreal man that I can't see him not come and I know it was kind of a meme like are we gonna get new broly too but I don't know man now it kind of feels like we are speculation over I think that is going to happen though okay now that's the for real end of the video thank you guys so much for watching I know the last part of this was a lot of speculation but but those are the important dates to look out for and the things that we can realistically expect to see maybe even some balanced patches thrown in there but you know that kind of depends on whether or not they want to continue supporting this game competitively as well which they think they will depending on how successful the world tour is other than that let me know down in the comments what you think about season two when you think we're going to see it and what you think is going to come along with it and just everything in between man I'm looking forward to discussing this with you guys down there as I'm just a super dragon ball fighters nerd and dragon ball nerd in general so I really want to talk about this other than that while you're down there if you like the video make sure to leave a like and subscribe to the channel so you never miss another video from me and I also have some other videos on the screen right now if any of those interests you give those a watch as well other than that I'm Dotto doya I'll see you in the next video


it got a little bit better from where it was in April and May, you know, in June and July. And then, you know, I feel like now, you know, getting guys back and healthy, you know, you know, the balance we have in the order, you know, some of the things we're doing on the base is playing much cleaner. But the quality of the advance from the whole group, you know, is making us a lot tougher. And, you know, I feel good about where we are offensively right now. Thanks. Can you go next? Go in to the other side.  Is that sort of what you envision for the sort of ideal way for your team to win these games, this draft structure? Yeah, I'm sorry. Yeah, sorry. No, just having, you know, having your AISCO six score list, having a lot of productions out of your lineup. Is this was like the ideal type of game you win that you had to be for? Yeah, I mean, look, I mean, we've played obviously a ton of close games and it's been really good, obviously, that we've won a lot of those close games and played well in those close games.   But the pull away there at the end, it was good to see us continue to add on and continue to, you know, get, get some separation there at the end of the games. That's some of something we haven't done a lot of, you know, when we've had a chance to put a team away. Tonight, we're able to do that. Being the next, can David off? Aaron, another big night for judge. He's just been so consistent this year, which is really he struggled out in the past as many big guys do. What have you seen this year that a lot of them just be so consistently productive? Yeah. You know, I think one thing is he's been overall pretty, pretty healthy. And I think that is exactly how I would describe his year. He has been very consistent. I feel like this year and obviously he's been pretty hot lately, but he's had, you know, less really, you know, stretches where he carries us and less of those couple of weeks where he really struggled. He's been really consistent all year. That's one thing that's kind of stood out to me. And now, you know, I think he's probably playing at his best right now. And I think he's, you know, physically in a good place. I think he's, I think he's really focused and, and from a game-planning standpoint, I think has a really good idea each and every day of, you know, what he's looking for in the box. I feel like he's executing his plan in the box and he's, he's playing really well. He's playing really well. Thanks, Aaron. We can go next to Brendan, Cuddy. Aaron, similar question to Ken's, but in terms of John Carlo, he's just been so hot lately and just what have you seen from him at the plate and just in general? Just, you know, really controlling the strike zone on time, balance, and then just making really good swing decisions. And when he does that, you know, you see the ball exit to bat at a really high, high rate of speed. And it's been really good to see. I think he's just in a really, like, judgy. I think he's in a really good place physically, really good frame of mind. And just, I think, really executing his game plan at a high level right now. Is this something similar to some of the zones you've seen him snap into the playoffs last year, certain times through his tenure here? Or do you think it more has to do with maybe what's going on around him, the lineup, just some of the new faces around? I don't know. I mean, every year is a little bit different.  And so anytime he goes through a stretch where he's swinging the way, it's not surprising. You know, he just, he's kind of different than anyone. You know, it looks different than anyone.  And I just, I think it has to do with being in a good spot physically and with what we're playing for as a group and just the overall team focus. I feel like a lot of guys are in a good mental state. Any other questions for Aaron? Thank you.