 We will discuss in this class design methods by means of federal highway administration whatever they have given step by step. Now first one is your external seismic stability the mass of retaining soil that is reinforced by horizontal layer it should be considered as a monolithic block that means it is as if a one unit soil plus reinforcing material as one unit the sliding weights failure mechanism is considered for the external stability analysis if you look at the figure this is your mass of soil mass which has been reinforced by means of geosynthetic layers it has been considered a one layer and the CG in the CG center of gravity unit weight means weight of your soil as well as weight of your of course geosynthetic materials are counted to be unitless so weight will be acted in vertically that is your a b f e that means a b f e if I consider this mass of the soil will be reinforced by means of geosynthetics then there are there are resultant force will how it will act at a distance this resultant force will act at a distance of eccentricity e from the CG and this sliding mass will be considered by means of external stability if you look at here this is your horizontal layer of the force acted this is your summation of v tan delta b delta b is your coefficient of friction of soil mass at the base with your soil at the top and earth pressure distribution diagram considered taking into consideration of surcharge if this is my surcharge say q is your surcharge so this diagram is your earth pressure because of your surcharge and this diagram is your earth pressure because of your soil mass the composite mass must be stable against sliding failure along the base of the structure as I said along the base this is your base of the structure it will be stable against your sliding failure base of the structure at the foundation and back field interface interface at the foundation and back field interface this is your back field this is your foundation so this is your foundation and back field interface eccentricity failure of resultant force this is my resultant force r and this is your eccentricity resultant force that means because of eccentricity it will be acted as you have discussed earlier overturning failure about the toe and the bearing failure of the supporting foundation soil as well that means bearing capacity of the entire soil mass because of your supporting foundation this is the supporting foundation bearing capacity calculations assume that the base of the reinforced zone act as an eccentrically loaded footing with an equivalent footing width with an equivalent footing width that means the assumption is an is a as if it is a eccentrically loaded footing this is a eccentrically loaded footing with an equivalent footing width that means this is your equivalent footing width so based on that if you are getting a b then you can find it out your bearing capacity calculations and this calculation has been made as per the mayor of sub approach whatever given mayor of sub approach so it is taken is f h w a 2001 the measure of relative stability against the external modes of failure is defined by the ratio of resisting force to the resisting forces that that measure of relative stability against the external mode of failure this is we are talking about external state seismic stability external modes of failure is defined by the ratio of resisting forces to the restraining forces if you look at here ratio of resisting forces what are your resisting forces this is your resisting forces and restraining forces that means lateral forces is your restraining forces as in design of conventional gravity wall structure then a free body diagram of wall showing different forces coming on to on to it from soil and due to seismic loading along with their respective points of application you have to draw a free body diagram if you see here this free body diagram has been drawn and you will have to show different forces coming or acted on the soil mass of or the reinforced earth wall so this is your weight that means acted at the c g this is your force p ir it is lateral force it will be acted in the lateral direction also seismic force all the forces at the respective point of application if I draw it here make it into suppose this is your earth pressure because of a short charge where is your resultant earth pressure will act in this case the resultant earth pressure will be act at a distance of 0.5 page from the base of the wall and this is your pressure distribution diagram similarly earth pressure because of your soil mass if you look at here this is your earth pressure the resultant pressure p a it will act at a distance of h by 3 similarly your seismic force it will act at a distance of 0.6 h from the base this is all about a bigger picture if you look at the earlier I have what I have discussed it is here in a bigger picture a soil mass has been reinforced with your geosynthetics and what are the forces what are the forces external load going to be acted assuming bearing means your surcharge is going to act then soil earth pressure then is your because of your seismic load it will act then you find it out your resultant where it will act whether it will act at the c g or at a distance of eccentricity e then find it out your l minus 2 e that means you will have to check it e should be less than equal to b by 6 then mass of resisting forces these are all mass of resisting forces then mass of inertia forces if you look at here this is mass of inertia forces then you can find it out sigma b p i r forces and earth pressure considered for external seismic stability analysis of geosynthetic reinforced earth wall g r s walls has been shown here the active coefficient of earth pressure is calculated for vertical walls defined as wall with a face battered less than 8 degree the moment i say it is a vertical wall it will be defined as face battered of less than 8 degree if this is the face of the wall it should be battered less than less than 8 degree that means it should be inclined both way less than 8 degree so that has been defined as a vertical wall so and horizontal back slope horizontal back slope means if this is my wall this is my wall if i say horizontal back slope means this is the horizontal back slope that means the soil retained on this behind the retaining wall this will be a horizontal back slope if i say inclined back slope then it will be in this form that means soil retained back face of your wall it will be inclined so this is called inclined back slope now k a is your active earth pressure so coefficient that is your tan square 45 degree earth pressure coefficient k a a means active that is your tan square 45 degree minus 5 by 2 for vertical wall with searcher slope that means if a vertical wall with searcher slope this is my vertical wall with a searcher slope this searcher slope is your beta that means with this searcher slope what is your value of k a this is your k a value earth pressure active earth pressure coefficient k a if it is a if it is horizontal back slope that means if this kind of horizontal back slope k a will be tan square 45 degree minus 5 by 2 if it is a inclined searcher or searcher slope or inclined back slope of beta that means in this case k a is equal to cos beta into cos beta minus cos square beta minus cos square phi root over divided by cos beta plus cos square beta minus cos square phi root over beta is your searcher slope slope angle your beta is your searcher slope angle that means this will be your beta searcher slope angle now it has been shown in this case what is your beta what is your theta and what is your delta delta is your angle between your soil and the wall that means your that means your frictional angle between your soil and wall and beta is your searcher angle theta is your with respect to how much it will be inclined that means k a is equal to if I take in terms of if wall is inclined that means in this case wall has been inclined at an angle theta this wall has been inclined at an angle theta also soil has been inclined at an angle beta also there is a coefficient of friction or between the soil as well as wall that is your delta then you can find it out k a in this term sin square theta plus phi prime means phi effective divided by sin square theta and sin theta minus delta into 1 plus sin phi prime plus delta into sin phi prime minus beta divided by sin theta minus delta into sin theta plus beta root over whole into square gamma prime is your effective unit weight that is your and phi prime is your effective angle of internal friction delta is your angle of wall friction that means angle between wall and the soil this is called angle of wall friction all angles are positive as it is shown that means if it is inclined like this that means it will be your positive if it is inclined like this this is also your positive and if it is inclined the direction has been shown in this direction this is termed to be as a positive angle now a simple reinforced soil mass if you take it we if you consider it this is your reinforced wall and this reinforced soil mass and v is equal to how much force it will be at gamma r into h into l h is equal to height l is equal to up to this and this is your earth pressure distribution diagram this is because of your surcharge this is earth pressure because of your soil and how much is your resultant force it will act at what distance this will h by 3 this will be your h by 2 so f 1 for this case it will be f 1 is equal to in this case f 1 is equal to half gamma h square into k f 2 is equal to q h k sigma prime f 2 is your this case so sliding stability that means check the preliminary sizing of sizing with respect to sliding at the base of the layer which is most critical depth as follows factor of safety against sliding we generally write f s and below subscript is your sliding it is ratio of summation of horizontal resisting forces that means p r horizontal resisting forces by horizontal driving forces horizontal resisting forces by horizontal driving forces and it should be greater than equal to 1.5 if your sliding if your factor safety is less than 1.5 then it is said to be unstable that means this factor of safety minimum factor of safety for stability of sliding it should be greater than 1.5 then come back to during an earthquake loading this geosynthetic reinforced wall is subjected to a dynamic thrust at the back of your reinforced zone and to inertia forces within the reinforced in addition to static forces that means there will be a in addition to static force there will be another force that is inertia force that has been considered for seismic the external seismic stability of wall can be analyzed by the following procedure as given by federal highway in 2001 the peak horizontal oscillation coefficient at the center of the reinforced zone that means peak horizontal oscillation coefficient it has been termed as a small k subscript of h is calculated from the given value of peak horizontal ground oscillation coefficient a maximum k h has been calculated from your a maximum as shown below so k h is equal to 1.45 minus a maximum into a maximum the additional dynamic earth pressure can be calculated from the following equation this is your calculation of k h the additional dynamic earth pressure you can calculate that means your delta p delta p into a e earthquake force a is your earthquake force delta p additional earth pressure because of your earthquake force this is your 0.5 gamma h square into k a e minus k a where gamma is equal to unit weight of reinforced backfill h is equal to height of gear s wall k a is the static earth pressure coefficient as you have discussed earlier k a e is the seismic active earth pressure coefficient this is your additional term k a e is your seismic active earth pressure coefficient so you can calculate k a e as given by your federal highway of united states of america in 2001. That is your cos square phi minus theta w minus psi divided by the whole term where psi is equal to tan inverse of k h theta w is equal to angle of slope with vertical angle of slope with a vertical it generally if it is purely vertical that means it is theta is equal to 0 degree and delta is equal to interface friction angle between reinforced and retained backfill. That means 0 for horizontal backfill for static case the location of resultant of soil pressure acting on the reinforced block as shown is it is generally taken as h by 3 if you come back to for static case it is generally taken as for static case this is your earth pressure distribution diagram it is generally taken as h by 3 total height by 3 is your at that height is your resultant active earth pressure will act to consider your to consider your earth quake effect 50 percent of additional dynamic earth pressure 50 percent of your additional dynamic earth pressure that means 0.5 delta p e component is assumed to act as 0.6 h from the bottom of the wall that means it is defined that to consider the earth quake effect 50 percent of additional dynamic earth pressure that means 0.5 delta p a e component is assumed to act at 0.6 h at a height of 0.6 h from the bottom of wall if I take it back to slightly before this what we have discussed you see this is the earth pressure distribution diagram for your considering your dynamic earth quake effect it is your 50 percent of your delta p a e and it will act at a distance from the base this is your 0.6 h h is your total height of the wall the effective inertia force that means p i r i r is your effective inertia force p is your force is a horizontal load acting at the center of gravity p i r is a horizontal load it will act at the c g or center of the gravity of effective mass of w a g i e and can be written as you can find it out p i r is equal to k h w capital w a g i e that means your k h into gamma h into 0.5 h external stability computation are made considering that the horizontal inertial force p i r act simultaneously remember here it is written act simultaneously with 50 percent of dynamic horizontal thrust that means is your 0.5 delta p a e in addition to your static force it will act simultaneously first one and second one is your with 50 percent of dynamic horizontal thrust that means is your 0.5 delta p a e in addition in addition to your static force whatever the static force is there in addition to that it will act now if you consider these forces and earth pressure this same this part is your for static this is your earth pressure distribution diagram as I said for your surcharge that means load acted above your soil mass this is your earth pressure distribution diagram because of your soil as well as reinforcement materials and this part is your additional dynamic force because of additional dynamic force earth pressure distribution diagram and it will act at a distance 0.6 h from the base of the wall then limit state function of function for sliding failure mode for stability against sliding failure along the base of g r s wall sum of horizontal resisting force generally it is written f r r is your capital F subscript of r r is your resistance force should be more than sum of the horizontal driving force d is your driving force that means resisting force should be more than your driving force the factor of safety against sliding failure considering considering your dynamic forces into picture f s sliding is equal to f r summation of f r by f d if I take f r is equal to summation of v tan delta b plus k 1 c l divided by p i r plus p a plus p q plus delta p 0.5 delta p a e summation of v is the normal force at the base that means that is your w a b f e w a b f e this is your summation of forces that will be acted at the base this is clearly written here then k 1 is equal to k 1 is equal to two-third it will act two-third p a is your static earth pressure p a is your static earth pressure p a is your static earth pressure this is your static earth pressure it will act at a distance of a h by 3 and this is your 0.5 gamma h square into k a p q is the seismic active earth pressure due to surcharge that is your q h k a e delta b is your interface friction angle between wall base and foundation soil expressed in terms of phi b phi b is equal to frictional friction angle of the soil below the base of the slab and the retaining wall q is equal to surcharge load acting on the backfield soil q is equal to surcharge load acting on the backfield soil this is your q surcharge load acting on the backfield soil substituting all these things you can find it out your factor safety sliding in terms of 2 into l by h tan delta b and 2 k 1 c by gamma h l by h divided by k h plus k a plus capital q k e 0.5 k e minus k a all so where q is equal to 2 q by gamma h is the surcharge coefficient c is the cohesion of foundation soil one sliding is over next step is your overturning or eccentricity failure this is all about the same figure we are repeating again and again taking into consideration showing all the forces static as well as dynamic forces earth pressure distribution diagram and resultant forces and the eccentricity overturning and eccentricity failure if you look at the f h w a 2001 reported that for stability of g r s wall in terms of eccentricity of resultant force the eccentricity should be lesser than one sixth of your base width of wall that means e should be for stable e should be less than equal to your b b by 6 or sometimes it is l by 6 l is your width or b is equal to width at the base of the wall under static condition this condition is your under starting condition and it should be less than equal to your l by 4 under seismic condition if seismic condition has been taken into consideration it should be less than l by 4 eccentricity of resulting force that means e on the base can be calculated it can be calculated by summing the movement of resisting forces and disturbing forces about the central line of the wall base that means summing of the forces but taking is summing of the forces or summing of the movement of the resisting forces as well as disturbing forces about the central line of the wall base so we can find it out r into e that means r into e the movement will be your clockwise r into e this is your clockwise then k h w a g i e into 0.5 h that means k h w h i 0.5 h in this region you to lack all this has been that means clockwise movement and anticlockwise movement means movement you can summation sum it up for clockwise as well as anticlockwise then equate it then you can find it out in terms of r is equal to resultant vertical forces and in terms of e by h you will get it in terms of e by h is equal to eccentricity divided by your total height of wall in terms of k h k a k a e into divided by all this sum q l h l and h then once this returning movement and eccentricity part is over next step of the next check is your bearing capacity failure bearing capacity refers to the ability of foundation soil to support the weight of your g r s wall placed upon it that means this foundation soil to support this whatever the wall constructed above the foundation soil mayer hops generally mayer hops distributions assume that eccentric loading result in a uniform redistribution of pressure over a reduced area at the base of the wall this area is defined by a width equal to your wall width less twice this eccentricity that means l minus two e here once there is an eccentricity this area will be reduced that means once there is eccentricity what will happen there will be a tension at the base of the wall that means one there is a tension in the base of the wall there is a gap between your reinforcing material and foundation soil as per the mayer of distribution you have to remove that eccentricity and the entire width will be l will be redistributed it will be as l minus two e so that is why it will be l minus two e the factor of safety against bearing capacity failure can be estimated as the ratio of ultimate bearing capacity that is your q u of a shallow foundation below the base slab of g r s wall and the vertical stress at the base that means sigma b calculated with the factor of safety against bearing capacity failure can be estimated as the ratio of ultimate bearing capacity of shallow foundation below the base slab of g r s wall and the vertical stress at the base that means your sigma b calculated with your mayer of types of distribution if i calculate the factor of safety bearing capacity failure that means q u ultimate bearing capacity by sigma b q e is your as per your mayer of calculation that means as per your this c n c plus 0.5 gamma b l minus two e into n gamma n c and n gamma is your bearing capacity factors and sigma b is your total vertical forces divided by l minus two e so gamma b is your unit weight of foundation soil as b is your base b generally termed as base unit weight of foundation soil n c and n gamma are bearing capacity factors as given by thus 1999 normalizing q u and sigma b with gamma h we can get q u by gamma h is equal to c by gamma h n c plus 0.5 gamma b by gamma l by h minus two e by h n gamma and from there you can find it out sigma b in terms of gamma h you will get it then next step is your internal stability that means is your inextensible reinforcement if wall face is battered if wall face is battered that means an offset of 0.3 h 1 is still required if it is battered that means an offset of 0.3 h 1 is still required and the upper portion of the zone of maximum stress should be parallel to your wall face upper upper portion of the zone of maximum stress should be parallel to your wall face that means it should be the zone of maximum stress or potential failure it should be parallel to your wall face and h 1 based on that h 1 should be calculated was h plus tan beta into 0.3 h by 1 minus 0.3 into tan beta so this is your zone of maximum stress or potential failure surface if I take it here this is my zone of maximum stress or potential failure surface with an angle psi, psi is equal to 45 degree plus phi by 2 and this as I have already discussed earlier this is your active zone and this is your resistant zone we will discuss about this internal stability of different air pressures tomorrow we will discuss all this detail thanks a lot.