so now let us move on to the last method of designing pre-stress concrete elements that we will study on this meeting which is called the load balancing method so actually the information that we got from the previous discussion especially the the case of the basic concept method can also be used to derive the underlying principles behind load balancing method also in the case of load balancing method it is usually used when the when analyzing a concrete element which is already in its service load condition meaning the pre-stressing force acting on it on the pre-stressing reinforcements are already the effective pre-stress force after losses so which is actually the reduced value of the initial pre-stressing force which is again diminishes in value because of what we call pre-stress losses and again those causes of pre-stress losses will be discussed later on on the later chapter and again additional to that we know that if the concrete element is already in the service load condition the bending moment being experienced by that particular concrete element is already due to the moment due to the dead load or the south weight itself due to the superimposed dead load and due to the live load that is being carried by the concrete element at service load so these moments or these loads are again actually called the service loads or the which are actually aggravated these loads are called service loads which are actually gravity loads so just like the client method the load balancing method is usually used to analyze a concrete element which is already in its service node condition but then again just like in the case of c line in my experience i i believe it can also be used on the other stages of pre-stressing even in initial pre-stressing and transfer or at initial pre-stressing a transfer with self-weight so in my in my experience uh the formula that we will derive here later on will can be also be used to analyze a concrete element at other stages of distressing but most of the time load balancing method is just used to analyze a concrete element which is already in service load condition so now to illustrate the concept behind load balancing method let us consider this concrete dim here that is already subjected to the total service loads that are intended to be carried by the element itself which are the dead loads the software itself the super purpose dead loads and the live nodes and we know for a fact that those loads impose to this concrete element will actually produce bending moments all throughout the length of the concrete beam especially in the case of simply supported beam and actually the magnitude of those bending moments that will be produced by this impulse loads the dead load live load and the superimposed deadlock the magnitude of those moments will actually vary from the one end of the simply supported beam to the other end of the simply supported beam with the maximum value occurring at mid span so we already know that for the case of simply supported beam which supports a uniformly distributed load we know for a fact that the variation of the moment or the moments being induced or produced inside the correct element of the simplest of the simply supported beam varies from zero to the to one support to a maximum value at mid span and then we'll again go down to a value of zero at the other end so if we will try to graph all of those magnitude all of those magnitudes all of those variations of magnitudes from one end to another the diagram will actually look like this with a maximum magnitude of moment occurring at midspan which is actually the bending moment induced by the applied loads so this bending moment you see here is the maximum moment acting on the concrete element which is actually the moment due to the combined effect of the that load suprempos dead load and live loop so that is the variation of the bending moment that occurs inside the concrete element as it supports the total service loads the dead load live load and the superimpose that load so now in the case of load balancing method just to start the formulation of the formulas underlying the load balancing method actually for the case of load balancing method we have a goal and actually we have two goals here and the first goal is actually this so goal number one in load balancing method we have to find a load w sub b which is the balanced load that we are talking about here that will just balance the moment due to the pre-stress eccentricity because we know we all know we know for a fact that the pre-stressing force acting on the pre-stressing reinforcement at an eccentricity from the neutral axis or from the centroid of the cross-section will actually produce what we call a centric moment with respect to the centroid or neutral axis which has actually a magnitude of p times e if we are talking about at mid span and actually that multitude of uh eccentric moment being produced by the pre-stressing force say for example at mid-span uh must be balanced by this uh by the moment that will be induced at midspan by this balanced load that we want to find on our first code so we have to find a load wsrp that will produce bending moments especially us at midspan which will just balance the eccentric movement that will be produced by the pre-stressing force and we know that the magnitude of that moment our eccentric moment produced by the pre-stress force at an eccentricity from the neutral axis or from the centroid will have a magnitude of p times e so the pre-stressing force times the eccentricity and that eccentric movement say for example acting at uh mid span of the simple supporting bin must be must be balanced by the bending moment that will be produced by this what we call balance load which is usually this balance load here is actually a fork this balanced load here is usually just a portion of the total service loads which are again the sum of the dead load live load and the super bowls that load so you can expect that the magnitude of this balance node here is just uh is you can expect that the magnitude of this balance load here is lesser than the sum of the total service loads which are again the dead load live load and the superposed deadlock so that is our first goal we have to find this balance load w that will just produce a bending moment at mid span say for example which will have a magnitude equal to say for example let's just call the bending moment that will be produced by the balanced load at mid span equal to m sub b the balance moment so that balance moment uh produced by this balanced load acting at midspan of a simply supported beam loaded by a uniformly distributed load must just balance the eccentric movement coming from the uh pre-stress force acting at the pre-stressing reinforcement atmosphere so that is our first goal here on the load balancing method we must find this balance load that will just produce a bending moment at mid-span say for example at midspawn that will just balance the eccentric moment coming from the pre-stressing force so now so again just take note that the again may just repeat the balance load here since we are in service load condition already we know for a fact that the our concrete element already supports the total service loads which are the dead load light bloat and the superimposed dead loads and again the sum of those that load live load and super posted loads will be will obviously be greater than this balanced load here that we will try to find our on our first goal or saying in other words this balanced load here will obviously have a lesser magnitude than the total service loads because like what i have said before this is just usually a portion of the total service loads so we will just find a portion of the total service load that will produce a bending moment a balance bending moment that will just balance the eccentric movement that is being produced by the pre-stress force acting say for example at the midspan so that is our first goal and actually that balance moment that is being endorsed by this balance load will again produce what we call flexural stresses because we know that on this on this condition the pre-stressing force acting on the pre-stressing reinforcement produces what we call action stresses combined with the flexural stresses due to the eccentricity of the pre-stress force so additional to that since we now add that a balanced load here balanced load w that produce a balanced moment at mid-span that balance movement acting at midspan will again induce additional stress on the concrete section at mid-span additional to these stresses induced by the pre-stressing force which will look like this if we will try to graph the flexural stresses that will be induced by this balance moment at midspan it will look something like this with an extreme uh fiber magnitude equal to mc over i so we will just uh change the magnitude of m with uh with uh mb we will just replace the magnitude of m with the balance moment because it is the one who produces this flexural stress as you see here and again let us just put a negative sign because we all know that the bending moment acting on mid-span produces a counterclockwise moment here at this section so if that is a counter-clockwise moment the resultant couple of the several forces must be equal to that clockwise moment with the same direction so with that to be consistent on the direction of the bending moment produced by the balanced load the flexural stresses on the upper portion of the neutral axis must point towards the concrete element which will induce what we call compressive force or for stresses and again our sign convention or compressive stresses we usually say we usually use negative sign to indicate compressive stresses while positive for tensile stresses so in this same manner if you want to find the extreme fiber stress or tensile stress acting here induced by the balance moment that is produced by the balance balance load it will have a magnitude similar to this mc over i so again we just change or replace m by the balance moment because again it is the one who produces this flexural stresses and you just have to take note that rc here must be with reference to the bottom fiber so our c here must be the distance from the neutral axis to the bottom fiber our c here must be the distance from the neutral axis to the extreme top fiber so again we put a positive sign because the stresses flexural stresses acting below the neutral axis points away or point away from the concrete element so they are what we call tensile tensile stresses because they point away from the concrete element itself so these are now the total ir i mean these are now the stresses acting on the uh concrete element at mid span when it is loaded by the prestressing force and the balance load that we have found from our first goal so if we will try to find the total stress acting at mid-span due to the pre-stressing force and the balance load so we just have to add this stresses you see here the actual stress due to the pre-stressing force the flexural stress due to the eccentricity of the pre-stressing force and the flexural stress due to the balance moment so if you will try to add them the total diagram of the total stress will look something like this so why is that so why why is the sum why does the sum of these stresses produce a what we call uniform stress acting at midspan because we all know that the bending moment here the mb just balances the eccentric moment coming from the pre-stress force acting at mid spine so therefore we can say that pe the exatic moment produced by the processing force at midspan is just equal to mb and since they have uh equal c equal i and equal magnitude of moment but they they just have opposite direction here because the stress is induced by the flexural i mean the flexural stresses induced by the eccentricity of the pre-stressing force produces tensile stresses on the upper portion while the structural stresses due to the balance movement produces compressive stresses on the upper portion of the neutral axis so since they just have a magnetic equal magnitudes but opposite effect or opposite directions they will just balance each other as well as on the as well as the stresses below the neutral axis they will just cancel each other since again pe the eccentric moment due to the crystal force acting at mid-spawn is just or i mean that centric moment is just balanced by the balance movement due to the applied balance loop so since this stresses and these stresses just have equal magnitudes but opposite direction they will just cancel out each other therefore the sum of these stresses you see here will just be coming from the actual stress coming from the application of the pre-stressing force at mid-spine so this is now the total stress acting at mid-span if we will impose a balanced load on the concrete element so actually that is the goal of load balancing method to produce a constant or uniformly distributed action stress acting at every section of the concrete element itself that is the goal of applying a balanced load that's why it's called load balancing that is the goal for this method to produce uniform stresses acting at every section of the concrete element although you will see here there might be a problem there there is a problem that will occur in order for us to address that goal of load balancing method which is to produce uh uniform stresses at every section of the concrete element itself and what what might be that problem that will occur that will prevent us from achieving our goal which is this to have this kind of stress at every section of the concrete element so the difficulty that we will encounter on achieving that goal is because due to the fact that the bending moments being produced by the balance load with respect to the simply supported beam varies from zero at the end at one end and then having a variation of moment with a maximum moment at midspawn and then will again vary up to or down to a magnitude of zero to the other end of the simply supported view so in other words the bending moment produced with the uplight balance load is not constant all throughout the length of the simply supported beam while in this case the centric movement produced by the pre-stressing force if you can notice the magnitude of dicedric moment produced by the eccentric force here or i mean the crystal's force along the distressing reinforcement will actually have a constant magnitude all throughout the length of the simply supported beam so why is that why is the centric movement due to the pre-stressing force will have a constant magnitude all throughout the length of the beam simply because the eccentricity of the pre-stressing reinforcement all throughout the length is just constant so if e is constant and obviously p is constant we assume that i'll throw out our discussion we assume that the tension force or the stress force acting on the pre-stressing reinforcement is constant all throughout the length of the distressing reinforcement itself so since c is cons since e is constant as well as p therefore you can imagine that the eccentric moment acting here at the end will just be equal to the centric moment acting at midspawn and again will just be equal on the eccentric moment acting at any other section or portion of the simply supported beam so therefore we can say that the only portion or section on the simply supported beam that will have a balanced condition meaning the centric moment from the crystal force is just balanced by the bending moment produced by the applied balance load that condition will just hold at mid span because again we know that the bending moments varies with respect to the length of the concrete element the bending moment produced by the balance load varies along the length of the concrete beam while the eccentric movement coming from the pre-stress force is just constant all throughout the leg of the beam so say for example if we will try to to find or investigate a particular section acting here so we at this particular section we have a centric moment due to the pre-stress equal to pe same magnitude as as the eccentric moment coming from the mid span so the magnitude of centric moment here at this particular section is equal to pe but if you will look on the corresponding magnitude of moment at that same particular section obviously it will has it it will have a magnitude which is obviously lesser than the magnitude at mid span and since the magnitude of bending moment at midspan just equals pe therefore we can say that the bending moment here can't mark can't balance the eccentric moment coming from the pre-stress force acting at this particular section so we failed to have a uniform stress acting on this particular section since the moment acting on that section which is this is not equal to this the flexural stresses that is being induced by the eccentricity of the pre-stress force so they will not cancel out each other so the total stress that will act on that particular section will not look like this it will not be a uniform uniform stress which is the goal of load balancing method so that is our main problem here for this particular condition for for having a tendon or pressurizing reinforcement with constant eccentricity so now to address that difficulty we will have our second goal and what is our second goal so our second goal here is to have a tendon profile which is the profile of the prestressing reinforcement which will have an eccentric movement which will that which will balance the movement due to external balance load so that the resultant stress at every section is a uniform stress only like this so that is now our goal we will try to to vary the eccentricity of the pre-stressing reinforcement so that the eccentric movement that will be produced at every section will just balance the corresponding bending moment produced by the applied balance load so that is our second goal for this method to address the to formulate the formulas underlying the load balancing method we have to find or we have to vary the eccentricity along the length of the uh along the length of the concrete element so now the problem will now be again there will be another problem that we look forward and that the problem now will be how will we but how will we vary the eccentricity in what way will we vary the eccentricity will we make the eccentricity at the end equal to zero or will we make the eccentricity at the other end to be above the neutral axis or above the centroid so how how how how in which in what way will we vary the eccentricity of the pre-stressing reinforcement and throughout the length of the concrete element so that the the corresponding bending moment will just have equal magnitude to the corresponding eccentric moment coming from the eccentricity of the pre-stress force so actually we can address that by simply by simply adapting the shape of the bending moment diagram that is produced by the balance node with with the opposite direction obviously the the shape of the tendon profile that we must put here to balance this kind of bending moment diagram must have a similar shape as this but in opposite direction meaning if this is concave downward the attendant profile that we must put here must be concave upward just like this so this is how we can achieve our our goal which is to have a uniform stress at every section of the concrete element by applying a balanced load so with that we can now make sure that the eccentric moments for example at this particular section will just be equal to the bending moment acting at that same particular section due to the balance load since the eccentricity is now uh different from the eccentricity at mid span so we can expect that that particular eccentric moment acting out here acting here at this particular section will now be equal to this bending moment acting at this same section due to the balance load so that is how that is how we can address the difficulty we encounter available so in that way we can assure that the total stresses acting at every section will just look like this because we all know that the total stresses acting at every section is just the sum of the actual stress coming from the pre-stressing force itself and the flexural stresses due to the eccentricity of the viscera's force plus the flexural stresses induced by the balance load so in this way by having this type of tender profile or profile of the pistachio red horse but we can make sure that at every section we will have a diagram of stresses that will look that will just look like this and with that we can make sure that this eccentric moment will always just be equals to will just be always equal to the corresponding bending moment here which is which are actually what we call uh balance moments so all of those moments here are now called uh are actually called balance moments so they are called balanced moments because they are just balancing the eccentric moment produced by the eccentricity of the pre-stressing force all throughout the length of the concrete beam so there you go that is how we can address our difficulty when countered a while ago so now we can achieve this we can achieve node a uniform stress acting at every section of the concrete element itself so now since we are already assured that all of this uh bending moments balance bending moments here will now just be balanced or counter uh counteracted by the pres by the eccentric movement due to the eccentricity of the pre-stressing force acting at the pre-stressing reinforcement now we can now say that the bending moment the balance bending moment say for example at midspan is now equal to the eccentric moment due to the eccentricity of the pre-stressing force acting at mid-spot or in equation form we can say now that mb the balance moment at midspan c for example is now be equal is now equal to the centric moment due to this pre-stress force acting at the prestressing reinforcement which have a magnitude of p times e so this is so we are now sure on this relationship we are now sure that the centric moment here at midspan is now equal to the demanding moment acting at mid span as well which is due to the application of the balanced load and we know on this case for a case of simply supported beam which supports a uniformly distributed load we know that the bending moment specifically at midspan is just equal to this formula wl squared over eight since we are investigating the mid-span section we know that the magnitude of the bending moment acting at midspan for a simply supported beam which supports a uniformly distributed load wb this is just equal to wl squared over eight therefore we can change this mb here by wl squared over eight which by also changing w with wb the balance load so that bending moment which is now again equal to wb l squared over 8 is now equal to the product of the pre-stressing force and end its eccentricity or if we will just express this equation in terms of wp we can now find the magnitude of that balanced load that we have to apply on this concrete element so that the the eccentric movement due to the eccentricity of the pre-stressing force will just be balanced by the bending moments that will be produced by that balanced node so we can now find the magnitude of that balanced load by using this formula here eight times the pre-stressing force times its eccentricity at midspan so since we we derived this formula by considering the section at midspan so the eccentricity so the eccentricity that we will talk about here uh will be at the midspan divided by l squared which is the total length of the beam itself doesn't support your beam although you have to take note that this is not always the case this is not always the the way to find the magnitude of the balance load that you have to apply on the concrete element itself to just balance the eccentric movement at every section being produced by the eccentricity of the president's force this is not the only way actually it depends on the it depends on the load that will be carried by the concrete element itself actually we can directly say that this formula is dependent on the moment diagram that will be produced by the load that will be applied to this particular concrete element say for example in this case if we will apply a concentrated load acting at mid-span that points downward it will have a moment diagram that will look like this although it is just inverted so in order for us to counter up all of those bending moments that are being produced by a concentrated load acting at midspan we must have a tendon profile same as this so we must have a actually what we call a heart and heart tendon profile such as this why because we know that if the tendon profile is similar to this the if and if you will try to graph the variation of the bending moment being produced by the eccentricity of the pre-stressing force all throughout the length of the concrete element the diagram of the or the graph of that diagram of the eccentric movement being produced all throughout the length of the wing will will just look like this which is exactly the opposite of the moment diagram which will produce by applying a concentrated load at mid-spine so again if you will have a this is how you and this is how you will decide what which tendon profile you have to use in order to have a balanced condition on the concrete element so say for example again if we have a concentrated load we know that the moment diagram for a concentrated concentrated load if the if the beam is simply supported we know that the moment diagram for a concentrated concentrated load applied downward at mid spot will look similar to this but in opposite direction it must be that it must be positive it must be positive so in order for us to counter all of those bending moments and throughout the length of the concrete element here we must have a tendon profile same same as this so that the the eccentric movements that will be produce all throughout the length will look like this what you what what you are seeing here so since this is similar to the bending moment being produced by the applied concentrated loaded with spine therefore we can we can be assured that they will cancel each other at every section at every section of the concrete element itself so that is how you can choose which tendon profile you have to use to balance the applied applied load on the concrete element or actually you can you can look it on a different way say for example if the given is the tendon profile itself so say for example we have here a tendon a parabolic tendon profile with an eccentricity zero at the ends say for example if you want to find a a load that will just balance the eccentric movement that is being produced by the eccentricity of the pre-stressing force acting at the pre-stressing reinforcement we can say that we have to find a load that will produce a bending moment diagram opposite this same magnitude opposite the same same magnitude of bending moment similarly to the bending moments but will but must have a opposite direction so we know that a uniformly distributed load just like this one produces a parabolic moment diagram same as this which will just counter up or counterbalance the eccentric moments being produced by the stressing force all throughout the length of the beam so therefore if you have if you will have a tender profile such as this we have to apply a uniformly distributed load so that the eccentric moments coming from the pre-stress force will just be balanced by the bending moments that will be produced by the applied uniformly distributed nodes so that is how you can find either the tendon profile or the balance load that you have to apply in order for you to have a balanced condition with respect to the moments or flexural stresses acting on the concrete element so we have different uh profile here depending on the type of load that you want to balance so if you want to balance a concentrated moment say for example the load that is being carried by the concrete element is a concentrated moment at the ends so we know that the bending bending moment diagram that is being produced by that kind of load is similar to this but opposite direction so therefore in order for up in order for us to counteract those bending moments or throughout the length of the beam we must have a tendon profile same as this with a constant eccentricity so that there will be also a constant eccentric movement due to the pre-stressing force all throughout the length of the beam which will look like this which will just balance the bending moments that will be produced by the applied concentrated movements at the ends of the concrete element so that is how you can find this wb so it it it doesn't have a a formula similar to this always so again it depends it depends on the load that will be carried by the concrete element if the load that will be coded by the congruent element is a uniform distributed load so you can use this formula but say for example if the load is that is that will be carried by the correct element is a concentrated load you can use other formula so just derive the formula for finding the say for example the bending moment at midspan for a concentrated and equate just equate that to the eccentric moment acting at that same location which is the midspan and then just express the equation in terms of the wb or the balanced node balanced load p that you have to apply here on the mid spine of the concrete element so in that way you can find the balance load that uh that must be imposed on the concrete element to have a balanced moments at every section so that you can you will be sure you will be assured that the stresses acting at every section will be similar to this will be just a uniformly distributed stress so again you have to remember you have to realize that what we are talking about here is service load condition stage of distressing so we are we are in the service load condition here everything that we are talking about here now for the load balancing method is in service load condition we're in the load being carried by the concrete element is the torque is the total service loads and again just like what i have said before the balance load here is just a portion of the total service loads so meaning there there is still a a remaining load that will be carried by this concrete element so let us just call that the remaining remaining load that that will be carried by this concrete element equal to w sub ub the unbalanced unbalanced load so again that advanced load is the remaining portion of the total service notes that will be carried by this concrete element since we are already in service load condition so you have to remember we are in service load condition because that that is where the load balancing load balancing method can be applied at service load condition so you can expect that the sum of the balance load here and that balance load here is just equal to the total service loads so again just like the balance load that balance load here will also produce a bending bending moment all throughout the comfort element or draw the length of the simplest portrait beam which will have a similar shape as this or in other words the bending moments at every section will just have additional bending moment coming from the applied and balanced load so let us just call the moment that will be produced by the unbalanced node here equal to mu sub m sub ub the unbalanced moment and because of that because there is an additional bending moment applied to this concrete element therefore there will be additional flexural stresses that will be induced inside the concrete element since before since before the application of the unbalanced load the total stress acting on the concrete element at every section is equal to this if you can remember so if we will just if we will try to apply additional load which is the unbalanced load here that will produce unbalanced movement therefore there will be additional flexural stresses that will be in in those inside the concrete element which will have a diagram similar to this with a maximum extreme fiber magnitude equal to this mc over i so since the moment that induces this flexural stress is muv so that is our m that we have to use on the formula mc over i as well as on the bottom fiber so again you have to take note of the sign because we know that this flexural stresses acting on the upper portion of the centroid or the neutral axis produces compressive stress as well on the bottom or the lower portion produces tensile stresses that's why it has a positive side so before again before the application of the unbalanced load the total stress acting at every section of the concrete element is just this a uniform stress only but since we now apply the remaining portion of the total service loads which is the unbalanced load here that produce an unbalanced moment here which produce again additional flexural stresses at every section of the concrete element with the flexural stress acting at midspan equal to this which have a extreme fiber stress equal to this and extreme fiber stress equal to this so now if we will try to find the total stress acting at every section now of the concrete element with the application of that balance force our unbalanced load here the total diagram of the total stress will now look like this we will try to add this stresses with a again an extreme fiber magnitude of stress equal to the sum of the stress here which is negative pi over ac and the and the stress acting here on the top fiber which is equal to this and similarly the total stress acting at the bottom fiber is just the sum of the stress here negative pi over c negative b over ac and this stress here which is mubct over ic so those are actually the formulas that that you will use to find the total stress acting on a concrete element using the load balancing method so you have you just have to take note of this formulas these are the formulas that you will use if you want to find the total stress acting on a concrete element at every section of a concrete element which is again in in service load condition so you have to meet that criteria first the concrete element must be already in service load condition since we derive this formulas considering that our concrete element is already in service load condition wherein the total loads that it will support is equal to the total service nodes which will which are due to the dead load or self-weight suprempos dead load and the level so if you will try to get the summation of this again submission of this flexural stresses here you can see that the total force acting on this particular concrete section of the concrete element is just a single for c acting at a certain distance from the neutral axis or centroid so actually that c is just equal to the the oppressive stress or compressive force we have derived from the c9 method and and the basic concept method so if you will try to investigate this formula the format the formulas that we have derived for load balancing method if you will try to compare that to the formulas we have derived from the basic concept method you can actually see that these formulas are just actually equal so how come how come they are equal so if you will try to uh express the total moment here on the case of the building in the case of the basic concept method if you will try to express the total bending moment acting at the concrete element at service loads which is again equal to the summation of the moment due to the dead load likelihood and the superimposed deadlock if you will try to express that as the sum of the balance movement here which is due to the balance load here and the unbalanced moment which is again due to the unbalanced load here and again take note that some of this are just equal to the total service loads so if you will just try to express the total moment here on the formula for the basic concept method in terms of the balance movement and the unbalanced moment where in the balance movement we have found out a while ago that the balance movement is just equal to the eccentric moment due to the eccentricity of the stressing force which is just equal to pe so we can change mb here by pe so if you can look on this equation if you will try to look on this equation uh if you will express this in terms of mub therefore you just have to transpose pe to the left side of the equation and try to factor out c over i c over c over i on these two terms so the remaining term inside the quantity will just be the uh negative or i mean the m minus p so try to and try to factor out c over i and c over i here together with the negative sign if you will factor out them you will have an expression equal to negative c over i times m minus p e and since m minus p is just equal to mu b you can just replace m minus p e here by mu b and the resulting equation will be equal to this m negative c over i times mu b which is again equal to m minus b so uh with that we have seen that the formula for the risky concept method and the formula for the load balancing method are just actually equal but uh just seen on a different perspective just like what we have what we have seen on the case of the c line method so the formula is different but if you will try to investigate it it is just equal to the formula for basic concept method just seen on a different perspective so again they are just equal if you will try to compare so technically you must have a similar answer if you will try to use the basic concept method and the load balancing method formula also in cases where you where the attendant profile or the profile of the pre-stressing reinforcement is varying with respect to the length you have to take note that the force p that you will use on this formula or even on this formula must be the horizontal component of the prestressing force just like what we did there because if you will try to imagine what is the direction of the pre-stressing force acting say for example at the end of this pre-stressing reinforcement we know that the pre-stressing force is always tangent to the profile of the pre-stressing reinforcement so if you will imagine what is the direction of the pre-stressing force acting at this end at this end of the stressing reinforcement it is a inclined force and since it is inclined it has components horizontal and vertical so that horizontal component that magnitude of force must be the one that you have to use on this formulas even on the other formulas that we have derived from the basic concept method and the client method although for cases where in the length of the simply supported beam is a very large length you will encounter no problem on using the magnitude of the pre-stressing force itself instead of the horizontal component because for long span beams the pre-stressing force is just approximately equal to the horizontal component of it for long span beams but for the cases of short short spine beams you can see that the horizontal component of the pre-stressing force is not any more equal to the pre-stressing force itself so in the in those cases you have to use the horizontal component instead of the instead of the restressing force itself so again take note of that you have to take note of that for four cases of having a profile with varying eccentricity that is always the case for those type of concrete elements we tend with intended profile or pre-stressing restressing reinforcement profile with varying eccentricity just you must always use the horizontal component of the pre-stressing force for the formulas for finding the total stress acting on the concrete element at every section at any section so that is how the formula for the load balancing method was derived