Load and Ultimate Moment of Prestressed ConcreteAction Under Overload-Cracking LoadIt has been shown that a variation in the external load acting on a prestressed beam results in a change in the location of the pressure line for beams in the elastic range.This is a fundamental principle of prestressed construction.In a normal prestressed beam,this shift in the location of the pressure line continues at a relatively uniform rate,as the external load is increased,to the point where cracks develop in the tension fiber.After the cracking load has been exceeded,the rate of movement in the pressure line decreases as additional load is applied,and a significant increase in the stress in the prestressing tendon and the resultant concrete force begins to take place.This change in the action of the internal moment continues until all movement of the pressure line ceases.The moment caused by loads that are applied thereafter is offset entirely by a corresponding and proportional change in the internal forces,just as in reinforced-concrete construction.This fact,that the load in the elastic range and the plastic range is carried by actions that are fundamentally different,is very significant and renders strength computations essential for all designs in order to ensure that adequate safety factors exist.This is true even though the stresses in the elastic range may conform to a recognized elastic design criterion.It should be noted that the load deflection curve is close to a straight line up to the cracking load and that the curve becomes progressively more curved as the load is increased above the cracking load.The curvature of the load-deflection curve for loads over the cracking load is due to the change in the basic internal resisting moment action that counteracts the applied loads,as described above,as well as to plastic strains that begin to take place in the steel and the concrete when stressed to high levels.In some structures it may be essential that the flexural members remain crack free even under significant overloads.This may be due to the structures’being exposed to exceptionally corrosive atmospheres during their useful life.In designing prestressed members to be used in special structures of this type,it may be necessary to compute the load that causes cracking of the tensile flange,in order to ensure that adequate safety against cracking is provided by the design.The computation of the moment that will cause cracking is also necessary to ensure compliance with some design criteria.Many tests have demonstrated that the load-deflection curves of prestressed beams are approximately linear up to and slightly in excess of the load that causes the first cracks in the tensile flange.(The linearity is a function of the rate at which the load is applied.)For this reason,normal elastic-design relationships can be used in computing the cracking load by simply determining the load that results in a net tensile stress in the tensile flange(prestress minus the effects of the applied loads)that is equal to the tensile strength of the concrete.It is customary to assume that the flexural tensile strength of the concrete is equal to the modulus of rupture of thewhen computing the cracking load.In should be recognized that the performance of bonded prestressed member is actually a function of the transformed section rather than the gross concrete section.If it is desirable to make a precise estimate of the cracking load,such as is required in some research work,this effect should be considered.Principles of Ultimate Moments Capacity for Bonded membersWhen prestressed flexural members that are stronger in shear and bond than in bending are loaded to failure,they fail in one of the following modes:Failure at cracking load In very lightly prestressed members,the cracking moment may be greater than the moment the member can withstand in the cracked condition and,hence,the cracking moment is the ultimate moment.This condition is rare and is most likely to occur in members that are prestressed concentrically with small amounts of steel.It can also occur in hollow or solid prestressed concrete members that have relatively low levels of reinforcing.Determination of the possibility of this type of failure is accomplished by comparing the estimated moment that would cause cracking to the estimated ultimate moment,computed as described below.When the estimated cracking load is larger than the computed ultimate load, this type of failure would take place if the member were subjected to the required loads.Because this type of failure is brittle failure,it occurs without warning–designs that would yield this mode of failure should be avoided.Failure due to rupture of steel In lightly reinforced members subjected to ultimate load,the ultimate strength of the steel may be attained before the concrete has reached a highly plastic state.This type of failure is occasionally encountered in the design of structures with very large compression flanges in comparison to the amount of prestressing steel,such as a composite bridge putation of the ultimate moment of a member subject to this type of failure can be done with a high precision. The method of computation,as well as the determination of which members are subject to this mode of failure,is described below.Failure due to strain The usual underreinforced,prestressed structure that are encountered in practice are of such proportions that,if loaded to ultimate,the steel would be stressed well into the plastic range and the member would evidence large deflection.Failure of the member will occur when the concrete attains the maximum strain that it is capable of withstanding.It is important to understand that research into the ultimate bending strength of reinforced and prestressed concrete has led most investigators to the conclusion that concrete,of the quality normally encountered in prestressed work fails when the limiting strain of0.003is attained in the concrete. Since the ultimate bending capacity is limited by strain rather than stress in the concrete,it is a function of the elastic moduli of the concrete and steel.The magnitude of the ultimate moment for members of this category can also be predicted,as a rule, within the normal tolerance expected in structural design.The ultimate moment of underreinforced sections cannot be predicted with the same precision as the lightly reinforced members described above,since the ultimate moments of underreinforcedare a function of the elastic properties of the steel and the effective stresses in the prestressing steel,whereas the ultimate moment capacities of lightly reinforced members are not.Failure due to crushing of the concrete Flexural members that have relatively large amounts of prestressing steel or relatively small compressive flanges are referred to as being overreinforced.Overreinforced members,when loaded to destruction,do not attain the large deflections associated with underreinforced members–the steel stresses do not exceed the yield point and failure is the result of the concrete being putation of the ultimate moments of overreinforced members is done by a trial and error procedure，involving assumed strain patterns,as well as by empirical relationships.It must be emphasized that there is no clear distinction between the different classifications of failure listed above.For convenience of design,certain parameters, which are a function of the percentage of steel,are used by different authorities to distinguish between the different types of failure that would be anticipated.In order to simplify the explanation of the theory related to the computation of the ultimate moments,a rectangular section will be assumed throughout the derivation, in order to eliminate the variable of flange width which is frequently encountered with I or T sections.In addition,the following assumptions,some of which differ slightly from those contained in ACI-318,are made:1)Plane sections are assumed to remain plane.2)The stress-strain properties of the steel are smooth curves without a definiteyield point.3)The limiting strain of the concrete is equal to0.0034,regardless of the strengthof the concrete.4)The steel and concrete are completely bonded.5)The sress diagram of the concrete at failure is such that the aveage concretestress is0.80ƒ΄c and the resultant of the stress in the concrete acts at a distance from the extreme fiber equal to0.42of the depth of the compression block.6)The strain in the top fiber under prestress alone is equal to zero.7)The section is subject to pure bending.8)The analysis is for the condition of static loads of short duration.As was stated above,the relationships that were developed are applicable to rectangular sections.These relationships are equally accurate for flanged sections, provided the neutral axis of the section at ultimate is within the limits of the flange.If the neutral axis falls outside of the flange area,the same strain distribution applies as in the case of rectangular sections,but due to the variable width of the section,the distance to the resultant of the compressive block must be calculated.To facilitate the calculation of the location of the resultant,the compression block can be assumed to be rectangular rather than curved without introducing significant error.When small quantities of non-prestressed reinforcement are used in combination with small quantities of prestressed reinforcement,the additional ultimate moment due to the non-prestressed reinforcement can be calculated.For larger amounts of non-prestressed reinforcement or for members with high steel indices,the moment shoulddetermined by trial and error from the basic strain patterns.Examination will show that small variations in the effective presstress have no significant effect on the ultimate strength of prestressed members.It is important to note that even if errors are made in estimating the losses of prestress,in estimating the stressing friction,or even if the stressing is not carried out to a high precision in the field due to poor workmanship,the effect on the ultimate moment is generally small for flexural members with bonded tendons.Principles of Ultimate Moment Capacity for Unbonded MembersBecause the prestressing tendons can slip(with respect to the concrete)during loading of an unbonded member,the relationships for ultimate moment capacity do not apply to unbonded beams.Because the tendons can slip with respect to the concrete,other variables affect the ultimate moment capacity of unbonded prestressed concrete members.Variables that affect the ultimate moment capacity of an unbonded beam,but which do not affect bonded beams in the same manner or not at all,include the following:1)Magnitude of the effective stress in the tendons.2)Span to depth ratio.3)Characteristics of the materials.4)Form of loading(shape of the bending moment diagram).5)Profile of the prestressing tendon.6)Friction coefficient between the prestressing steel and duct.7)Amount of bonded non-prestressed reinforcing.A method of computing the ultimate strength of prestressed members(with unbonded tendons)that takes into account the variables listed above has been proposed by Pannell.This method is based upon experimental data and is considered slightly conservative.It should be recognized that the ultimate moment capacity of a member stressed with unbonded tendons,unlike members with bonded tendons,may be adversely affected by unintentional variations in the effective prestress.Hence,it is considered prudent to exert more care in estimating the losses of prestress and in supervising the stressing of unbonded members than would be considered necessary for bonded members,in order to assure the desired results are obtained.。