Comparison of Methods of Estimating Prestress Losses for Bridge Girders

Transcription

1 Comparison o Methods o stimatg Prestress Losses or Bridge Girders by Sott Walton, M.S.C..,.I.T. 1 and Timothy. Bradberry, M.S.., P.. 2 ABSTRACT A new method or estimatg prestress losses has been published NCHRP Report 496, Prestress Losses Pretensioned High-Strength Conrete Bridge Girders. The new proedure, whih is the basis o provisional hanges to be published the 2005 Interim Revisions o the 2004 AASHTO LRFD Bridge Design Speiiations, Setion 5.9.5, Loss o Prestress, is signiiantly more omplex and alulation tensive than that o the urrent design speiiations. The authors ompare the two prestress loss estimation methods and ompare the results o the proposed detailed loss alulation proedure with the results o the urrent AASHTO LRFD loss alulation proedure or three prestressed onrete bridge girders, representg a range o typial bridge girders. The authors lude a disussion o how use o transormed setion properties (i.e. lude prestressg steel as part o the omposite setion resistg loads and deormation) aets proedurally and numerially the estimation o elasti losses and gas. INTRODUCTION With today s destop omputg power, engeerg equations that were reated to simpliy omplex relationships and mae hand designs possible are beg replaed by muh more alulation tensive equations and proedures that are presumed to be more aurate. These new proedures are beg orporated to design speiiations, simply beause the tehnology exists to implement them the design proess. The implementation o these state-othe-art proedures essentially requires design engeers to either purhase or develop sotware, and learn how to use it, order to be ompetitive. This is assumed to be a good thg. However, the ase o loss o prestress, i the design engeer really believes that the prestress losses and girder stresses that an AASHTO LRFD Bridge Design Speiiations (2004) 3 ompliant omputer program alulates are any way exat, suh an implementation would be unortunate. Also, i the design engeer eels that the prestress loss provisions o the design speiiations are too ompliated to implement by hand s/he may eel pressured by wor load to orgo understandg the provisions and stead may bldly aept the omputer analysis as at without developg any engeerg judgment as to the limitations o the provisions. This too would be unortunate. Together, these two unortunate possibilities ould result a ompromise o the eonomy, onstrutability, and/or the servieability o a prestressed onrete bridge. In rare ases the saety o suh a bridge may also be at ris. 1 ngeerg Assistant, Bridge Division, Texas Department o Transportation, 125 ast 11 th Street, Aust, Texas ; voie: (512) ; ax: (512) ; swalton@dot.state,tx.us. 2 Support Branh Manager, Bridge Division, Texas Department o Transportation, 125 ast 11 th Street, Aust, Texas ; voie: (512) ; ax: (512) ; tbradber@dot.state,tx.us. 3 Amerian Assoiation o State Highway and Transportation Oiials: AASHTO-LRFD Bridge Design Speiiations. (2004) 1

2 The Modiied Compression Field Theory (MCFT) or shear design o strutural onrete, as implemented the AASTHO LRFD Speiiations, requires iteration o a amily o somewhat omplex ormulae and is another example o a ompliated design proedure that lends itsel to omputerization but not to human tuition (exeptg the opion o the small band o engeers volved its development). While suh analysis and design methods as the new prestress loss provisions and the MCFT shear provisions may be diiult to use or understand, many o these improvements are thought to allow engeers to design more eiient strutures while matag uniorm strutural reliability. Reent researh perormed as part o NCHRP Projet (FY 1999), Prestress Losses Pretensioned High-Strength Conrete Bridge Girders, resulted the development o revised alulations whih allow or more aurate preditions o onrete modulus o elastiity, reep, shrage, amber, and prestress losses. The results o this researh are published NCHRP Report 496, Prestress Losses Pretensioned High-Strength Conrete Bridge Girders. 4 The purportedly more aurate prestress loss preditions ome at a signiiant prie, however. There is an reased proedural omplexity o the alulations. While the alulations are more omplex than the urrent provisions they may still be perormed by hand, but are more eiiently implemented with a omputer spreadsheet or other omputer based tool. The reommendations o NCHRP Report 496 have been airmatively voted to the provisions o the 2005 Interim Revisions to the AASHTO LRFD Bridge Design Speiiation by the AASHTO Standg Committee on Bridges and Strutures (SCOBS), replag the reed method the urrent speiiations. Se the pratig design engeer has no hoie but to implement provisions o design odes or speiiations when designg appliable strutures some explanation o this substantially dierent method is warranted. The authors are not amiliar with any explanation o the implementation o the NCHRP Report 496 proedures that they would onsider satisatory to the pratig bridge design engeer. The ollowg is tended to provide suh an explanation. TH ANATOMY OF PRSTRSS LOSSS The hand alulation proedures or estimatg loss o prestress have evolved over the ity-our year history o the preast prestressed onrete dustry the United States rom simple lump sums, suh as the 35,000 psi or losses rom all auses exept rition, whih was speiied the 1961 edition o the AASHO Standard Speiiations or Highway Bridges, to the omplex design proedures o NCHRP Report 496. Durg that time period, virtually the entire United States Interstate Highway System was built (and a signiiant amount o lane miles added or rebuilt) with a large part o that onstrution utilizg preast prestressed onrete and with very ew, i any, signiiant problems traed to designer engeers aurate prestress loss alulations. At least, that has been the experiene Texas. It is highly probable that i hand alulation proedures where still employed bridge design oies, rather than digital omputer aided design proedures, the simple prestress losses estimation proedures o a deade or so ago ould ontue to be employed to design sae and servieable preast prestressed onrete members or as long as onrete and prestressg steel materials remaed similar harateristis to the materials used regularly today. Nonetheless, this setion the authors 4 Tadros, Al-Omaishi, Seguirant, and Gallt: Prestress Losses Pretensioned High-Strength Conrete Bridge Girders. (2003) 2

3 disuss state-o-the-art prestress loss alulation proedures along with the phenomenon o eah loss omponent, thus exposg the anatomy o prestress losses. LRFD Reed Method The 2004 AASHTO LRFD Bridge Design Speiiations allow or three dierent methods to ompute prestressed losses the lump sum method, the reed method, and the time step method. The lump sum method volves usg a sgle equation to estimate time dependant losses. This equation is a untion o the partial prestressg ratio, and some ases the al onrete strength, '. The reed method uses a separate equation or eah o the dierent, terdependent omponents o the time dependant losses shrage, reep, and relaxation. The most signiiant hange the loss alulation between the urrent AASHTO LRFD Speiiations and those proposed NCHRP Report 496 are to the reed loss estimate equations. These equations are thereore disussed more detail the appropriate setions below. The speiiations onta no guidane or, or mention o, the time step method. They stead reer to a more detailed analysis, whih is typially a time step method. In a time step analysis, the lie o the prestressed onrete girder is divided to small steps, over whih the stra the onrete and steel are assumed to be onstant. Knowg the onditions at the begng o the step, the total loss due to reep, shrage, and relaxation over a given time terval an be determed. This proess is repeated or eah step, and the sum o the losses over eah step yields the total time dependant loss. The time step method will not hange with the orporation o the NCHRP Report 496 proedures to the speiiations; however, the values used or ultimate reep stra and ultimate shrage will hange. NCHRP Report 496 Method The NCHRP Report 496 prestress loss alulation method alls somewhere between the time step method and the reed method the AASHTO LRFD Speiiations. Losses are alulated by summg omponents, just as the reed method, but the time-dependant losses are now alulated stages, rom release to de plaement and rom de plaement to al. The ma loss equations are: = + (LRFD ) pt ps plt = ) (LRFD ) plt ( psr + pcr + pr2 ) id + ( psr + pcr + pr3 pss d where: ps is the sum o all losses or gas due to elasti shorteng or extension at the time o appliation o prestress and/or external loads (si) ( psr + pcr + pr2 ) is the sum o all time dependant losses between release and de plaement (si) psr pcr pr id ( ) is the sum o all time dependant losses between de plaement and al (si) pss d 3

4 Not only have the overall loss equations or the reed method hanged, but the equations or eah o the dividual loss omponents have also hanged, as well as those or the modulus o elastiity, shrage stra, reep oeiient, and the lump sum estimate or I beams. An -depth desription and explanation o eah o these hanges and reasons or them is beyond the sope o this paper. The authors reer the reader to NCHRP Report 496 or more ormation on items not overed here. The NCHRP Report 496 approah to the alulation o prestress losses or high strength onrete, as published, is dependent to a large degree on the modulus o elastiity, reep oeiient, and shrage o the onrete. These onrete material properties are turn dependent on loal onrete onstituent materials and mix design praties. Similarly, TxDOT Researh Report 381-1, Time Dependant Deletions o Pretensioned Beams, 5 (a report o TxDOT Researh Study ) proposes a amber predition proedure that would beneit rom a more reed haraterization o the stantaneous and time dependant behavior o onrete used the prodution o prestressed onrete bridge beams rom disparate loations Texas. Reent researh has also highlighted the relationship between aggregate properties a onrete mix and the modulus o elastiity o the onrete. 6 For this reason, the authors have submitted to TxDOT s Researh and Tehnology Implementation oie a researh problem statement to ailitate the implementation o the new AASHTO LRFD prestress loss provisions and state-o-the-art amber predition to TxDOT prestressed onrete girder design sotware. The dgs o the NCHRP Report 496, as implemented the 2005 Interim revisions to the AASHTO LRFD Speiiations, however, has removed rom the ormulae the ators or the onsideration o loal material properties. The disussion here is thereore a review o the prestress loss alulation proedure rom NCHRP Report 496 as it will appear the Interim AASTHO LRFD Speiiations, not neessarily as it appeared the report. Results rom TxDOT s pendg researh study ould signiiantly alter the author s dgs, onlusions, and reommendations with regard to implementation the state o Texas. Comparison o Prestress Loss Calulation Methods The dierenes between the AASHTO LRFD Speiiations and the NCHRP Report 496 are detailed Table 1 through Table 9, with the LRFD Speiiations provisions shown on the let and the orrespondg NCHRP Report 496 proedures on the right eah table. Followg is a disussion o these dierenes by prestress loss omponent. Creep and Shrage A quote rom a prestressed onrete text boo is useul to illustrate the overall magnitude o reep and shrage losses omparison with jag stress and eetive stress prestressg tendons: Ater beg preompressed, onrete ontues to shorten with time a proess alled reep. Loss o moisture with time also auses shorteng o the onrete by shrage. Creep and shrage together may ause a total shorteng o the onrete o nearly 1 part per thousand High-strength steel wire an be elongated about 7 parts per thousand the prestressg operation. ven with a loss o 1 part per thousand, six-sevenths o the prestressg 5 Kelly, Bradberry, and Breen: Time-Dependent Deletions o Pretensioned Beams. (August 1987) 6 Myers, and Carrasquillo: Prodution and Quality Control o High Perormane Conrete Texas Bridge Strutures. (1999) 4

5 would rema. 7 It is this type o appreiation or relative magnitudes that an be lost when the omplexity o engeerg alulations ditate their automation. Though the design engeer maes use o the omputer to perorm both simple and omplex alulations eiiently, suh an appreiation or relative magnitudes o loss omponents should be ultivated, rather than shunned. Creep Creep is deed as the time-dependent deormation o a material due to sustaed stress. Creep (plasti low) onrete is aeted by a number o ators ludg, the magnitude o the sustaed stress, the age o the onrete upon appliation o the reep dug stress, ambient temperature and relatively humidity o the onrete, the water-ement ratio (w/), the aggregate-ement ratio, the type and eness o ement, the size and shape o speimen, et. Table 1 - Creep Coeiient Calulations Comparison AASHTO LRFD Speiiations NCHRP Report 496 ψ 0.6 H ( t ti ) ( t, ti ) = ti ( t ti ) where: ( ) t 0.36( V / S) 0.54( V / S) + = e e t t t (C ) 1 = ' ( ) t = maturity o onrete (days) t i = age o the onrete when load is itially applied (days) where: ψ ( t, ti ) = 1.95vsh td ti t 0.36( V / S) 0.54( V / S) + = e e t vs t t or vs = ( V / S) 1.0 h = H td t = 61 4 ' i + t 5 = 1 + ' i t = maturity o onrete (days) t i = age o the onrete when load is itially applied (days) Unlie many o the loss alulations, the estimation o the reep oeiient has not signiiantly reased omplexity rom the LRFD Speiiations to the NCHRP Report 496 proedures. A simpliied equation or the eet o the volume-to-surae ratio has even been luded. However, stead o simply alulatg a sgle ultimate reep oeiient, the design 7 Colls, and Mithell: Prestressed Conrete Strutures, p. 3 5

6 engeer will need to alulate the oeiient or several dierent situations at de plaement relative to transer, at al relative to transer, or al relative to de plaement, and or the de. The reep oeiient alulation proedures are ompared Table 1. Table 2 - Shrage Stra Calulation Comparison AASHTO LRFD Speiiations NCHRP Report 496 A) For moist ured onrete ε sh = s h t t ( ) B) For steam-ured onrete where: ε sh t = dryg time (days) t 3 = sh t ( ) t 0.36( V / S) = ( V / S e t s t t (C ) or H < 80% 140 H h = 70 or H 80% (C ) 3(100 H ) h = 70 (C ) 3 ) ε 3 sh = vshs td I the onrete is exposed to dryg beore ive days o urg have elapsed, the shrage as diated above should be redued by 20 perent. where: t = dryg time (days) t 0.36( V / S) = ( V / S) e t vs t t or vs = ( V / S) 1.0 hs = H td 5 = 1 + ' i = 61 Shrage Lie reep, ultimate shrage and the resultg loss o prestress is aeted by everythg rom the ambient relative humidity, to the onrete temperature history, to the w/ ratio, to the type, size, quality and gradation o the aggregate, to the volume-to-surae ratio, et. The amount o shrage varies widely, dependg on the dividual onditions. For the purpose o design, an average value o shrage stra would be to Furthermore, the rate o shrage is highly dependant on weather onditions, with very dry onditions aeleratg t 4 ' i + t 8 L, and Burns: Design o Prestressed Conrete Strutures (1981), p. 47 6

7 shrage suh that ultimate shrage is ahieved a ew months, and with wet onditions stoopg or reversg shrage altogether. 9 Similar to the reep oeiient alulations, the shrage stra ormula has not signiiantly reased omplexity. However, there is the ompliation o onsiderg the eets o the shrage o de onrete, a onsideration not made the LRFD Speiiations. Shrage stra alulation proedures are ompared Table 2. Perspetive on Creep and Shrage Losses Considerg all the unertaties herent the predition o reep and shrage o onrete and assoiated prestress losses, as disussed this setion, it is impossible or the design engeer to use reep and shrage ormulae to predit the atual reep and shrage loss, se s/he has little or no ontrol over the onstituent mae-up o the onrete, over the loadg history o the prestressed onrete element beg designed, and over the environment whih the prestressed onrete element is abriated and lives out its servie lie. This at should always rema the md o the design engeer. Modulus o lastiity The modulus o elastiity (MO) o the onrete is a ator the alulation o elasti losses and gas as well as the alulations o reep, shrage, and relaxation losses. Hene, its magnitude has wide rangg eets on prestress losses. Lie reep and shrage, the predition o MO is wrought with unertaty. The only hange the predition o the MO luded the NCHRP Report 496 revisions is the lusion o a orretion ator, K 1, whih an be used to adjust the MO to address dierenes due to aggregate soure, available MO test results, ield experiene, or other more aurate ormation when available. Similar multipliers were luded the NCHRP Report 496, as well as a multipliers to establish upper and lower bounds, or both reep and shrage. These have been removed or the implementation o the proedures the 2005 Interim Speiiations. Table 3 - Modulus o lastiity Calulation Comparison AASHTO LRFD Speiiations NCHRP Report 496 = 33,000w 1.5 ' = 33,000K w ' ( ) lasti Losses The elasti loss as it is the urrent LRFD Speiiations is a airly simple alulation, that only requires the engeer to mae what is essentially an estimate o the elasti loss and then to he to see i the estimate was orret, or at least lose enough. The equation remas unhanged the NCHRP Report 496 proedures. However, verbiage is added to desribe the speial treatment o elasti losses when transormed setion properties are used the analysis. The authors believe that the tent o addg this verbiage is to warn the designer not to mae the error o onsiderg the elasti loss twie when transormed setion properties are used. It also 9 L, and Burns: Design o Prestressed Conrete Strutures (1981), p

8 seems to give redene to, or suggest that the engeer use, transormed setion properties beause o the ease with whih the elasti losses an be determed ( no iteration required ). However, the previous method, whih required the engeer to estimate the loss and then iterate until suiient auray is ahieved, probably requires less eort than determg the transormed setion properties, thus redug or elimatg any omputational beneit that may have been realized. This is disussed more detail the ollowg setion. Table 4 - lasti Loss quations AASHTO LRFD Speiiations NCHRP Report 496 p p ps = gp ps = gp i ( a-1) i Transormed Setion Analysis One o the hanges that seems reasonable enough, but is liely to ause a great deal o onusion as it represents a major hange the beam stress analysis ideology, is the use o transormed setion properties the beam analysis, reerred to here as transormed setion analysis. One beneit o transormed setion analysis is that no iteration is required to determe elasti losses or gas. The prestressg ore an be oneptualized as externally applied ompression that the strands resist, rather than an ternal ore that is transerred rom the strands to the onrete. I it is oneptualized this ashion and the strands are transormed to an equivalent area o onrete (i.e. a onrete area equal stess to the steel area), then stra ompatibility between the strands and the onrete is automatially satisied. The mathematial explanation or this is that the derivation o the ormula or alulatg transer o prestress or a transormed setion assumes ompatibility o stras between the prestressg steel and the onrete suh that the prestress ore beore transer is utilized the alulation rather than the ore ater transer. I stead the engeer oneptualizes the prestressg proess as the strands transerrg their ore to the net or gross onrete setion, as is typially done, iteration is needed to ensure stra ompatibility. This is beause the prestress ore dereases as the onrete stras, and as the prestress ore dereases the stra the onrete is less, iteration is required to d the pot where these two stras balane. The same proess is required to ensure stra ompatibility or elasti gas. The elasti loss and gas are determed the same way or both methods, it is simply the stress at the level o the strands times the modular ratio between the strands and the onrete. One dierene between the net and transormed setion analysis is that the ore the strands used to determe the stress the onrete will not be redued or reased due to elasti eets when perormg a transormed setion analysis. These hanges must be luded, however, when perormg a net or gross setion analysis. To support this statement, Table 5 ontas alulations or elasti loss usg both net and transormed setion properties or omparison. Net setion properties are used to ensure that the same results are obtaed both examples. The results usg gross setion properties are also luded or omparison. 8

9 Table 5 - lasti Loss Calulation with Net and Transormed Setion Properties Setion Properties Gross A n = I n = e CL = Net A n_n = I n_n = e CL_n = Transormed A n_ti = I n_ti = e CL_ti = Net Setion Other Knowns M Bm = ps = 28000si i = 3834si ip t NumStrands = 20 Guess itial prestressg stress ater transer: pr1 = 1.548si pj = 202.5si A ps = Low relaxation strands, t = 12 hrs Transormed Setion Stress strands just beore transer: pat := si P := NumStrands pat pat = 189si A ps P = 578.3ip pbt := pj pr1 P := pbt = 201.0si NumStrands pbt A ps P = 614.9ip P gp := A n_n + 2 Pe CL_n I n_n M Bm e CL_n I n_n P gp := A n_ti + 2 Pe CL_ti I n_ti M Bm e CL_ti I n_ti gp = 2.135si ps ps := (LRFD a) gp i ps = 15.59si pat := pj ps pr1 pat = 185.4si Not equal to itial guess, must repeat ps = 15.22si gp = 2.085si ps := While the lasti Loss has been determed, it must not be luded the prestressg ore applied to the transormed setion, when determg onrete stresses due to prestress. pat := pbt gp = 2.089si ps gp i (LRFD a) ps = 15.26si pat = 201.0si pat = 185.7si ps = 15.26si gp = 2.089si Closer, repeat until onverge with 0.1 si gp = 2.090si ps = 15.26si pat = 185.7si Converged pat = 201.0si Note: Gross Setion analysis yeilds the ollowg results ps = 15.08si gp = 2.064si pat = 185.9si 9

10 lasti Gas The NCHRP Report 496 method allows or the diret lusion o elasti gas due to superimposed dead loads and live loads. Aountg or the elasti gas allows the engeer to rely on onsiderably redued losses, whih ould redue the number o strands required or a given beam. These gas have onservatively been negleted the past and the wordg o the revised speiiations will allow engeers to ontue this pratie should they so hoose. The lusion o elasti gas also requires that at eah termediate stage o loadg or stress he only realized elasti gas may be onsidered. In other words, the engeer must exerise due diligene to sure that only the gas resultg rom loads applied at the stage onsidered be relied upon. Thus, eah load ase onsidered will have its unique assoiated prestress ore. This is a signiiant departure rom urrent pratie. Shrage Loss As shown Table 6, the shrage loss estimation the AASTHO LRFD Speiiations is a very simple equation that is a untion only o relative humidity. The new equation presented the NCHRP Report 496 may appear just as simple at irst glane, but it is now a untion o the shrage stras, whih are turn untions o time, beam material properties, beam ross setion properties, and relative humidity. This signiiantly reases the amount o omputation, and it requires the designer to mae estimates or the age o the beam at release, de plaement, and at the end o servie. Most o the rease omplexity is hidden a new variable, reerred to as the transormed setion oeiient. This variable must be alulated or two dierent time steps, and it is a untion o time, beam material properties, beam ross setion properties, area o prestressg steel, and reep oeiients. These equations are shown Table 6. Table 6 - Shrage Loss Calulations AASHTO LRFD Speiiations NCHRP Report 496 Transer to de plaement psr = H = ε K ( ) psr bid where: 1 Kid = p A ps Age pg i A g I g De plaement to al 2 p id ( ψ b ( t, t )) i where: K d = 1 + p i psd = ε bd 1 A ps A e 1 + A I 2 p p K d ( ψ b ( t, t )) i 10

11 These omplex equations, relyg on variables that are mostly unnown at design time, ould add signiiant omplexity to the design o prestressed onrete beams. Whereas the prestress loss the AASHTO LRFD equations are, rom a design standpot, simply a untion o the number o strands and the onrete strength at transer, these new equations orporate a dependene on the al onrete strength. This adds a new layer o iteration to the design proess. Currently the last variable determed the design o prestressed onrete beams would typially be the al onrete strength required. I the al onrete strength required must be reased, then the losses must be realulated, whih ould hange the number o strands required, puttg the design o the beam ba to the very begng. Creep Loss The AASHTO LRFD Speiiations had a simple method to determe the reep loss, whih was only a untion o the stress at the enter o strands due to loads and prestress. Inherent these equations are the reep oeiients and the itial and al modular ratios. The method employed the NCHRP Report 496, as shown Table 7, maes the reep loss a untion o these values. The reep loss equation is very similar to the shrage equation disussed above, and thereore suers rom the same problems and ompliations. It is also worth notg that the vestigations disussed here, the dierene prestressed loss due to reep represented the largest dierene between the AASHTO LRFD Speiiation and the NCHRP Report 496. This dierene beame more pronouned as the number o strands reased. Table 7 - Creep Loss Calulations AASHTO LRFD Speiiations NCHRP Report 496 pcr = gp ( ) dp Transer to De Plaement p pcr = gpψ b ( td, ti ) Kid i De Plaement to Fal ( ψ b ( t, ti ) ψ b ( td, ti ) K d p pcd = gp ) i + p ψ ( t, t ) K 0 d b d d Relaxation The relaxation loss is typially the smallest omponent o the total loss, and an area where one might be willg to trade auray or simpliity, although the relaxation loss equation the urrent AASHTO speiiations is already quite simple. The relaxation loss equations the NCHRP Report 496 do deed trade o auray or simpliity. There is one equation and it is not a untion o time, nor is it tied to the eets o reep, shrage, or elasti losses and gas onsidered. One problem with the new relaxation loss provisions is the removal o the equation used to determe the trsi relaxation losses between itial stressg and transer. Intrsi relaxation is a logarithmi untion, and as suh a majority o the relaxation losses our 11

12 beore transer. For a number o years no guidane on how to alulate this loss has been provided the AASHTO Speiiations, but there are now equations the AASHTO LRFD Speiiations that give guidane. The proposed revisions unortunately remove this guidane rom the Speiiations. Shrage o De Conrete The ga prestress due to shrage o de onrete has not been aounted or previous AASHTO Speiiations, though it may seem rational to aount or this. Its eet however is quite mimal (less than a 1% rease) but the alulations required to determe this ga are not trivial. Table 8 - Relaxation Loss Calulations AASHTO LRFD Speiiations NCHRP Report 496 From Initial Stressg to Transer A) or stress-relieved (SR) strands: log(24.0t) pj pr1 = py ( b-1) B) or low-relaxation (LR) strands: log(24.0t) pj pr1 = pj 40.0 py ( b-1) where: t = estimated time rom stressg to transer (days) From Transer to Fal (LRFD) or stress-relieved strands ( ): = ( pr2 ps psr pcr or low-relaxation strands: = ( pj + + pr2 ps psr pcr ) ) From Initial Stressg to Transer No guidane given From Transer to De Plaement where: pt pt = pr1 K L py (NCHRP Report 496) pt = stress prestressg strands immediately ater transer, not to be taen less than 0.55 py this equation From De Plaement to Fal = pr2 pr1 xample Prestress Losses Analysis The authors seleted prestressed onrete beams rom a bridge urrently the design stage to use to illustrate and ompare the prestress loss alulations. They hose these beams beause they represent a range o onrete strengths and levels o prestressg. They evaluated three dierent beams usg both o the dierent loss methods and both gross and transormed setion properties. The three beams are Texas Department o Transportation standard type C beams, with spans o 60, 75, and 90 t. Type C beam properties and strand layout details are shown Figure 1, and the details o the three beam designs are shown Table

13 Table 9 - Prestress Loss Due to Shrage o De Conrete AASHTO LRFD Speiiations NCHRP Report 496 p Not onsidered = K ( + 0.7ψ ( t, t )) pss d d 1 b d where: d = hange onrete stress at entroid o prestressg strands due to shrage o de onrete d ε dd Ad = ( ψ ( t d d 1 e ped +, td )) A I Beam Type A B C D F G H W Yt Yb Area 2 I 4 Wt/L lbs C ½ 16 3 ½ , Figure 1 - Type C Beam 13

14 Table 10 - xample Beams Span eet No. o Strands e CL hes e nd hes ' i psi ' psi 60' ' ' These beams were designed aordane with the AASHTO LRFD Speiiation. Both the LRFD Speiiations method and the NCHRP Report 496 method were used to determe the losses or these ixed beam designs order to provide a diret omparison, even though it is liely that the beam designs would be dierent i the NCHRP Report 496 method were used their design. Results Comparison The authors employed both the urrent AASHTO LRFD provisions and the NCHRP Report 496 proedures, onsiderg gross and transormed setion properties, to determe the total loss eah o the three beam designs. The resultg our ombations o prestress loss methodology and type o setion properties are shown below along with justiiation or eah identiied ombation: 1) AASHTO LRFD usg gross setion properties This is the urrent method ommonly used or determg prestressed losses. 2) AASHTO LRFD usg transormed setion properties This was done to see the eet o usg transormed setion properties while orretly ludg the elasti shorteng the prestress loss alulations. lasti loss o prestress ertaly does our when analyzg a beam usg transormed setion properties, but the subtration o suh loss rom the applied prestress ore and the subsequent appliation o that ore to the beam setion to alulate onrete stresses aounts or elasti shorteng twie, whih is not orret. 3) NCHRP Report 496 usg gross setion properties This will most liely be the most ommon implementation o this method one it is orporated to the speiiations 4) NCHRP Report 496 usg transormed setion properties This was the tent o the revised speiiations as outled the NCHRP Report 496. Net setion properties were used where appropriate (time-dependant losses, and order to exlude elasti gas due to superimposed dead and live loads). For the most part the dierenes the total losses alulated usg the dierent methods is relatively small, rangg rom about 5% to 25%. This is, however, only i the elasti gas are negleted. I the elasti gas due to superimposed dead loads and live load are luded as the NCHRP Report 496 method allows, the dierene beomes muh more signiiant, rangg rom about 25% to 50% less than that predited with the AASHTO LRFD ormulae. The largest redution prestress loss, rom the LRFD method to the NCHRP method, ourred the longest span with the greatest number o strands and greatest onrete strength, as shown Figure 2. The majority o this redution is due to a redution the reep loss, partiularly or the member with the most prestressg strands. Where the reep loss predited usg the LRFD method reases as the number o strands reases, the reep loss predited by the NCHRP method stays relatively the same, but atually dereases slightly. 14

15 Total Fal Prestress Loss Prestress Loss (si) LRFD (Gross) LRFD (Trans.) NCHRP (Gross) NCHRP (Trans.) NCHRP (Gross)* NCHRP (Trans.)* *Considers elasti gas Span Length (t) Figure 2 - Total Fal Prestress Loss Comparison

16 Fal Bottom Fiber Stress at Midspan Due to Servie Loads and Prestress Stress (si) LRFD (Gross) LRFD (Trans.) NCHRP (Gross) NCHRP (Net/Trans.) NCHRP (Gross)* NCHRP (Net/Trans.)* *Considers elasti gas Span Length (t) Figure 3 - Bottom Fiber Stress Due to Servie Loads and Prestress at Midspan

17 The losses estimated usg the LRFD method reased as the number o strands and onrete strength reased, while the NCHRP loss alulations estimated approximately the same loss or all three beams. Figure 2 also shows that when used orretly, transormed setion analysis and gross setion analysis produe essentially the same prestress loss. Any rease prestressg ore, by either reased number o strands or a redution losses, results redued bottom iber tension. Figure 3 shows the bottom iber tension at midspan due to ull servie loads (ludg 100% o the HL-93 load). This tension, that oten ontrols the number o strands, is redued by up to 50% when usg the NCHRP method losses. An even greater redution, up to 90%, is seen when the elasti gas are luded. These redutions loss, when used design, ould substantially redue the number o strands required, espeially when the elasti gas are luded. The pattern that appears usg the somewhat limited vestigation o three beams is that longer beams will have ewer strands and shorter beams will require more usg the new loss estimation proedures. This igure also shows that the dierene between a gross setion analysis and its orrespondg transormed or net setion analysis are at negligible, when the elasti deormations eet on the loss is handled properly. The le representg the LRFD Speiiations results usg transormed setion properties shows the eet o improperly aountg or elasti shorteng when usg transormed setions. CONCLUSIONS AND RCOMMNDATIONS Se these new loss provisions have been approved by the AASHTO SCOBS or orporation the 2005 Interims to the AASHTO LRFD Speiiations, a reommendation on whether or not this new loss method should be used would be irrelevant. However, reommendations whih guide the design engeer the appliation o the new prestress losses proedures are useul; the ollowg reommendations are thereore oered: Contued neglet o elasti gas: It is not reommended that the design engeer tae advantage o the elasti gas due to superimposed dead and live loads. This reates a substantial redution the loss ompared to equations that have been used or many years without problems. Further researh should be done, usg beams designed with these new loss alulations beore engeers should routely tae advantage o this their loss alulations. Contued use o gross setion properties: The use o transormed setion properties the analysis is not reommended, due to its reased omplexity and the onusion that is herent with the nuanes o its use. Any small beneit that may be obtaed by the reased auray is easily oset by the potential or mistaes and the la o understandg that is liely to result. Additionally, the use o transormed setions the analysis typially preludes the ability to neglet the elasti gas as reommended above. Implementation usg nown material properties: I more speii ormation about the onrete material properties is available durg the design stage, the prestressed loss method as outled the NCHRP Report 496, ludg the adjustment ators or properties suh as reep, shrage, and modulus o elastiity, may be used. The loss predition method as outled the report provides a good ramewor or tag advantage o this ormation when available. However, the reommendations above would still stand. Though it is admirable that the researhers and authors o NCHRP Report 496 are pursug more aurate loss alulation proedures that may, i enough ormation is available at design time, allow bridge design engeers to reate more eiient designs, suh auray is 17

18 not somethg that lends itsel to route bridge design. There are a large number o variables that are unnown at design time, but that nevertheless have a substantial eet on the prestress losses. The modulus o elastiity o the onrete, the urg onditions, and the atual onrete strength are typially not nown with any level o auray when the losses are alulated. The at that these variables are not urrently well nown at design time should not prelude eorts to obta a better understandg o these variables, suh as the onrete modulus o elastiity and reep properties, whih have suh a signiiant eet on the behavior and design o prestressed onrete strutural members. Design engeers and design speiiations writers may be well served to reognize that the preision that is diated the omplexity o the prestress loss alulations should be ept le with the degree o nowledge o the material properties whih the alulations are dependant upon. RFRNCS Amerian Assoiation o State Highway and Transportation Oiials, AASHTO-LRFD Bridge Design Speiiations, Third dition, Washgton, DC (2004) Colls, M. P. and Mithell, D., Prestressed Conrete Strutures, Response Publiations, Canada (1997), 765 pp. Kelly, D.J., Bradberry, T.., and Breen, J.., Time-Dependent Deletions o Pretensioned Beams. Researh Report 381-1, Researh Projet , Center or Transportation Researh, Bureau o ngeerg Researh, The University o Texas at Aust, Aust, TX (August 1987) 211 pp. L, T.Y. and Burns, N.H., Design o Prestressed Conrete Strutures, John Wiley and Sons, 3rd dition (1981), 646 pp. Myers, J. J., and Carrasquillo, R. L., Prodution and Quality Control o High Perormane Conrete Texas Bridge Strutures. Researh Report 580/589-1, Center or Transportation Researh, University o Texas, Aust, TX (1999) 176 pp. Tadros, M. K., Al-Omaishi, N., Seguirant, S. J., and Gallt, J. G. Prestress Losses Pretensioned High-Strength Conrete Bridge Girders. NCHRP Report 496, Transportation Researh Board, Washgton, DC, (2003) 63 pp. APPNDIX NOTATION A = net area o the omposite setion with the slab transormed to beam onrete usg the slab-to-beam modular ratio A d = area o de onrete A g = gross area o the beam d = modulus o elastiity o de onrete i = modulus o elastiity o onrete at transer (si) t = modulus o elastiity o onrete at transer or time o load appliation (si) p = modulus o elastiity o prestressg steel (si) 18

19 e d = eentriity o de with respet to the net omposite setion with transormed de, taen negative ommon onstrution e p = eentriity o strands with respet to entroid o omposite setion e pg = eentriity o strands with respet to entroid o girder ' = ompressive strength o onrete at 28 days gp (LRFD) = sum o onrete stresses at the enter o gravity o prestressg tendons due to the prestressg ore at transer and the sel-weight o the member at the setions o maximum moment (si) gp (NCHRP) = sum o onrete stresses at the enter o gravity o prestressg tendons due to the prestressg ore immediately ater transer and the sel-weight o the member at the setions o maximum moment (si). Also, hange o onrete stress at the enter o gravity o prestressg tendons due to subsequent applied loads, when onsidered. ' i = ompressive strength o onrete at time o prestressg pj = itial stress the tendon at the end o stressg (si) py = speiied yield strength o prestressg steel (si) H = relative humidity (%) I = moment o ertia o net omposite setion with the slab transormed to beam onrete usg the slab-to-beam modular ratio K 1 = orretion ator or soure aggregate to be taen as 1.0 unless determed by physial test, and as approved by authority o jurisdition K d = transormed setion oeiient that aounts or time-dependent teration between onrete and bonded steel the setion beg onsidered or the time period between de plaement and al time K id = transormed setion oeiient that aounts or time-dependent teration between onrete and bonded steel the setion beg onsidered or the time period between transer and de plaement K L = ator or strand type, 30 or low-relaxation strands and 7 or other prestressg steel, unless more aurate manuaturer s data are available = ator or the eet o the volume-to-surae ratio = ator or the eet o onrete strength h = humidity ator h = humidity ator or reep hs = humidity ator or shrage s = size ator 19

20 td = time development ator vs = ator or the eet o the volume-to-surae ratio o the omponent t d = age at de plaement (days) t = al age (days) V / S = volume-to-surae-ratio w = unit weight o onrete (ip/t 3 ) d = hange onrete stress at entroid o prestressg strands due to longterm losses between transer and de plaement, ombed with de weight and superimposed loads d = hange onrete stress at entroid o prestressg strands due to shrage o de onrete psr = prestress loss due to shrage o girder onrete between transer and de plaement pcd = prestress loss due to reep o girder onrete between time o de plaement and al time pcr = prestress loss due to reep o girder onrete between transer and de plaement ps (LRFD) = loss due to elasti shorteng ps (NCHRP) = sum o all losses and gas due to elasti deormations aused by applied loads and prestress plt = prestress loss due to long-term shrage and reep o onrete, and relaxation o prestressg steel pr1 (LRFD) = prestress loss due to relaxation o prestressg strands between itial stressg o the strands and transer pr1 (NCHRP) = prestress loss due to relaxation o prestressg strands between time o transer and de plaement pr2 (LRFD) = prestress loss due to relaxation o prestressg strands rom transer to al time pr2 (NCHRP) = prestress loss due to relaxation o prestressg strands omposite setion between time o de plaement and al time psd = prestress loss due to shrage between de plaement and al time psr = prestress loss due to shrage between transer and de plaement pss = prestress loss due to shrage o omposite de pt = total prestress loss 20

21 ε bd = onrete shrage stra o girder between time o de plaement and al time ε bid = onrete shrage stra o girder between time o transer and de plaement ε dd = shrage stra o de onrete between plaement and al time ψ b( td, ti) = girder reep oeiient at the time o de plaement due to loadg trodued at transer ψ t, t ) = girder reep oeiient at al time due to loadg trodued at de b( d plaement ψ t, t ) = girder reep oeiient at al time due to loadg trodued at transer b ( i 21