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STRUCTURAL SYSTEMS RESEARCH PROJECT Report No. SSRP– 06/ 21 A SIMPLIFIED METHOD FOR PREDICTION OF LONG TERM PRESTRESS LOSS IN POSTTENSIONED CONCRETE BRIDGES by SAMER YOUAKIM VISTASP M. KARBHARI Final Report Submitted to the California Department of Transportation Under Contract No. 59A0337. July 2006 Department of Structural Engineering University of California, San Diego La Jolla, California 92093 0085 University of California, San Diego Department of Structural Engineering Structural Systems Research Project Report No. SSRP– 06/ 21 A Simplified Method for Prediction of Long Term Prestress Loss in Post Tensioned Concrete Bridges by Samer Youakim Assistant Project Scientist Vistasp M. Karbhari Professor of Structural Engineering Final Report Submitted to the California Department of Transportation Under Contract No. 59A0337 Department of Structural Engineering University of California, San Diego La Jolla, California 92093 0085 July 2006 Technical Report Documentation Page 1. Report No. 2. Government Accession No. 3. Recipient’s Catalog No. 4. Title and Subtitle A Simplified Method for Prediction of Long Term Prestress Loss in Post Tensioned Concrete Bridges 5. Report Date July 2006 6. Performing Organization Code 7. Author( s) Samer Youakim and Vistasp M. Karbhari 8. Performing Organization Report No. UCSD / SSRP 06/ 21 9. Performing Organization Name and Address Department of Structural Engineering School of Engineering 10. Work Unit No. ( TRAIS) University of California, San Diego La Jolla, California 92093 0085 11. Contract or Grant No. 59A0337 12. Sponsoring Agency Name and Address California Department of Transportation 13. Type of Report and Period Covered Final Report Engineering Service Center 1801 30th St., West Building MS 9 Sacramento, California 95807 14. Sponsoring Agency Code 15. Supplementary Notes Prepared in cooperation with the State of California Department of Transportation. 16. Abstract Creep and shrinkage of concrete and relaxation of prestressing steel cause time dependent changes in the stresses and strains of concrete structures. These changes result in continuous reduction in the concrete compression stresses and in the tension in prestressing steel. Reasonably accurate estimate of the long term prestress losses are needed to avoid any serviceability problems of the structure ( due to cracking and/ or excessive deflection). An analytical method is presented to predict the long term prestress losses in continuous cast in place post tensioned bridges. The method is based on the basic principles of solid mechanics and satisfies the requirements of equilibrium and compatibility of the bridge cross section. The proposed method for a section in concrete girder reduces to a single equation with three coefficients, which are functions of the modulus of elasticity and creep coefficient of concrete, location and amount of prestressing and non prestressed steel, and geometry of the cross section. To expedite the use of the method and to make it more appealing to practicing engineers, design aids are provided to estimate these three coefficients. The method is further extended to continuous bridge girders by using the force method to calculate the change in connecting moments at intermediate supports and hence the increase or decrease in prestressing losses. The predictions of the proposed method are compared with the current provisions of design standards and codes of practice. It is shown that the present empirical equations of the bridge standards can overly underestimate or overestimate the long term prestress losses, depending on the concrete creep and shrinkage properties as well as prestressing and non prestressed steel ratios. 17. Key Words Prestress loss, creep, shrinkage 18. Distribution Statement No restrictions 19. Security Classification ( of this report) Unclassified 20. Security Classification ( of this page) Unclassified 21. No. of Pages 72 22. Price Form DOT F 1700.7 ( 8 72) Reproduction of completed page authorized Disclaimer The contents of this report reflect the views of the authors who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California. This report does not constitute a standard, specification or regulation. 1 i Abstract Creep and shrinkage of concrete and relaxation of prestressing steel cause timedependent changes in the stresses and strains of concrete structures. These changes result in continuous reduction in the concrete compression stresses and in the tension in prestressing steel. Reasonably accurate estimate of the long term prestress losses are needed to avoid any serviceability problems of the structure ( due to cracking and/ or excessive deflection). An analytical method is presented to predict the long term prestress losses in continuous cast in place post tensioned bridges. The method is based on the basic principles of solid mechanics and satisfies the requirements of equilibrium and compatibility of the bridge cross section. It is assumed that prestressing and dead load are applied at the same instant, shrinkage starts to take place at the application of loads, and one concrete type for the cross section. The proposed method for a section in concrete girder reduces to a single equation with three coefficients, which are functions of the modulus of elasticity and creep coefficient of concrete, location and amount of prestressing and non prestressed steel, and geometry of the cross section. To expedite the use of the method and to make it more appealing to practicing engineers, design aids are provided to estimate these three coefficients. The method is further extended to continuous bridge girders by using the force method to calculate the change in connecting moments at intermediate supports and hence the increase or decrease in prestressing losses. The predictions of the proposed method are compared with the current provisions of design standards and codes of practice. It is shown that the present empirical equations of the bridge standards can overly underestimate or overestimate the long term prestress losses, depending on the concrete creep and shrinkage properties as well as prestressing and non prestressed steel ratios. The proposed method is applied to a number of continuous post tensioned concrete bridges currently under construction in San Diego County. It was found that the prestressing losses due to continuity could be ignored without affecting the accuracy of the method. ii Table of Contents Abstract....................................................................................................................... ....... i Table of Contents .............................................................................................................. ii List of Figures................................................................................................................... iii List of Tables ..................................................................................................................... v List of Symbols ................................................................................................................. vi 1. Introduction................................................................................................................... 1 2. Long term Material Properties ................................................................................... 2 2.1 Shrinkage of Concrete............................................................................................... 2 2.2 Creep of Concrete ..................................................................................................... 2 2.3 Relaxation of Prestressing Steel................................................................................ 4 3. Provisions of Bridge Codes for Long term Prestress Losses .................................... 6 3.1 AASHTO LRFD Refined Method5 .......................................................................... 6 3.2 AASHTO LRFD Approximate Method5.................................................................. 6 3.3 CEB FIP Model Code6 ............................................................................................. 7 3.4 Canadian Highway Bridge Design Code7 ( CHBDC) ............................................... 7 4. Proposed Method of Analysis ...................................................................................... 8 4.1 Assumptions.............................................................................................................. 8 4.2 Sign Convention........................................................................................................ 8 4.3 Steps of Analysis....................................................................................................... 8 4.4 Derivation of Method of Analysis .......................................................................... 10 4.5 Prestress Loss Due to Relaxation............................................................................ 14 4.6 Design Aids............................................................................................................. 15 4.7 Effect of Continuity ................................................................................................ 18 5. Comparisons with Bridge Design Specifications...................................................... 21 5.1 AASHTO LRFD Refined and Approximate Methods ........................................... 21 5.2 CEB FIP Model Code............................................................................................. 29 5.3 Canadian Highway Bridge Design Code ( CHBDC)............................................... 31 6. Examples...................................................................................................................... 34 6.1 Lake Hodges Bridge ............................................................................................... 34 6.2 Duenda Road Overcrossing .................................................................................... 37 7. Conclusions.................................................................................................................. 40 References..................................................................................................................... .. 41 Appendix A: Coefficients kA , kI and kh ...................................................................... 43 Appendix B: Prestress Loss Due to Continuity............................................................ 63 iii List of Figures Fig. 2.1 Development of concrete shrinkage with time. ..................................................... 2 Fig. 2.2 Development of concrete strains with time due to: ( a) stress applied at time t0 and sustained to a later time t; and ( b) stress applied gradually from t0 to t...................... 3 Fig. 4.1 Four steps for the analysis of time dependent effects ( after Ghali et al. 2002)..... 9 Fig. 4.2 Typical strain distribution in a bridge girder at transfer...................................... 12 Fig. 4.3 Geometric dimensions and reinforcement in a typical bridge cross section. ...... 15 Fig. 4.4 Two span continuous prestressed beam. ............................................................. 18 Fig. 4.5 Locations of integration points ( sections) in a two span beam. .......................... 18 Fig. 4.6 Released structure and coordinate system for a two span beam. ........................ 19 Fig. 4.7 Moment diagram due to unit value of connecting moment................................. 19 Fig. 5.1 Assumed stress and strain profiles at time of transfer. ........................................ 22 Fig. 5.2 Comparison between proposed method and AASHTO LRFD refined method for long term prestress losses due to creep: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ............... 24 Fig. 5.3 Comparison between proposed method and AASHTO LRFD refined method for long term prestress losses due to shrinkage: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ........ 26 Fig. 5.4 Comparison between proposed method and AASHTO LRFD refined and approximate methods for total prestress losses: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ... 28 Fig. 5.6 Comparison between proposed method and CEB FIP method for long term prestress losses due to shrinkage ( ρ ps = 0.8%)................................................................ 30 Fig. 5.7 Comparison between proposed method and CEB FIP method for total long term prestress losses ( ρ ps = 0.8%). .......................................................................................... 31 Fig. 5.8 Comparison between proposed method and CHBDC method for long term prestress losses due to creep ( ρ ps = 0.8%)....................................................................... 32 Fig. 5.9 Comparison between proposed method and CHBDC method for long term prestress losses due to shrinkage ( .................................................................................... 33 Fig. 5.10 Comparison between proposed method and CHBDC method for total long term prestress losses ( ρ ps = 0.8%). .......................................................................................... 33 Fig. 6.1 Lake Hodges Bridge: ( a) Half elevation; and ( b) Half cross section................... 35 Fig. 6.2 Duenda Road Overcrossing: ( a) Elevation; and ( b) Cross section. ..................... 38 Fig. A. 1 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. ...... 45 Fig. A. 2 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. ...... 46 Fig. A. 3 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. ...... 47 Fig. A. 4 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. ....... 48 Fig. A. 5 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. ....... 49 iv Fig. A. 6 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. ....... 50 Fig. A. 7 kA , kI and kh for the case ΣBw B = 0.1, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. ....... 51 Fig. A. 8 kA , kI and kh for the case ΣBw B = 0.2, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. ....... 52 Fig. A. 9 kA , kI and kh for the case ΣBw B = 0.3, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. ....... 53 Fig. A. 10 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%...... 54 Fig. A. 11 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%...... 55 Fig. A. 12 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%...... 56 Fig. A. 13 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%....... 57 Fig. A. 14 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%....... 58 Fig. A. 15 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%....... 59 Fig. A. 16 kA , kI and kh for the case ΣBw B = 0.1, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%....... 60 Fig. A. 17 kA , kI and kh for the case ΣBw B = 0.2, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%....... 61 Fig. A. 18 kA , kI and kh for the case ΣBw B = 0.3, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%....... 62 Fig. B. 1 Coordinate system and locations of integration points ( sections) for continuous beams: ( a) two spans; ( b) three spans; ( c) four spans; and ( d) five spans. ....................... 63 v List of Tables Table 4.1 Survey of bridges under construction in the State of California....................... 16 Table 6.1 Lake Hodges Bridge: Concrete Dimensions..................................................... 35 Table 6.2 Lake Hodges Bridge: Reinforcement and Prestressing .................................... 35 Table 6.3 Lake Hodges Bridge: Analysis results.............................................................. 36 Table 6.4 Lake Hodges Bridge: Comparison with design specifications ......................... 37 Table 6.5 Duenda Road Overcrossing: Concrete Dimensions.......................................... 38 Table 6.6 Duenda Road Overcrossing: Reinforcement and Prestressing ......................... 39 Table 6.7 Duenda Road Overcrossing: Analysis results................................................... 39 Table 6.8 Duenda Road Overcrossing: Comparison with design specifications .............. 39 vi List of Symbols A = area Bt B = ratio of top slab width to bottom slab width E = modulus of elasticity ( ) Ec t, t0 = age adjusted elasticity modulus of concrete fc′ = concrete compressive strength fcgp = concrete stress at center of gravity of prestressing steel at transfer f ij = age adjusted flexibility coefficient. Change in displacement at coordinate i due to unit action applied gradually at coordinate j f pu = ultimate strength of prestressing steel f py = yield strength of prestressing steel f y = yield strength of non prestressed steel hb h = ratio of bottom slab thickness to total depth of cross section ht h = ratio of top slab thickness to total depth of cross section I = second moment of area l = span length M = bending moment N = axial force PPR = partial prestress ratio RH = relative humidity ( in percent) t = final time ( end of service life of concrete member) ts = concrete age at end of curing period, which marks beginning of development of shrinkage strains V/ S = volume to surface ratio of concrete member y = vertical distance measured positive ( downwards) from the centroid of the ageadjusted transformed section α = modular ratio vii χ = aging coefficient r χ = reduced relaxation coefficient ΔD = change in angular discontinuity with time Δfcdp = change in concrete stress at center of gravity of prestressing steel due to permanent loads applied after transfer ΔF = change in connecting moment with time ( ) Δε c t, t0 = change in concrete strain between time t0 and t Δε O = change in axial strain at reference point O μ = curvature coefficient ( ) Δσ c t, t0 = stress applied gradually from time t0 to its full amount at time t Δσ pr = intrinsic relaxation Δσ pr = reduced relaxation Δσ ps = total long term prestress loss Δσ ps( cr) = prestress loss due to creep Δσ ps( es) = prestress loss due to elastic shortening Δσ ps( fr) = prestress loss due to friction Δσ ps( sh) = prestress loss due to shrinkage Δσ ps( relax) = prestress loss due to relaxation Δψ = change in curvature ε cs = shrinkage strain of concrete ( ) ε c t0 = instantaneous strain at time t0 ε c ( t) = total strain at time t ε O = axial strain at reference point O ( ) ϕ t, t0 = creep coefficient between t0 and t ( ) σ c t0 = stress applied at time t0 and sustained to a later time t σ p0 = initial stress of prestressing steel viii Σ Bw = summation of web thicknesses ρ = steel ratio ψ = curvature Subscripts c = net concrete section i = coordinate number (= 1, 2, …) or section integration point (= A, B, …) ns = non prestressed steel ps = prestressing steel 1 1. Introduction Creep and shrinkage of concrete and relaxation of prestressing steel cause long term prestress losses in concrete structures. The effects of these factors are interdependent and it is usually difficult to isolate the effect of each factor. While it is generally accepted that long term losses do not affect the ultimate capacity of a prestressed concrete member, a reasonably accurate prediction of long term losses is important to ensure satisfactory performance of the concrete member under service loads. If prestress losses are underestimated, the tensile strength of concrete could be exceeded at critical sections ( mid spans and over supports) under full service loads and thereby causing cracking and large deflections. On the other hand, overestimating prestress losses leads to excessive camber and uneconomic design because of using large amounts of prestressing steel. The error in predicting the long term prestress losses can be due to two sources: ( 1) inaccuracy of the long term material properties ( creep and shrinkage of concrete and relaxation of prestressing steel); and ( 2) inaccuracy of the method of analysis used. The objective of this report is to address the second source of inaccuracy by presenting a simple, yet comprehensive, analytical method to estimate long term prestress losses in continuous cast in place post tensioned concrete bridges. The method satisfies the requirements of equilibrium and compatibility of the bridge cross sections and avoids the use of any empirical equations, as suggested in most bridge codes, which cannot be accurate in all cases. The inaccuracy in the material properties used can be mitigated by varying the input material properties to the proposed method and determine upper and lower bounds on the prestress losses. 2 2. Long term Material Properties 2.1 Shrinkage of Concrete As curing of concrete ends, concrete starts to lose moisture and undergoes change in volume as a result of chemical reactions between cement paste and water. This phenomenon is known as shrinkage and it starts to develop rapidly after time ts , the age of concrete at the end of the curing period, as shown in Fig. 2.1. Shrinkage of concrete ε cs is influenced by the concrete strength fc′ ( or concrete mix proportions), the method of curing, the relative humidity of the environment RH, the volume to surface ratio of the concrete member V/ S ( or the area of the cross section divided by the perimeter exposed to the atmosphere), and the interval of time t − ts . ts t cs time Fig. 2.1 Development of concrete shrinkage with time. 2.2 Creep of Concrete Creep of concrete is defined as the increase of strain under sustained stress. This increase in strain can be several times the elastic ( or instantaneous) strain at first loading. As shown in Fig. 2.2( a), a stress ( ) σ c t0 is applied at time t0 and sustained to a later time t, the creep coefficient ( ) ϕ t, t0 ( Fig. 2.2( b)) is defined as the ratio of the creep strain ( ) ( ) ε c t − ε c t0 to the instantaneous strain ( ) ε c t0 : 3 ( ) ( ) ( ) [ ( )] 0 0 0 1 t, t E t t t c c c ϕ σ ε = + ( 2.1) where ( ) Ec t0 is the modulus of elasticity of concrete at age t0 . Concrete structures are often subjected to stresses that vary with time from zero at t0 to some value ( ) Δσ c t, t0 at later time t, as depicted in Fig. 2.2( b). This is typical of some cases such as settlement of supports in continuous concrete beams and long term prestress losses in prestressed concrete beams. Since the stress in this case is applied gradually, the creep strain at time t will be less than that in the case when stress is applied at full value at time t0 . To account for this, a dimensionless multiplier χ ( smaller than unity) referred to as the aging coefficient 1,2 is used to express the total strain at time t: ( ) ( ) ( ) [ ( )] 0 0 , 0 1 t, t E t t t t c c c χ ϕ σ ε + Δ = ( 2.2) ( t0 ) t time c stress t 0 t 0 c ( t, t 0 ) stress t time Δ t 0 c ( t0 ) strain c ( t) t time t 0 t ( t) c ( t0 ) c strain time ( a) ( b) Fig. 2.2 Development of concrete strains with time due to: ( a) stress applied at time t0 and sustained to a later time t; and ( b) stress applied gradually from t0 to t. 4 The value of χ ranges from 0.6 to 0.9 and depends on the development of Ec ( t) with time, the creep coefficient ϕ, the time period t − t0 , and the true shape of ( ) Δσ c t, t0 , which is normally not known in advance. For most practical cases, χ can be taken equal to 0.8. It follows from Eq. 2.2 that in the analysis of stresses due to gradually developed forces during a period t0 to t, ( ) Δσ c t, t0 can be related to the change in strains ( ) Δε c t, t0 using the age adjusted elasticity modulus ( ) Ec t, t0 : ( ) ( ) ( ) 0 0 0 1 , , t t Ec t t Ec t + χϕ = ( 2.3) Creep is influenced by the same factors that affect shrinkage in addition to the age of concrete at loading. It is generally accepted that for stress levels below 0.4 fc′ , which is the common range of concrete stresses at service conditions, the creep coefficient ϕ is independent of the applied stress. For most practical situations, ϕ varies between 2 and 4. 2.3 Relaxation of Prestressing Steel Similar to concrete, prestressing steel subjected to stresses more than 50% of its ultimate strength f pu exhibits some creep. In practice, steel used for prestressing is usually subjected to stresses between 0.5 to 0.8 of its ultimate strength. When a prestressing tendon is stretched between two points, it will be subjected to a constant strain. Because of creep, the stress in the tendon decreases ( or relaxes) with time to maintain the state of constant strain. This reduction in stress is known as intrinsic relaxation Δσ pr . It depends on the type of prestressing tendons ( stress relieved or low relaxation), the ratio of the initial stress σ p0 to the yield stress f py and the time t from initial stressing. An equation that is widely used in North America3 for Δσ pr in low relaxation strands is given by 0 0 0.55 40 log( 24 ) p py p pr f t σ σ σ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ = − ( 2.4) 5 In prestressed concrete members, the two ends of the prestressing tendons constantly move toward each other because of the creep and shrinkage effects of concrete, thereby reducing the tensile stress in the tendons. This reduction in tension has a similar effect as if the tendons were subjected to lesser initial stress. Thus, a reduced relaxation value Δσ pr has to be used in the analysis of long term effects in prestressed members: Δσ pr = χ r Δσ pr ( 2.5) where r χ is a dimensionless coefficient less than unity. Ghali and Trevino4 presented a graph to evaluate r χ that depends on, among other factors, the total prestress loss Δσ ps . Since Δσ ps is not known in advance, normally a trial and error procedure is required. A common value for r χ that is used in practice is 0.7. 6 3. Provisions of Bridge Codes for Long term Prestress Losses 3.1 AASHTO LRFD Refined Method5 The total long term prestress loss Δσ ps is expressed as the summation of prestress loss due to creep Δσ ps( cr) , prestress loss due to shrinkage Δσ ps( sh) , and prestress loss due to relaxation Δσ ps( relax) , as follows: Δσ ps( cr) = 12 fcgp − 7Δfcdp ( 3.1) Δσ ps( sh) = ( 13.5 − 0.123RH) ( 3.2) { ( )}Δσ ps( relax) = 0.3 20 − 0.3Δσ ps( fr) − 0.4Δσ ps( es) − 0.2 Δσ ps( sh) + Δσ ps( cr) ( 3.3) where fcgp = concrete stress at center of gravity of prestressing steel at transfer; Δfcdp = change in concrete stress at center of gravity of prestressing steel due to permanent loads applied after transfer; RH = relative humidity in percent; and Δσ ps( fr) and Δσ ps( es) = prestress losses due to friction and elastic shortening, respectively. 3.2 AASHTO LRFD Approximate Method5 For post tensioned concrete bridges with spans up to 160 ft ( 50 m), stressed at concrete age of 10 to 30 days with low relaxation strands and subjected to average exposure conditions, the following equations are given by AASHTO LRFD for box girder bridges: Δσ ps = 17 + 4PPR ( Upper bound) ( 3.4) Δσ ps = 15 + 4PPR ( Average) ( 3.5) PPR is the partial prestress ratio given by: ps py s y ps py A f A f A f PPR + = ( 3.6) where Aps and As = area of prestressing and non prestressed steel, respectively; and 7 f py and f y = yield strength of prestressing and non prestressed steel, respectively. 3.3 CEB FIP Model Code6 The following equation was suggested to estimate the long term prestress losses: ( ) ( ( )) 0 2 0 1 1 1 , , 0.8 t t I A y A A t t f E c c ps c ps ps ps cgp ps cs pr ps α χϕ α ϕ ε σ σ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + + Δ Δ = ( 3.7) where α ps = Eps Ec = ratio of modulus of elasticity of prestressing steel to that of concrete; Aps and Ac = areas of prestressing steel and net concrete section, respectively; Ic = second moment of area of net concrete section; and yps = y coordinate of prestressing steel measured downwards from centroid of net concrete section. Equation 3.7 was derived assuming a single layer of prestressing steel; the effect of nonprestressed steel was not taken into account. The reduced relaxation coefficient r χ ( see Eq. 2.5) was taken 0.8. 3.4 Canadian Highway Bridge Design Code7 ( CHBDC) For post tensioned concrete bridges with low relaxation strands and As Aps ratio less than unity, CHBDC recommends the use of the following equations: [ ] ( ) Δσ ps cr = − RH 2 α ps fcdg − fcdp ( ) 1.61.37 0.77( 0.01 ) ( 3.8) Δσ ps( sh) = 13.6 − 0.12RH ( 3.9) pu pu pu ps cr ps sh pu p ps relax f f f f 0.002 1.25 3 0 0.55 0.34 ( ) ( ) ) ( ≥ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ + Δ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ = − σ σ σ σ ( 3.10) 8 4. Proposed Method of Analysis 4.1 Assumptions • One layer of prestressing steel • One concrete type for the entire cross section • Any cross sectional shape • Prestressing and dead load are applied at the same time t0 to the concrete section ( this may not reflect normal construction operations and will be researched at a later date) • The wearing surface load is ignored or could be considered at an earlier time t0 • Assume the prestressing is applied at the same time of end of curing of concrete; i. e., shrinkage will start to take place at ts = t0. 4.2 Sign Convention The following sign convention will be used throughout the report. Axial force N is positive when it is tensile. Bending moment M and its associated curvature ψ are positive when they produce tension at the bottom fiber of the cross section. Stress σ and strain ε are positive when they produce tension. Positive vertical distance y from the centroid of the cross section is measured downward. It follows that the concrete shrinkage, ε cs ; the reduced steel relaxation, Δσ pr ; and the total prestress loss, Δσ ps , are always negative quantities. However, for convenience of illustration, the absolute value of Δσ ps will be used in graphs. 4.3 Steps of Analysis The procedure of time dependent analysis of a concrete section can be summarized in four steps, 8 as shown in Fig. 4.1. Step 1: Determine the distribution of instantaneous strain at time t0 due to dead load and prestressing force after immediate losses ( friction, anchorage slip and elastic shortening). 9 The distribution of strain can be defined by the strain at an arbitrary point O, ε O( t0 ) and the curvature ( ) ψ t0 . The transformed section properties at time t0 should be used. Since at this stage the prestressing steel is not yet bonded to concrete, its area should not be included in the analysis. Note that the assumption of introducing the prestressing and dead load at the same instant is fairly acceptable since prestressing causes the member to camber and the gap that forms between the member and formwork is usually sufficient to activated the dead load. Fig. 4.1 Four steps for the analysis of time dependent effects ( after Ghali et al. 2002). Step 2: Determine the hypothetical change in strain distribution in the period t0 to t due to creep and shrinkage. The change in strain is ( ) ϕε O t0 + ε cs and the change in curvature is ( ) ϕψ t0 . Ans 2 Ans1 Aps O εO( t0) ψ( t0) Step 1: Instantaneous strain shrinkage ϕψ( t0) ϕεO( t0) Step 2: Free shrinkage and creep creep εcs ΔM ΔN Δψ( t, t0) ΔεO( t, t0 Step 4: Restraining forces applied in reversed directions Δσ Restraint Step 3: Artificial restraint of concrete deformations ΔM ΔN 10 Step 3: Apply artificial restraining stresses gradually on the cross section during the period t0 to t to counteract the hypothetical strains calculated in Step ( 2). The restraining stress Δσ restrained at any fiber y can be calculated by { [ ( ) ( ) ] } Δσ restrained = − Ec ϕ ε O t0 + ψ t0 y + ε cs ( 4.1) where Ec is given by Eq. 2.3. Note that Δσ restrained is applied only on the net concrete section. The change in concrete strains due to relaxation of prestressing steel can be artificially prevented by application of a force equal to Apsσ pr at the centroid of prestressing steel. Step 4: Integrate the artificial stresses determined in Step 3 to get a normal force ΔN and a moment ΔM at point O. To eliminate the artificial restraint, apply ΔN and ΔM in reversed direction on the age adjusted transformed section to determine the long term changes in strains and curvatures of the cross section. Since the prestressing ducts are shortly grouted after prestress transfer, the properties of the cross section include the area of the prestressing steel at this stage. 4.4 Derivation of Method of Analysis In the four steps presented in the previous section, an arbitrary reference point O was selected to perform all the calculations. This is most suited for computer programming and for structures built in stages; however, the equations become quite involved since point O is not the centroid of the cross section. Since the objective of this report is to present a method that is simple enough for use by practicing engineers, the centroid of the cross section will be determined in each step. 11 ( a) Step 1: Instantaneous strains Determine the instantaneous strain and curvature at time t0 due to the dead load and prestressing forces after immediate losses: O1 ( t0 ) Nequivalent [ Ec ( t0 ) A] ε = ; ψ ( t0 ) = Mequivalent [ Ec ( t0 ) I ] ( 4.2) where O1 = centroid of transformed section at time t0 ; Nequivalent = equivalent normal force due to dead weight and prestressing; Mequivalent = equivalent moment due to dead load and prestressing at centroid of the transformed section at time t0 ; ( ) 1 0 ε O t = axial strain at O1 due to applied loads at time t0 ; ( ) ψ t0 = curvature due to applied loads at time t0 ; A = area of the transformed section at time t0 ; I = second moment of area of the transformed section at time t0 about O1; and ( ) Ec t0 = modulus of elasticity of concrete at time t0 . Since only single layer of prestressing is assumed, the elastic deformation of concrete takes place when the jacking force is applied, and there is automatic compensation for the elastic shortening loss. The steel stress in prestressing steel immediately after transfer is equal to the initial stress minus the immediate losses ( friction plus anchorage slip). 12 ps A / 2 ps y y Δ y 2 Δ 1 A ns 1 A / 2 ps O O O 1 3 2 A ns 2 0 ( t ) 1O O 0 2 0 ( t ) O ( Centroid of net concrete section) O ( Centroid of age adjusted transformed section) O ( Centroid of transformed area at time t ) 3 1 2 0 Fig. 4.2 Typical strain distribution in a bridge girder at transfer. ( b) Step 2: Free creep and shrinkage of concrete Determine the axial strain at the centroid of the net concrete section O2 : ( ) ( ) ( ) 2 0 1 0 0 1 ε O t = ε O t + ψ t Δy ( 4.3) where Δy1 = vertical distance between O1 and O2 . For most practical applications, Δy1 is very small compared to the section depth h and can be neglected: ε O2 ≈ ε O1 . The hypothetical change in strain and curvature would be ( ) ϕε O2 t0 + ε cs and ( ) ϕψ t0 . ( c) Step 3: Calculation of artificial force necessary to prevent creep and shrinkage Calculate artificial forces ΔN and ΔM at the centroid of the net concrete section O2 necessary to prevent free creep and shrinkage: 2 ( 0 ) ΔNcreep = − Ec ϕ Ac ε O t ; ( ) ΔMcreep = − Ec ϕ Icψ t0 ( 4.4) ΔNshrinkage = − Ec ε cs Ac ( 4.5) [ ( ) ] ΔN = − EcAc ϕ ε O2 t0 + ε cs ( 4.6) ( ) ΔM = − Ec Icϕ ψ t0 ( 4.7) where Ac = area of the net concrete section; 13 Ic = second moment of area of the net concrete section about O2 ; Ec = age adjusted elasticity modulus of concrete = Ec ( t0 ) ( 1+ χϕ ); χ = aging coefficient; ϕ = creep coefficient between t0 and t; and ε cs = shrinkage strain between t0 and t. ( d) Step 4: Application of the artificial forces in reversed direction Transfer ΔN and ΔM from O2 to the centroid of the age adjusted transformed section O3 : ΔN* = ΔN ( 4.8) ( ) ΔM* = ΔM − ΔN Δy2 − Δy1 ( 4.9) Δy2 = vertical distance between O1 and O3 ( usually positive value for a section at midspan). Usually Δy1 is very small compared to Δy2 : Δy = Δy2 − Δy1 = Δy2 ΔM* = ΔM − ΔN Δy ( 4.10) Apply ΔN* and ΔM* in reversed direction on the age adjusted transformed section: O N ( Ec A) Δε = − Δ * ( 4.11) M ( Ec I ) Δψ = − Δ * ( 4.12) where A = area of the age adjusted transformed section; I = second moment of area of the age adjusted transformed section; Δε O = change in the axial strain between t0 and t at O3; and Δψ = change in curvature between t0 and t Substituting from Eqs. 4.6 through 4.10, Δε O and Δψ can be expressed as: Δε O = kA Δε free ( 4.13) Δψ = kI Δψ free − kh Δε free / h ( 4.14) where 14 A k Ac A = ; I k Ic I = ; I A h k c y h Δ = ( 4.15) ( ) Δε free = ϕ ε O2 t0 + ε cs ; ( ) Δψ free = ϕ ψ t0 ( 4.16) The long term prestress loss Δσ ps between t0 and t can be given by [ ] Δσ ps = Eps Δε O+ Δψ yps ( 4.17) Δσ ps = Eps{ kA Δε free+ yps [ kI Δψ free − khΔε free / h]} ( 4.18) where yps = y coordinate of prestressing steel with respect to the centroid of the ageadjusted transformed section O3 ( Fig. 4.2). 4.5 Prestress Loss Due to Relaxation Following the same procedure to evaluate prestressing losses due to creep and shrinkage, the prestress loss due to relaxation can be determined. The artificial forces to be applied at the centroid of the age adjusted transformed section O3 to prevent relaxation of prestressing steel: { ΔN} relax = Aps Δσ pr ; { ΔM} relax = Aps yps Δσ pr ( 4.19) Apply the artificial forces in reversed direction on the age adjusted transformed section to evaluate the change in axial strain ( ) Δε O relax and curvature ( ) Δψ relax due to relaxation: E A A c ps pr O σ ε − Δ Δ = ; E I A y c ps σ pr ps ψ − Δ Δ = ( 4.20) The long term prestress loss due to relaxation ( ) Δσ ps relax can be computed as ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ = Δ − + I A y A A E E ps ps ps c ps ps( relax ) pr 2 σ σ 1 ( 4.21) Equation 4.21 can be rewritten as ( )⎥⎦ ⎤ ⎢⎣ ⎡ Δ = Δ − Aps + y c ps ps( relax ) pr k k E E σ σ 1 ( 4.22) where 15 A A k ps Aps = ; I A y k ps ps y 2 = ( 4.23) For all practical applications ( see survey in Sec. 4.6), the term ( )⎥⎦ ⎤ ⎢⎣ ⎡ − Aps + y c ps k k E E 1 was found to vary within a narrow range between 0.98 and 0.69. Therefore, an average value of 0.85 can be assumed without sacrificing the accuracy of the method: Δσ ps( relax ) = 0.85Δσ pr ( 4.24) 4.6 Design Aids To evaluate the long term prestress loss Δσ ps , the terms Δε free and Δψ free ( Eq. 4.16) can be easily determined: ( ) 2 0 ε O t and ( ) ψ t0 can be calculated from the initial loading conditions; ϕ and ε cs can be estimated from any design standards, such as ACI 2099 and CEB FIP MC 90.6 The coefficients kA , kI , and kh can be calculated from the geometric dimensions, prestressing and non prestressed steel ratios and the creep coefficient of the cross section or by using design aids, as will be illustrated in the following. A = ps Σ h B B 2w B ns A 2 A t ns1 wB 3 hb wB 4 h t Bw1 Fig. 4.3 Geometric dimensions and reinforcement in a typical bridge cross section. Fig. 4.3 shows a typical cross section of a post tensioned box girder bridge. The number of cells in the figure is chosen arbitrary. The geometric dimensions are defined in Fig. 4.3. Note that the widths of the inclined external webs ( Bw1 and Bw4 ) have to be measured parallel to the horizontal. To be able to determine the practical range of variation of these variables, a survey of bridges currently under construction or soon to be 16 constructed in California was conducted. Table 4.1 summarizes the results of the survey, where: ΣBw = Bw1 + Bw2 + ... = summation of web widths of the cross section; ρ ns1 = Ans1 Bt ht = ratio of non prestressed steel in top slab; ρ ns2 = Ans2 B hb = ratio of non prestressed steel in bottom slab; and ρ ps = Aps h ΣBw = ratio of prestressing steel with respect to web areas. Table 4.1 Survey of bridges under construction in the State of California Variable Range Selected values in parametric studies No. of cells 2 ~ 10 Any number of cells Span length 26 ~ 80 m N/ A ΣBw B 0.1 ~ 0.3 0.1, 0.2, 0.3 ht h 0.05 ~ 0.15 0.05, 0.10, 0.15 hb h 0.05 ~ 0.20 0.05, 0.10, 0.15 Bt B 1.0 ~ 2.0 1.0, 1.5, 2.0 ρ ns1 0.5% ~ 3.0% 0.2, 1.5, 3.0% ρ ns2 0.2% ~ 1.6% 0.2, 1.5, 3.0% ρ ps 0.8% ~ 1.0% 0.8, 1.2% RH 40% ~ 90% 40% – 90% V/ S 5 ~ 7 in. ( 130 – 180 mm) 5, 7 in. ( 130, 180 mm) χϕ 1 ~ 3 1, 2, 3 ε sc 200 ~ 600 × 10 6 500 × 10 6 The National Oceanic and Atmospheric Administration ( NOAA) provides, through its web site, statistical data about the monthly and average annual relative humidity RH 17 values ( morning and afternoon) for major cities and resorts in the United Sates. 10 For the State of California, the lowest average annual RH was reported in Bakersfield ( 39% in the afternoon); Santa Maria has the highest RH ( 87% in the morning). Therefore, a range for RH between 40% to 90% was considered. It should be mentioned that RH values as low as 20% were reported for the community of Bishop but was excluded from the study for scarce bridge construction in this area. In order to determine the common range of V/ S ratios for post tensioned bridges in California, typical bridge cross sections for spans between 26 and 80 m were assumed. The number of cells varied between 2 and 10; the overhang slab was either taken equal to 4 ft ( 1200 mm) or 6.5 ft ( 2000 mm). The bridge dimensions for each span range were taken similar to respective bridges currently under construction. As per AASHTO LRFD recommendations for poorly ventilated enclosed cells ( Article 5.4.2.3.2), only 50% of the interior perimeter is used in calculating the surface area S. The V/ S ratio was found to vary between 5 in. ( 130 mm) and 7 in. ( 180 mm). The RH and V/ S ratios were used to determine the upper and lower bound values of the creep coefficient ϕ ( and accordingly χϕ ) and the shrinkage coefficient ε sc listed in Table 4.1 using varies the empirical models included in ACI 209,9 CEB FIP MC 90,6 AASHTO LRFD, 5 and NCHRP Report 496.11 A spreadsheet was developed to calculate the variation of the coefficients kA , kI , and kh with χϕ for each of the selected geometric dimensions and steel ratios listed in Table 4.1. The following two assumptions were made: ( 1) Ans1 = Ans2 and their centroids are located at mid depth of the top and bottom flanges, respectively; and ( 2) depth of prestressing steel d ps = 0.8h ( for section at mid span) and = 0.2h ( for section at support). The coefficients kA , kI , and kh are presented in graphs in Appendix A. It should be noted that for some graphs, for clarity of presentation, the upper and lower bound values of a specific variable were only used instead of the entire range listed in Table 4.1; linear interpolation can be used for intermediate values not shown in the graphs. 18 4.7 Effect of Continuity Consider a two span continuous beam, as shown in Fig. 4.4. The variation of the tendon profile is parabolic in each span. Other assumptions are as listed in Sec. 4.1. Solve the statically indeterminate beam by any method of structural analysis ( such as force method) to determine the moment diagram at time t0 due to dead load and prestressing ( after immediate losses). 1 l 2l Prestressing tendon Fig. 4.4 Two span continuous prestressed beam. Perform the time dependant sectional analysis as shown previously in Sec. 4.4 for each of the three sections shown in Fig. 4.5 and determine ( Δψ ) i for each section, where i = A, B and C. 2l / 2 l / 2 2 l / 2 1 l / 2 1 A B C Fig. 4.5 Locations of integration points ( sections) in a two span beam. Use the force method to determine the change in internal forces and displacements in the continuous beam. The released structure in Fig. 4.6 with the shown coordinate system shown can be used. Assume the change in angular discontinuity at middle support between t0 and t is ΔD1 and the unknown change in the connecting moment is ΔF1. The change in angular discontinuity ΔD1 can be evaluated as the summation of the two end rotations of each of the simple spans l1 and l2 . Using the method of elastic weights and assuming parabolic variation of curvature in each span, ΔD1 can be expressed as12 [( ) ( ) ] [( ) ( ) ] A B C B ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 1 2 1 ( 4.25) 19 l1 l 2 Δ D1 Fig. 4.6 Released structure and coordinate system for a two span beam. Due to unit load of the connecting moment ΔF1 = 1 that is to be applied gradually on the released structure from zero at time t0 to unity at time t ( Fig. 4.7), determine the change in curvature at each section ( Δψ u1 ) i : ( ) ( Δψ u1) i = − ( ΔF1) i Ec I i ( 4.26) A B C 0.5 Δ( F ) = A 1 1 ( F ) = F = 1 Δ Δ 1.0 B 0.5 1 C Δ( F ) = Fig. 4.7 Moment diagram due to unit value of connecting moment. Evaluate the age adjusted flexibility coefficient f 11 : [ ] [ ( )] u A u B u C u B f l l 1 1 2 1 1 1 11 ( 2( ) 6 2( ) ( ) 6 = Δψ + Δψ + Δψ + Δψ ( 4.27) The change in connecting moment ΔF1 can be computed by solving the compatibility equation 11 1 1 0 f ΔF + ΔD = : 11 1 1 f F D − Δ Δ = ( 4.28) The prestress loss at each section due to continuity at each section ( ) Δσ ps( cont) i can be given by ( ) i ps c ps ps cont i y I F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 σ ( ) ( 4.29) 20 where ( ΔF1) i is the change in moment at each section. For instance, ( F1) A ( F1) B F1 2 Δ = Δ = Δ . 21 5. Comparisons with Bridge Design Specifications A brief summary of the current design equations for long term prestress losses was given in Section 3. Most specifications give separate equations for each prestress loss component due to creep, shrinkage, and relaxation. Therefore, the predictions of the proposed method will be compared with the design equations for prestress loss due to each component individually; the comparison for total prestress loss is presented afterwards. For low relaxation strands, the long term prestress loss due to relaxation is usually a small quantity of the total prestress loss and, therefore, will only be included ( using Eq. 4.22) in the comparison for total prestress losses. In Eq. 4.22, Δσ pr was evaluated from Eq. 2.7 with pr σ Δ and r χ taken equal to 3 ksi and 0.8, respectively. In the proposed method, the prestress loss due to creep Δσ ps( cr) and due to shrinkage Δσ ps( sh) can be given, respectively, by substituting ε cs and ϕ equal to zero in Eq. 4.18: ( ) ϕ ε σ ε ψ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥⎦ ⎤ ⎢⎣ Δ = + ⎡ − h E k y k t k O ps ( cr) ps A O ps I 0 h ( 5.1) cs h ps sh ps A ps h σ E k y k ε ⎭ ⎬ ⎫⎩ ⎨ ⎧ Δ ( ) = − ( 5.2) In all of the comparisons presented in this chapter, the following geometric properties of the bridge cross section are assumed: ΣBw B = 0.2; Bt B = 1.5; and ht h = hb h = 0.1. The depth of the prestressing steel d ps is assumed 0.8h and centroids of the nonprestressed steel in top and bottom slabs are assumed at mid slab depth. The coefficients kA , kI and kh can be evaluated from either Eq. 4.15 or from the graphs in Appendix A. 5.1 AASHTO LRFD Refined and Approximate Methods The AASHTO LRFD refined method for prestress loss due to creep ( Eq. 3.1) is a function of the concrete stress at the center of gravity of prestressing tendons at transfer fcgp and the elastic stress due to additional permanent loads applied after transfer Δfcdp . For post tensioned bridges, almost all the permanent loads ( the dead weight of the bridge 22 and the prestressing) are introduced at transfer. The wearing surface may never be applied, or if applied that would after long time when most of the long term deformations have taken place. Therefore, the term Δfcdp is taken equal to zero and Eq. 3.1 reduces to Δσ ps ( cr) = 12 fcgp ( 5.3) To be able to compare Eq. 5.3 with the proposed method, two stress profiles across the depth of the section at transfer have been assumed, as shown in Fig. 5.1. At transfer, the AASHTO LRFD limits the compression stress at bottom fiber to 0.55 fci and permits no tensile stresses at top fiber. These stress limits were used for stress profile ( 1) ( Fig. 5.1( a)) assuming a specified concrete strength fc′ at 28 days of 4.35 ksi ( 30 MPa). The concrete strength at transfer fci was taken 0.7 fc′ . In the stress profile ( 2) shown in Fig. 5.1( b), the concrete stress at the center of gravity of prestressing steel was kept the same ( 1.33 ksi), but a compression stress of 0.46 ksi ( 0.15 fci ) was assumed at the top fiber. It is believed that stress profiles ( 1) and ( 2) represent common boundary limits for stress states at transfer. Strain profiles ( 1) and ( 2) are obtained from their respective stress profiles by dividing by the concrete modulus of elasticity at transfer. transformed section at time Centroid of net concrete section or Location of prestressing steel h h 1.33 ksi 1.33 ksi = f Strain Profile ( 2) h 0.8 O ( t ) 0 Strain Profile ( 1) 1.67 ksi ( 0.55 ) ci f O ( t ) 0 1.55 ksi cgp Stress Profile ( 2) Stress Profile ( 1) 0.8 h 0.46 ksi ( 0.15 ) 0t ci f Fig. 5.1 Assumed stress and strain profiles at time of transfer. ( a) ( b) 23 The predictions of the proposed method and AASHTO LRFD refined method for longterm prestress losses due to creep are compared in Figs. 5.2( a) and ( b) for prestressing steel ratios of 0.8% and 1.2%, respectively. The independent variable was chosen to be χϕ . The aging coefficient χ is assigned a constant value of 0.8; therefore, variation in χϕ essentially reflects variation in ϕ. The effect of the non prestressed steel is studied by assuming ratios of ρ ns , equal to 0.2%, 1.5%, and 3.0%, for each strain profile. As shown in Fig. 5.2, the AASHTO prediction is a straight horizontal line since it is only a function of the concrete stress at the center of gravity of prestressing steel ( Eq. 5.3). Compared with the proposed method, the AASHTO prediction changes from underestimating to overestimating creep losses as the creep coefficient increases. The AASHTO equation does not take into account the effect of non prestressed steel. While this could be acceptable for pretensioned girders since they contain little or no such reinforcement, this cannot be neglected in post tensioned bridges. As can be seen in Fig. 5.2, the effect of non prestressed steel reduces the absolute value of prestress loss. Among other factors, the long term prestress loss due to creep is dependent on the strain profile of the concrete cross section at time t0 ( application of post tensioning and dead weight). The strain profile can be determined by the strain at an arbitrary reference point ( ) ε O t0 and the slope of the strain profile ( ) ψ t0 . The AASHTO LRFD refined method ( Eq. 5.3) is a function of the concrete stress ( and hence the concrete strain) at the centroid of prestressing steel and therefore recognizes only the effect of the first parameter. As can be seen from both parts in Fig. 5.2, changing the strain profile results in insignificant variation in the prestress loss. It appears that increasing the strain at the net concrete section ( ) ε O t0 in strain profile ( 2) is offset by its reduced slope ( ) ψ t0 ( see Eq. 5.1). Increasing the ratio of prestressed steel ρ ps results in very modest decrease in prestress loss, as can be seen by comparing Fig. 5.2( a) and ( b). 24 0 10 20 30 40 1 2 3 χϕ Prestress Loss, Δσps( cr) ( ksi) AASHTO Refined Method ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Strain profile 2 0 10 20 30 40 1 2 3 χϕ Prestress Loss, Δσps( cr) ( ksi) AASHTO Refined Method ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method Strain profile 2 Fig. 5.2 Comparison between proposed method and AASHTO LRFD refined method for long term prestress losses due to creep: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ( b) ( a) 25 The current AASHTO LRFD equation ( Eq. 3.2) for prestress loss due to shrinkage is only a function of the relative humidity, RH. Although relative humidity is one of the major factors that affect concrete shrinkage, it is not the sole factor ( see Section 2.1). As mentioned in Section 4.6, post tensioned bridges in California have a V/ S ratio that ranges from 5 to 7 in. For moist cured concrete at a relative humidity of 40%, the AASHTO LRFD shrinkage model predicts an ultimate shrinkage strains of 500 × 10 6 and 330 × 10 6, respectively, for bridge cross sections having V/ S = 5 in. and 7 in. ( Note that this difference margin could even be higher when comparing with pretensioned bridges, which normally have V/ S ratio of 3 in. to 4 in.) In other words, the AASHTO equation predicts the same amount of shrinkage losses for two bridge cross sections subjected to different shrinkage strains. As shown in Eq. 5.2, the prestress loss due to shrinkage Δσ ps( sh) varies directly with the shrinkage strain ε cs . It is evident that an equation for prestress loss due to shrinkage should be a function of shrinkage strain ε cs rather than relative humidity RH. Figure 5.3 compares the predictions of the proposed method with the current AASHTO equation ( Eq. 3.2) for prestress losses due to shrinkage. In applying Eq. 5.2, kA and kh depend upon χϕ . Given the narrow range of variation of ϕ according to the AASHTO creep model ( see Fig. 5.2), a constant value of 1.5 was assumed for χϕ . This corresponds to a creep coefficient ϕ of 1.9, which is an approximate mean value for creep coefficients predicted by AASHTO. The shrinkage strain ε cs in Eq. 5.2 was taken according the AASHTO shrinkage model. The apparent kink in the proposed method curves at RH = 80% is due to the fact that AASHTO shrinkage model uses different coefficients for RH higher than 80%. For most of the range studied for RH, the AASHTO prediction underestimates shrinkage losses for members with V/ S = 5 in., whereas changes from overestimating to underestimating the losses as RH increases for members with V/ S = 7 in. The increase in the ratio of non prestressed steel ρ ns reduces the amount of prestress loss, but this reduction is more pronounced in the case of V/ S = 5 in. As in the case with losses due to creep, the effect of prestressed steel ratio ρ ps is insignificant. 26 0 2 4 6 8 10 12 14 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps( sh) ( ksi) AASHTO Refined Method ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method { V/ S ={ 5 in. V/ S = 7 in. 0 2 4 6 8 10 12 14 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps( sh) ( ksi) AASHTO Refined Method ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method { { V/ S = 5 in. V/ S = 7 in. Fig. 5.3 Comparison between proposed method and AASHTO LRFD refined method for long term prestress losses due to shrinkage: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ( a) ( b) 27 The total long term prestress losses are those due to the combined effects of creep, shrinkage of concrete and relaxation of prestressing steel. To evaluate the total long term prestress losses using the proposed method, Eq. 4.18 ( or adding up Eqs. 5.1 and 5.2) was used for the prestress losses due to creep and shrinkage; Eq. 4.22 was used for the prestress loss due to steel relaxation. The creep coefficient ϕ was varied to give χϕ values between 1 and 3 and the concrete shrinkage ε sc was assumed a constant value 500 × 10 6. Equations 3.1, 3.2, and 3.3 were used, respectively, to calculate the prestress losses due to creep, shrinkage, and relaxation in the AASHTO LRFD refined method. In Eq. 3.3, the terms Δσ ps( fr) and Δσ ps( es) ( prestress losses due to friction and elastic shortening, respectively) were not taken into account because: ( 1) Δσ ps( fr) depends on the profile of prestressing tendons and the duct material, factors that are not considered in the present study and are believed to have little or no impact on long term prestress losses; and ( 2) There are no elastic losses ( Δσ ps( es) = 0) for post tensioned girders with one layer of prestressing, as assumed in the present study. In the AASHTO LRFD approximate method ( Eqs. 3.4 and 3.5), the partial prestress ratio PPR ( Eq. 3.6) was calculated assuming yield strength of prestressing steel f py = 243 ksi ( 1675 MPa) and yield strength of non prestressed steel f y = 58 ksi ( 400 MPa). The AASHTO LRFD refined and approximate methods are compared with the proposed method in Fig. 5.4( a) and ( b). Regardless the ratio of non prestressed steel ρ ns , the average and upper bound approximate methods consistently underestimate the total prestress losses. The predictions of the AASHTO refined method change from underestimating to overestimating the total prestress losses as the ratio χϕ increases. The effect of the shape of strain profile at time of post tensioning ( Fig. 5.2) and prestresed steel ratio ρ ps ( Fig. 5.2 through 5.4) are shown to be insignificant. Therefore, these variables will be excluded in further comparisons with design specifications in this section. 28 10 20 30 40 50 1 2 3 χϕ Prestress Loss, Δσps ( ksi) AASHTO Refined Method AASHTO ( Approximate) Upper Bound AASHTO ( Approximate) Average ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% { Proposed method 10 20 30 40 50 1 2 3 χϕ Prestress Loss, Δσps ( ksi) AASHTO Refined Method AASHTO ( Approximate) Upper Bound AASHTO ( Approximate) Average ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Fig. 5.4 Comparison between proposed method and AASHTO LRFD refined and approximate methods for total prestress losses: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ( b) ( a) 29 5.2 CEB FIP Model Code The CEB FIP predictions for long term prestress losses due to the individual effects of creep and shrinkage can be given by Eqs. 5.4 and 5.5, respectively, considering only their respective terms in the numerator of Eq. 3.7: ( ) ( ( )) 0 2 0 ( ) 1 1 1 , , t t I A y A A t t f c c ps c ps ps ps cgp ps cr α χϕ α ϕ σ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + Δ = ( 5.4) ( ( )) 0 ( ) 2 1 1 1 t, t I A y A A E c c ps c ps ps ps cs ps sh α χϕ ε σ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + Δ = ( 5.5) The CEB FIP MC predictions for creep losses ( Eq. 5.4) are evaluated against the results of the proposed method ( Eq. 5.1) in Fig. 5.5. As shown in the figure, the CEB FIP overestimates the prestress losses since it does not take the effect of non prestressed steel into account. As the non prestressed steel ratio ρ ns increases, the difference between the proposed method and the CEB FIP becomes greater, as shown by the reduced slopes of the prediction curves of the proposed method compared with the CEB FIP curves. On the other hand, the difference between the two methods increases with the increase in the creep coefficient ϕ. A comparison between the proposed method ( Eq. 5.2) and the CEB FIP MC method ( Eq. 5.5) for prestress losses due to shrinkage strain of 500 × 10 6 is shown in Fig. 5.6. Since both equations are linear functions in shrinkage, the conclusions outlined here is valid for any value of shrinkage strain. As expected, creep alleviates shrinkage effects and therefore shrinkage losses decrease with the increase in ϕ. The total prestress losses using the CEB FIP method ( Eq. 3.7) and the proposed method ( Eq. 4.18 plus Eq. 4.22) are compared in Fig. 5.7. The concrete shrinkage ε cs was assumed 500 × 10 6. The general trend of the curves in Fig. 5.7 is quite similar to those in Fig. 5.5. The CEB FIP consistently overestimates the total prestress losses, with increasing divergence from the proposed method with the increase in ϕ and ρ ns values. 30 0 10 20 30 40 1 2 3 χϕ Prestress Loss, Δσps( cr) ( ksi) CEB FIP MC ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Fig. 5.5 Comparison between proposed method and CEB FIP method for long term prestress losses due to creep ( ρ ps = 0.8%). 0 5 10 15 20 1 2 3 χϕ Prestress Loss, Δσps( sh) ( ksi) CEB FIP MC ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method Fig. 5.6 Comparison between proposed method and CEB FIP method for long term prestress losses due to shrinkage ( ρ ps = 0.8%). 31 10 20 30 40 50 1 2 3 χϕ Prestress Loss, Δσps ( ksi) CEB FIP MC ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Fig. 5.7 Comparison between proposed method and CEB FIP method for total long term prestress losses ( ρ ps = 0.8%). 5.3 Canadian Highway Bridge Design Code ( CHBDC) The comparisons between the CHBDC and the proposed method with will be performed with respect to the relative humidity RH since it is the main variable in the CHBDC equations ( Eq. 3.8 through 3.10). For each value of RH, the creep and shrinkage coefficients ( to be used in the proposed method) were calculated using the empirical models of the CHBDC and assuming a V/ S ratio of 5 in. It should be noted that an increase in RH essentially reduces the creep and shrinkage values. The prestress losses due to creep according to both methods are shown in Fig. 5.8. The CHBDC equation considerably underestimates prestress losses due to creep for all ratios of non prestressed steel ρ ns ; the difference decreases with the increase in relative humidity RH. Figure 5.9 shows the CHBDC and the proposed method predictions for shrinkage losses. In case of ρ ps = 0.2%, the CHBDC equation compares very well with the proposed 32 method. For ρ ns = 1.5 and 3.0%, the CHBDC equation underestimates the shrinkage losses for RH values less than 70%; for higher values of RH, the CHBDC predictions compare well with the proposed method for all values of ρ ns . The total prestress losses using the proposed method ( summation of Eq. 4.18 and 4.22) and CHBDC method ( summation of Eq. 3.8 through 3.10) are compared in Fig. 5.10. In Eq. 3.10, f pu was taken 270 ksi and the ratio of steel stress at transfer to ultimate strength σ p0 f pu was assumed 0.7. The CHBDC predictions significantly underestimate the total losses, except for sections with ρ ns = 0.2% and RH greater than 75%. Similar to the observations made for creep losses, the difference between the two methods decreases with increasing RH values. 0 10 20 30 40 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps( cr) ( ksi) CHBDC 2000 ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method Fig. 5.8 Comparison between proposed method and CHBDC method for long term prestress losses due to creep ( ρ ps = 0.8%). 33 0 2 4 6 8 10 12 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps( sh) ( ksi) CHBDC 2000 ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Fig. 5.9 Comparison between proposed method and CHBDC method for long term prestress losses due to shrinkage ( ρ ps = 0.8%). 0 10 20 30 40 50 60 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps ( ksi) CHBDC 2000 ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method Fig. 5.10 Comparison between proposed method and CHBDC method for total long term prestress losses ( ρ ps = 0.8%). 34 6. Examples To further illustrate the use of the proposed method, two examples of cast in place posttensioned bridges currently under construction in San Diego County will be analyzed for long term prestress losses; the results from the proposed model will be compared with the current specifications of bridge codes. Since the construction drawings for these bridges were produced in SI units, the input data and results in this section, unlike the rest of the report, will be given in SI units only. The following parameters are assumed in the analysis of both bridges: ϕ = 2.5; χ = 0.8; ε cs = − 500 × 10 6; ( ) Ec t0 = 22.5 GPa; Eps = 195 GPa; Ens = 200 GPa. To account for immediate losses, the curvature friction coefficient μ and anchor set are assumed 0.2 and 10 mm, respectively. Since only one layer of prestressing is assumed, there are no immediate losses due to elastic shortening. 6.1 Lake Hodges Bridge Fig. 6.1 shows a half elevation and a half cross section of the Lake Hodges Bridge. Sections A to E shown in Fig. 6.1( a) are analyzed. The concrete dimensions of the bridge at the analyzed sections are shown in Fig. 6.1( b) and Table 6.1. The non prestressed steel ratios at top and bottom slabs, ρ ns1 and ρ ns2 , respectively, and the prestressed steel ratio ρ ps , are listed in Table 6.2. The variation of the prestressing force P after transfer, as a result of immediate losses, and its location d ps are also given in Table 6.2; posttensioning is performed from both ends. A B 40.7 m C D 45.8 m E 21.35 m Fig. 6.1( a) ( a) 35 C. L. 305 Typ Bt = 12480 mm ht hb h= 1850 mm B = 11260 mm Ans 2 Ans 1 Fig. 6.1 Lake Hodges Bridge: ( a) Half elevation; and ( b) Half cross section. Table 6.1 Lake Hodges Bridge: Concrete Dimensions Section ht hb Bt B ΣBw B ht h hb h A 215 180 1.11 0.12 0.12 0.1 B 215 305 1.11 0.12 0.12 0.16 C 215 180 1.11 0.12 0.12 0.1 D 215 305 1.11 0.12 0.12 0.16 E 215 180 1.11 0.12 0.12 0.1 Table 6.2 Lake Hodges Bridge: Reinforcement and Prestressing Section Ans1 ( mm2) ρ ns1 (%) Ans2 ( mm2) ρ ns2 (%) Aps ( mm2) ρ ps (%) d ps ( mm) d ps h P ( kN) A 32290 0.6 22060 0.54 42280 1.06 1510 0.82 55576 B 67410 1.25 46111 0.67 42280 1.16 355 0.19 55732 C 32290 0.6 22060 0.54 42280 1.06 1510 0.82 53472 D 67410 1.25 46111 0.67 42280 1.16 355 0.19 51303 E 32290 0.6 22060 0.54 42280 1.06 1510 0.82 49079 ( b) 36 The coefficients kA , kI , and kh ( Eq. 4.15) are evaluated for each section and listed in Table 6.3. The computer program CPF13 is used to analyze the bridge due to its own weight and prestressing. The output results from the program were used to determine the strain at the centroid of net concrete section ( ) ε O t0 and the slope of the strain diagram ( ) ψ t0 . The prestress loss due to creep and shrinkage Δσ ps is calculated using Eqs. 4.16 and 4.18. The prestress loss due to continuity Δσ ps( cont ) is evaluated using Eqs. B. 17 through B. 23. As can be seen from Table 6.3, Δσ ps( cont ) is a very small amount compared to Δσ ps . The total pestress loss Δσ ps( total ) is given is the right column of the table. The prestress loss for Lake Hodges Bridge using various code predictions are listed and compared to those of the proposed method in Table 6.4. While the AASHTO approximate method predictions are close to those of the proposed method, the AASHTO detailed method underestimates the total losses by an average of 24%. On the other hand, the CEB FIP and the CHBDC predictions are, respectively, 13% higher and 11% less than the predictions of the proposed method. It should be noted that the comparison results in Table 6.4 are intended only to this particular bridge along with the creep and shrinkage values considered and should not be taken as a general conclusion. Even without changing the creep and shrinkage values, these ratios could change with changing the bridge cross section, as will be seen in the next comparison. Table 6.3 Lake Hodges Bridge: Analysis results Section kA kI kh ( ) ε O t0 ( x 10 6) ( ) ψ t0 ( x 10 6 / m) Δσ ps ( MPa) Δσ ps( cont ) ( MPa) Δσ ps( total ) ( MPa) A 0.84 0.83 0.13  180 34 139 1 140 B 0.79 0.77  0.15  149  65 112  1 111 C 0.84 0.83 0.13  173 14 142 0 142 D 0.79 0.77  0.15  137  37 114 1 115 E 0.84 0.83 0.13  159  6 142  2 140 37 Table 6.4 Lake Hodges Bridge: Comparison with design specifications AASHTO Approximate AASHTO Detailed CEB FIP CHBDC Section Δσ ps Ratio* Δσ ps Ratio* Δσ ps Ratio* Δσ ps Ratio* A 128 0.91 102 0.73 151 1.08 121 0.86 B 123 1.11 89 0.80 134 1.21 102 0.92 C 128 0.90 103 0.73 154 1.09 123 0.87 D 123 1.08 90 0.79 136 1.19 104 0.91 E 128 0.90 103 0.73 154 1.08 123 0.87 Average 0.98 0.76 1.13 0.89 * Ratio of prestress loss using design specification to that of the proposed method. 6.2 Duenda Road Over crossing Fig. 6.2 shows an elevation and a cross section of the Duenda Road Over crossing. Sections A to C shown in Fig. 6.2( a) are analyzed. The concrete dimensions of the bridge at the analyzed sections are shown in Fig. 6.1( b) and Table 6.5. Non prestressed and prestressing steel data are given in Table 6.6. Post tensioning is performed from the longspan end. The analysis results and comparisons with design specifications are reported in Tables 6.7 and 6.8, respectively. The reason for choosing this bridge for analysis is to determine whether a significant difference in the length of two adjacent spans in a continuous bridge could have any effect on the prestress losses due to continuity. As shown in Table 6.7, the continuity effect is quite insignificant and could be ignored. This could be explained by referring to Eq. 4.17, which shows that the prestress loss depends on the change in axial strain at the centroid of the net concrete section Δε O and the change in curvature Δψ . For most practical applications, the change in axial strain is the most dominant factor. The analysis for continuity effects only calculates the additional change in Δψ . In addition, in the calculation of angular discontinuity ΔD1 at one end of intermediate support ( see for instance Eq. 4.25 for two spans) ( ) Δψ A and ( ) Δψ B are of 38 opposite signs therefore reduce ΔD1 , which in turn reduces the long term change in continuity moment ΔF1 . 58.8 m 38.2 m A B C h= 2350 mm 305 Typ Bt = 12510 mm B = 7880 mm hb Ans 1 ht Ans 2 Fig. 6.2 Duenda Road Over crossing: ( a) Elevation; and ( b) Cross section. Table 6.5 Duenda Road Over crossing: Concrete Dimensions Section ht hb Bt B ΣBw B ht h hb h A 220 185 1.51 0.16 0.09 0.08 B 220 305 1.51 0.16 0.09 0.13 C 220 255 1.51 0.16 0.09 0.11 ( a) ( b) 39 Table 6.6 Duenda Road Over crossing: Reinforcement and Prestressing Section Ans1 ( mm2) ρ ns1 (%) Ans2 ( mm2) ρ ns2 (%) Aps ( mm2) ρ ps (%) d ps ( mm) d ps h P ( kN) A 11148 0.43 10001 0.69 36000 1.19 1890 0.80 44788 B 27837 1.07 14910 0.57 36000 1.19 460 0.20 44411 C 24401 1.21 14910 0.74 36000 1.19 1145 0.49 43855 Table 6.7 Duenda Road Over crossing: Analysis results Section kA kI kh ( ) ε O t0 ( x 10 6) ( ) ψ t0 ( x 10 6 / m) Δσ ps ( MPa) Δσ ps( cont ) ( MPa) Δσ ps( total ) ( MPa) A 0.81 0.80 0.27  343  45 162 1 163 B 0.78 0.79  0.20  122  123 136  2 134 C 0.78 0.83  0.005  335 63 175 0 175 Table 6.8 Duenda Road Over crossing: Comparison with design specifications AASHTO Approximate AASHTO Detailed CEB FIP CHBDC Section Δσ ps Ratio* Δσ ps Ratio* Δσ ps Ratio* Δσ ps Ratio* A 129 0.79 129 0.79 173 1.06 160 0.98 B 129 0.96 107 0.80 154 1.15 129 0.96 C 129 0.74 130 0.74 199 1.14 162 0.93 Average 0.83 0.78 1.12 0.96 * Ratio of prestress loss using design specification to that of the proposed method. 40 7. Conclusions Based on the analytical studies presented in the present section, the following conclusions can be made: • The long term behavior of concrete bridges is a rather involved procedure that depends on many parameters. The prediction of long term prestress losses from equations that are function of only one or two parameters, as in the case of all the equations of bridge codes, cannot produce accurate results for all cases. • Although relative humidity is one of the major factors that affect the shrinkage and creep strains, it is not the only one. Equations for prestress losses that are functions of only the relative humidity can lead to misleading results. It is recommended that the prediction equations be functions of the creep and shrinkage coefficients as determined from codes of practice. • Accounting for the effect of non prestressed steel is very essential to produce reliable results for prestressing losses. Neglecting this effect, as in the case of the CEB FIP method, can greatly overestimate the prestress losses. Taking this effect into account in an empirical fashion, as in the case of the other prediction equations, can produce predictions that do not follow the actual trend of prestress losses. • The AASHTO LRFD upper bound approximate method is in fact a lower bound. The approximate average method lies outside the range of prestress loss predictions. • The CHBDC gives reasonable and better predictions for prestress losses compared with the AASHTO LRFD and CEB FIP methods. 41 References 1. Trost, H., “ Auswirkungen des Superpositionspringzips auf Kriech und Relaxationsproblems bei Beton und Apannbeton,” Beton und Stahlbetonbau, V. 62, No. 10, 1967, pp. 230 238; No. 11, 1967, pp. 261 269 ( in German). 2. Bazant, Z. P., “ Prediction of Concrete Creep Effects Using Age Adjusted Effective Modulus,” ACI Journal, V. 69, No. 4, 1972, pp. 212 217. 3. Magura, D. D., Sozen, M. A., and Siess, C. P., “ A Study of Stress Relaxation in Prestressing Reinforcements,” PCI Journal, V. 9, No. 2, 1964, pp. 13 57. 4. Ghali, A., and Trevino, J., “ Relaxation of Steel in Prestressed Concrete,” PCI Journal, V. 30, No. 5, 1985, pp. 82 94. 5. American Association of State Highway and Transportation Officials, “ AASHTOLRFD Bridge Design Specifications,” Third Edition, Washington, DC, 2004. 6. Comité Euro International du Beton − Fédération Internationale de la Précontrainte, “ Model Code for Concrete Structures,” CEB FIP MC 90, London, UK, 1993. 7. Canadian Highway Bridge Design Code, CAN/ CSA S6 00, Rexdale, Canada, 2000. 8. Ghali, A., Favre, R., and Elbadry, M. M., “ Concrete Structures: Stresses and Deformations,” Third Edition, Spon Press, London & New York, 2002, 584 pp. 9. ACI Committee 209, “ Prediction of Creep, Shrinkage, and Temperature Effects in Concrete Structures,” Committee Report 209R 92, American Concrete Institute, Detroit, MI, 1992, 47 pp. 10. http:// www. ncdc. noaa. gov/ oa/ climate/ online/ ccd/ avgrh. html 11. Tadros, M. K., Al Omaishi, N., Serguirant, S. J., and Gallt, J. G., “ Prestress Losses in Pretensioned High Strength Concrete Bridge Girders,” NCHRP Report 496, Transportation Research Board, Washington, DC, 2003, 63 pp. 12. Ghali, A., Neville, A. M., and Brown, T. G., “ Structural Analysis: A Unified Classical and Matrix Approach,” Fifth Edition, Spon Press, London & New York, 2004, 844 pp. 42 13. Elbadry, M. M., and Ghali, A., “ User’s Manual and Computer Program CPF: Cracked Plane Frames in Prestressed Concrete,” Research Report CE 85 2, Department of Civil Engineering, University of Calagry, Calgary, Alberta, Canada, 1990, 82 pp. 43 Appendix A: Coefficients kA , kI and kh Figures A. 1 to A. 18 give the coefficients kA , kI and kh included in Eq. 4.18 to calculate the long term prestress loss Δσ ps . The symbols in the following figures are defined below ( see Fig. 4.3): Bt and B = widths of top and bottom flanges, respectively; ht and hb = depths of top and bottom flanges, respectively; h = total depth of concrete section; Ans1 and Ans2 = areas of non prestressed steel in top and bottom flanges, respectively; Aps = area of prestressing steel; ΣBw = Bw1 + Bw2 + ... = summation of web widths of the cross section; ρ ns1 = Ans1 Bt ht = ratio of non prestressed steel in top flange; ρ ns2 = Ans2 B hb = ratio of non prestressed steel in bottom flange; and ρ ps = Aps h ΣBw = ratio of prestressing steel with respect to web areas. In the graphs below, the modulus of elasticity of prestressing steel Eps and nonprestressed steel Ens were assumed 27.5 ksi ( 190 GPa) and 29 ksi ( 200 GPa), respectively. The modulus of elasticity of concrete Ec at time of post tensioning was assumed 3.6 ksi ( 25 GPa); the aging coefficient χ was assumed 0.8. Two cases of top and bottom thicknesses were assumed: Case 1 ht h = 0.05 and hb h = 0.05; and Case 2 ht h = 0.15 and hb h = 0.15. The following should be noted when using these graphs: • The value of kA is the same for sections with d ps = 0.8h or 0.2h; therefore, no distinction was indicated on the graphs. • The value of kI is the same for sections with d ps = 0.8h or 0.2h and having Bt B = 1. For sections having Bt B = 2, two curves ( in dashed lines) are shown for kI . 44 • The value of kh for sections with d ps = 0.2h and having Bt B = 1 is the same ( but with opposite sign) for sections with d ps = 0.8h. For sections having Bt B = 2, two curves ( in dashed lines) are shown for kh . 45 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 1 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. 46 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 2 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. 47 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = Fig. A. 3 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. 48 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 4 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. 50 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 6 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. 51 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 7 kA , kI and kh for the case ΣBw B = 0.1, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. 52 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.2 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.2 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 8 kA , kI and kh for the case ΣBw B = 0.2, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. 53 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 9 kA , kI and kh for the case ΣBw B = 0.3, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. 54 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 10 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%. 55 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.2 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.2 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 11 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%. 56 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.8  0.6  0.4  0.2 0.0 0.2 0.4 0.6 0.8 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 12 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%. 57 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 13 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%. 58 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 14 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%. 59 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 15 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%. 60 0.2 0.4 0.6 0.8 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.1 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 16 kA , kI and kh for the case ΣBw B = 0.1, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%. 61 0.2 0.4 0.6 0.8 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 17 kA , kI and kh for the case ΣBw B = 0.2, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%. 62 0.2 0.4 0.6 0.8 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 18 kA , kI and kh for the case ΣBw B = 0.3, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%. 63 Appendix B: Prestress Loss Due to Continuity The approach to estimate prestress losses due to continuity in two unequal span continuous beam was outlined in Section 4.6. In the following, the approach will be extended to symmetric beams with two to five spans. The coordinate system and locations of integration points ( sections) are illustrated in Figs. B. 1( a) to ( d). The sections are located either at intermediate supports or at mid spans. In case of four or five spans, the equations are to be solved for two unknows: the connecting moments over intermediate supports ΔF1 and ΔF2 . The prestress loss due to continuity at each section is evaluated for the combined effect of the values of ΔF1 and ΔF2 at this section; see Eqs. B. 16 and B. 23. For definition of symbols used in the equations, refer to Sec. 4.6. l 1 l 1 A B A D 1 l 1 l 2 A B C D 1 l 1 D 1 B A l 1 l 2 A B C D 1 l 2 D 2 D C l 1 B A D 1 l 1 l 2 A B C D 1 l 3 D 2 D E l 1 D C D 2 B A D 1 l 1 Fig. B. 1 Coordinate system and locations of integration points ( sections) for continuous beams: ( a) two spans; ( b) three spans; ( c) four spans; and ( d) five spans. ( a) ( b) ( c) 64 ( a) Two Spans: [( ) ( ) ] A B ΔD = l 2 Δψ + Δψ 3 1 1 ( B. 1) [( ) ( )] u A u B f l 1 1 1 11 2 3 = Δψ + Δψ ( B. 2) 11 1 1 f F D − Δ Δ = ( B. 3) ( ) i ps c ps ps cont i y I F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 σ ( ) ( B. 4) ( b) Three Spans: [( ) ( ) ] [( ) ( ) ] A B C B ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 1 2 1 ( B. 5) [( ) ( )] [( ) ( )] u A u B u C u B f l l 1 1 2 1 1 1 11 2 6 2 6 = Δψ + Δψ + Δψ + Δψ ( B. 6) 11 1 1 f F D − Δ Δ = ( B. 7) ( ) i ps c ps ps cont i y I F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 σ ( ) ( B. 8) ( c) Four Spans: [( ) ( ) ] [( ) ( ) ] A B C B ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 1 2 1 ( B. 9) [( ) ( ) ] C D ΔD = l 2 Δψ + Δψ 3 2 2 ( B. 10) [( ) ( )] [( ) ( )] u A u B u C u B f l l 1 1 2 1 1 1 11 2 6 2 6 = Δψ + Δψ + Δψ + Δψ ( B. 11) [( ) ( ) ] u C u D f l 2 2 2 22 2 3 = Δψ + Δψ ( B. 12) [ ( ) ] u C f l 1 2 21 2 3 = Δψ ; [ ( ) ] u C f l 2 2 12 2 6 = Δψ ( B. 13) 65 ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Δ Δ = − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Δ Δ ⎥⎦ ⎤ ⎢⎣ ⎡ 2 1 2 1 21 22 11 12 D D F F f f f f ( B. 14) ( ) ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − Δ − Δ ⎥⎦ ⎤ ⎢⎣ ⎡ − − − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Δ Δ 2 1 21 11 22 12 2 11 22 12 21 1 1 D D f f f f F f f f f F ( B. 15) ( ) i ps c ps ps cont i y I F F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ + Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 2 σ ( ) ( B. 16) ( d) Five Spans: [( ) ( ) ] [( ) ( ) ] A B C B ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 1 2 1 ( B. 17) [( ) ( ) ] [( ) ( ) ] C D C E ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 2 3 2 ( B. 18) [( ) ( )] [( ) ( )] u A u B u C u B f l l 1 1 2 1 1 1 11 2 6 2 6 = Δψ + Δψ + Δψ + Δψ ( B. 19) [( ) ( ) ] [( ) ( )] u C u D u C u E f l l 2 2 3 2 2 2 22 2 6 2 6 = Δψ + Δψ + Δψ + Δψ ( B. 20) [ ( ) ] u C f l 1 2 21 2 6 = Δψ ; [ ( ) ] u C f l 2 2 12 2 6 = Δψ ( B. 21) ( ) ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − Δ − Δ ⎥⎦ ⎤ ⎢⎣ ⎡ − − − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Δ Δ 2 1 21 11 22 12 2 11 22 12 21 1 1 D D f f f f F f f f f F ( B. 22) ( ) i ps c ps ps cont i y I F F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ + Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 2 σ ( ) ( B. 23)
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Title  A simplified method for prediction of longterm prestress loss in posttensioned concrete bridges 
Subject  TG340.Y68 2006 compu/d; Concrete bridgesMaintenance and repair.; Posttensioned prestressed concrete construction.; T989.P76 CD 
Description  Title from PDF title page.; "July 2006."; Includes bibliographical references (p. 4142).; Final report.; Electronic text (PDF: viii, 65 p. : col. ill.); Submitted to the California Department of Transportation, Engineering Service Center, under contract no. 
Creator  Youakim, Samer Amir, 1970 
Publisher  Dept. of Structural Engineering, University of California, San Diego 
Contributors  Karbhari, Vistasp M.; Structural Systems Research Project.; University of California, San Diego. Dept. of Structural Engineering.; California. Dept. of Transportation. Engineering Service Center. 
Type  Text 
Language  eng 
Relation  http://worldcat.org/oclc/213416902/viewonline 
DateIssued  [2006] 
FormatExtent  1 CDROM : col. ; 4 3/4 in. 
RelationRequires  System requirements: Adobe Acrobat Reader; CDROM drive. 
RelationIs Part Of  Report / Structural Systems Research Project ; no. SSRP06/21; Report (Structural Systems Research Project) ; no. SSRP06/21. 
Transcript  STRUCTURAL SYSTEMS RESEARCH PROJECT Report No. SSRP– 06/ 21 A SIMPLIFIED METHOD FOR PREDICTION OF LONG TERM PRESTRESS LOSS IN POSTTENSIONED CONCRETE BRIDGES by SAMER YOUAKIM VISTASP M. KARBHARI Final Report Submitted to the California Department of Transportation Under Contract No. 59A0337. July 2006 Department of Structural Engineering University of California, San Diego La Jolla, California 92093 0085 University of California, San Diego Department of Structural Engineering Structural Systems Research Project Report No. SSRP– 06/ 21 A Simplified Method for Prediction of Long Term Prestress Loss in Post Tensioned Concrete Bridges by Samer Youakim Assistant Project Scientist Vistasp M. Karbhari Professor of Structural Engineering Final Report Submitted to the California Department of Transportation Under Contract No. 59A0337 Department of Structural Engineering University of California, San Diego La Jolla, California 92093 0085 July 2006 Technical Report Documentation Page 1. Report No. 2. Government Accession No. 3. Recipient’s Catalog No. 4. Title and Subtitle A Simplified Method for Prediction of Long Term Prestress Loss in Post Tensioned Concrete Bridges 5. Report Date July 2006 6. Performing Organization Code 7. Author( s) Samer Youakim and Vistasp M. Karbhari 8. Performing Organization Report No. UCSD / SSRP 06/ 21 9. Performing Organization Name and Address Department of Structural Engineering School of Engineering 10. Work Unit No. ( TRAIS) University of California, San Diego La Jolla, California 92093 0085 11. Contract or Grant No. 59A0337 12. Sponsoring Agency Name and Address California Department of Transportation 13. Type of Report and Period Covered Final Report Engineering Service Center 1801 30th St., West Building MS 9 Sacramento, California 95807 14. Sponsoring Agency Code 15. Supplementary Notes Prepared in cooperation with the State of California Department of Transportation. 16. Abstract Creep and shrinkage of concrete and relaxation of prestressing steel cause time dependent changes in the stresses and strains of concrete structures. These changes result in continuous reduction in the concrete compression stresses and in the tension in prestressing steel. Reasonably accurate estimate of the long term prestress losses are needed to avoid any serviceability problems of the structure ( due to cracking and/ or excessive deflection). An analytical method is presented to predict the long term prestress losses in continuous cast in place post tensioned bridges. The method is based on the basic principles of solid mechanics and satisfies the requirements of equilibrium and compatibility of the bridge cross section. The proposed method for a section in concrete girder reduces to a single equation with three coefficients, which are functions of the modulus of elasticity and creep coefficient of concrete, location and amount of prestressing and non prestressed steel, and geometry of the cross section. To expedite the use of the method and to make it more appealing to practicing engineers, design aids are provided to estimate these three coefficients. The method is further extended to continuous bridge girders by using the force method to calculate the change in connecting moments at intermediate supports and hence the increase or decrease in prestressing losses. The predictions of the proposed method are compared with the current provisions of design standards and codes of practice. It is shown that the present empirical equations of the bridge standards can overly underestimate or overestimate the long term prestress losses, depending on the concrete creep and shrinkage properties as well as prestressing and non prestressed steel ratios. 17. Key Words Prestress loss, creep, shrinkage 18. Distribution Statement No restrictions 19. Security Classification ( of this report) Unclassified 20. Security Classification ( of this page) Unclassified 21. No. of Pages 72 22. Price Form DOT F 1700.7 ( 8 72) Reproduction of completed page authorized Disclaimer The contents of this report reflect the views of the authors who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views or policies of the State of California. This report does not constitute a standard, specification or regulation. 1 i Abstract Creep and shrinkage of concrete and relaxation of prestressing steel cause timedependent changes in the stresses and strains of concrete structures. These changes result in continuous reduction in the concrete compression stresses and in the tension in prestressing steel. Reasonably accurate estimate of the long term prestress losses are needed to avoid any serviceability problems of the structure ( due to cracking and/ or excessive deflection). An analytical method is presented to predict the long term prestress losses in continuous cast in place post tensioned bridges. The method is based on the basic principles of solid mechanics and satisfies the requirements of equilibrium and compatibility of the bridge cross section. It is assumed that prestressing and dead load are applied at the same instant, shrinkage starts to take place at the application of loads, and one concrete type for the cross section. The proposed method for a section in concrete girder reduces to a single equation with three coefficients, which are functions of the modulus of elasticity and creep coefficient of concrete, location and amount of prestressing and non prestressed steel, and geometry of the cross section. To expedite the use of the method and to make it more appealing to practicing engineers, design aids are provided to estimate these three coefficients. The method is further extended to continuous bridge girders by using the force method to calculate the change in connecting moments at intermediate supports and hence the increase or decrease in prestressing losses. The predictions of the proposed method are compared with the current provisions of design standards and codes of practice. It is shown that the present empirical equations of the bridge standards can overly underestimate or overestimate the long term prestress losses, depending on the concrete creep and shrinkage properties as well as prestressing and non prestressed steel ratios. The proposed method is applied to a number of continuous post tensioned concrete bridges currently under construction in San Diego County. It was found that the prestressing losses due to continuity could be ignored without affecting the accuracy of the method. ii Table of Contents Abstract....................................................................................................................... ....... i Table of Contents .............................................................................................................. ii List of Figures................................................................................................................... iii List of Tables ..................................................................................................................... v List of Symbols ................................................................................................................. vi 1. Introduction................................................................................................................... 1 2. Long term Material Properties ................................................................................... 2 2.1 Shrinkage of Concrete............................................................................................... 2 2.2 Creep of Concrete ..................................................................................................... 2 2.3 Relaxation of Prestressing Steel................................................................................ 4 3. Provisions of Bridge Codes for Long term Prestress Losses .................................... 6 3.1 AASHTO LRFD Refined Method5 .......................................................................... 6 3.2 AASHTO LRFD Approximate Method5.................................................................. 6 3.3 CEB FIP Model Code6 ............................................................................................. 7 3.4 Canadian Highway Bridge Design Code7 ( CHBDC) ............................................... 7 4. Proposed Method of Analysis ...................................................................................... 8 4.1 Assumptions.............................................................................................................. 8 4.2 Sign Convention........................................................................................................ 8 4.3 Steps of Analysis....................................................................................................... 8 4.4 Derivation of Method of Analysis .......................................................................... 10 4.5 Prestress Loss Due to Relaxation............................................................................ 14 4.6 Design Aids............................................................................................................. 15 4.7 Effect of Continuity ................................................................................................ 18 5. Comparisons with Bridge Design Specifications...................................................... 21 5.1 AASHTO LRFD Refined and Approximate Methods ........................................... 21 5.2 CEB FIP Model Code............................................................................................. 29 5.3 Canadian Highway Bridge Design Code ( CHBDC)............................................... 31 6. Examples...................................................................................................................... 34 6.1 Lake Hodges Bridge ............................................................................................... 34 6.2 Duenda Road Overcrossing .................................................................................... 37 7. Conclusions.................................................................................................................. 40 References..................................................................................................................... .. 41 Appendix A: Coefficients kA , kI and kh ...................................................................... 43 Appendix B: Prestress Loss Due to Continuity............................................................ 63 iii List of Figures Fig. 2.1 Development of concrete shrinkage with time. ..................................................... 2 Fig. 2.2 Development of concrete strains with time due to: ( a) stress applied at time t0 and sustained to a later time t; and ( b) stress applied gradually from t0 to t...................... 3 Fig. 4.1 Four steps for the analysis of time dependent effects ( after Ghali et al. 2002)..... 9 Fig. 4.2 Typical strain distribution in a bridge girder at transfer...................................... 12 Fig. 4.3 Geometric dimensions and reinforcement in a typical bridge cross section. ...... 15 Fig. 4.4 Two span continuous prestressed beam. ............................................................. 18 Fig. 4.5 Locations of integration points ( sections) in a two span beam. .......................... 18 Fig. 4.6 Released structure and coordinate system for a two span beam. ........................ 19 Fig. 4.7 Moment diagram due to unit value of connecting moment................................. 19 Fig. 5.1 Assumed stress and strain profiles at time of transfer. ........................................ 22 Fig. 5.2 Comparison between proposed method and AASHTO LRFD refined method for long term prestress losses due to creep: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ............... 24 Fig. 5.3 Comparison between proposed method and AASHTO LRFD refined method for long term prestress losses due to shrinkage: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ........ 26 Fig. 5.4 Comparison between proposed method and AASHTO LRFD refined and approximate methods for total prestress losses: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ... 28 Fig. 5.6 Comparison between proposed method and CEB FIP method for long term prestress losses due to shrinkage ( ρ ps = 0.8%)................................................................ 30 Fig. 5.7 Comparison between proposed method and CEB FIP method for total long term prestress losses ( ρ ps = 0.8%). .......................................................................................... 31 Fig. 5.8 Comparison between proposed method and CHBDC method for long term prestress losses due to creep ( ρ ps = 0.8%)....................................................................... 32 Fig. 5.9 Comparison between proposed method and CHBDC method for long term prestress losses due to shrinkage ( .................................................................................... 33 Fig. 5.10 Comparison between proposed method and CHBDC method for total long term prestress losses ( ρ ps = 0.8%). .......................................................................................... 33 Fig. 6.1 Lake Hodges Bridge: ( a) Half elevation; and ( b) Half cross section................... 35 Fig. 6.2 Duenda Road Overcrossing: ( a) Elevation; and ( b) Cross section. ..................... 38 Fig. A. 1 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. ...... 45 Fig. A. 2 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. ...... 46 Fig. A. 3 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. ...... 47 Fig. A. 4 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. ....... 48 Fig. A. 5 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. ....... 49 iv Fig. A. 6 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. ....... 50 Fig. A. 7 kA , kI and kh for the case ΣBw B = 0.1, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. ....... 51 Fig. A. 8 kA , kI and kh for the case ΣBw B = 0.2, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. ....... 52 Fig. A. 9 kA , kI and kh for the case ΣBw B = 0.3, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. ....... 53 Fig. A. 10 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%...... 54 Fig. A. 11 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%...... 55 Fig. A. 12 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%...... 56 Fig. A. 13 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%....... 57 Fig. A. 14 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%....... 58 Fig. A. 15 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%....... 59 Fig. A. 16 kA , kI and kh for the case ΣBw B = 0.1, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%....... 60 Fig. A. 17 kA , kI and kh for the case ΣBw B = 0.2, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%....... 61 Fig. A. 18 kA , kI and kh for the case ΣBw B = 0.3, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%....... 62 Fig. B. 1 Coordinate system and locations of integration points ( sections) for continuous beams: ( a) two spans; ( b) three spans; ( c) four spans; and ( d) five spans. ....................... 63 v List of Tables Table 4.1 Survey of bridges under construction in the State of California....................... 16 Table 6.1 Lake Hodges Bridge: Concrete Dimensions..................................................... 35 Table 6.2 Lake Hodges Bridge: Reinforcement and Prestressing .................................... 35 Table 6.3 Lake Hodges Bridge: Analysis results.............................................................. 36 Table 6.4 Lake Hodges Bridge: Comparison with design specifications ......................... 37 Table 6.5 Duenda Road Overcrossing: Concrete Dimensions.......................................... 38 Table 6.6 Duenda Road Overcrossing: Reinforcement and Prestressing ......................... 39 Table 6.7 Duenda Road Overcrossing: Analysis results................................................... 39 Table 6.8 Duenda Road Overcrossing: Comparison with design specifications .............. 39 vi List of Symbols A = area Bt B = ratio of top slab width to bottom slab width E = modulus of elasticity ( ) Ec t, t0 = age adjusted elasticity modulus of concrete fc′ = concrete compressive strength fcgp = concrete stress at center of gravity of prestressing steel at transfer f ij = age adjusted flexibility coefficient. Change in displacement at coordinate i due to unit action applied gradually at coordinate j f pu = ultimate strength of prestressing steel f py = yield strength of prestressing steel f y = yield strength of non prestressed steel hb h = ratio of bottom slab thickness to total depth of cross section ht h = ratio of top slab thickness to total depth of cross section I = second moment of area l = span length M = bending moment N = axial force PPR = partial prestress ratio RH = relative humidity ( in percent) t = final time ( end of service life of concrete member) ts = concrete age at end of curing period, which marks beginning of development of shrinkage strains V/ S = volume to surface ratio of concrete member y = vertical distance measured positive ( downwards) from the centroid of the ageadjusted transformed section α = modular ratio vii χ = aging coefficient r χ = reduced relaxation coefficient ΔD = change in angular discontinuity with time Δfcdp = change in concrete stress at center of gravity of prestressing steel due to permanent loads applied after transfer ΔF = change in connecting moment with time ( ) Δε c t, t0 = change in concrete strain between time t0 and t Δε O = change in axial strain at reference point O μ = curvature coefficient ( ) Δσ c t, t0 = stress applied gradually from time t0 to its full amount at time t Δσ pr = intrinsic relaxation Δσ pr = reduced relaxation Δσ ps = total long term prestress loss Δσ ps( cr) = prestress loss due to creep Δσ ps( es) = prestress loss due to elastic shortening Δσ ps( fr) = prestress loss due to friction Δσ ps( sh) = prestress loss due to shrinkage Δσ ps( relax) = prestress loss due to relaxation Δψ = change in curvature ε cs = shrinkage strain of concrete ( ) ε c t0 = instantaneous strain at time t0 ε c ( t) = total strain at time t ε O = axial strain at reference point O ( ) ϕ t, t0 = creep coefficient between t0 and t ( ) σ c t0 = stress applied at time t0 and sustained to a later time t σ p0 = initial stress of prestressing steel viii Σ Bw = summation of web thicknesses ρ = steel ratio ψ = curvature Subscripts c = net concrete section i = coordinate number (= 1, 2, …) or section integration point (= A, B, …) ns = non prestressed steel ps = prestressing steel 1 1. Introduction Creep and shrinkage of concrete and relaxation of prestressing steel cause long term prestress losses in concrete structures. The effects of these factors are interdependent and it is usually difficult to isolate the effect of each factor. While it is generally accepted that long term losses do not affect the ultimate capacity of a prestressed concrete member, a reasonably accurate prediction of long term losses is important to ensure satisfactory performance of the concrete member under service loads. If prestress losses are underestimated, the tensile strength of concrete could be exceeded at critical sections ( mid spans and over supports) under full service loads and thereby causing cracking and large deflections. On the other hand, overestimating prestress losses leads to excessive camber and uneconomic design because of using large amounts of prestressing steel. The error in predicting the long term prestress losses can be due to two sources: ( 1) inaccuracy of the long term material properties ( creep and shrinkage of concrete and relaxation of prestressing steel); and ( 2) inaccuracy of the method of analysis used. The objective of this report is to address the second source of inaccuracy by presenting a simple, yet comprehensive, analytical method to estimate long term prestress losses in continuous cast in place post tensioned concrete bridges. The method satisfies the requirements of equilibrium and compatibility of the bridge cross sections and avoids the use of any empirical equations, as suggested in most bridge codes, which cannot be accurate in all cases. The inaccuracy in the material properties used can be mitigated by varying the input material properties to the proposed method and determine upper and lower bounds on the prestress losses. 2 2. Long term Material Properties 2.1 Shrinkage of Concrete As curing of concrete ends, concrete starts to lose moisture and undergoes change in volume as a result of chemical reactions between cement paste and water. This phenomenon is known as shrinkage and it starts to develop rapidly after time ts , the age of concrete at the end of the curing period, as shown in Fig. 2.1. Shrinkage of concrete ε cs is influenced by the concrete strength fc′ ( or concrete mix proportions), the method of curing, the relative humidity of the environment RH, the volume to surface ratio of the concrete member V/ S ( or the area of the cross section divided by the perimeter exposed to the atmosphere), and the interval of time t − ts . ts t cs time Fig. 2.1 Development of concrete shrinkage with time. 2.2 Creep of Concrete Creep of concrete is defined as the increase of strain under sustained stress. This increase in strain can be several times the elastic ( or instantaneous) strain at first loading. As shown in Fig. 2.2( a), a stress ( ) σ c t0 is applied at time t0 and sustained to a later time t, the creep coefficient ( ) ϕ t, t0 ( Fig. 2.2( b)) is defined as the ratio of the creep strain ( ) ( ) ε c t − ε c t0 to the instantaneous strain ( ) ε c t0 : 3 ( ) ( ) ( ) [ ( )] 0 0 0 1 t, t E t t t c c c ϕ σ ε = + ( 2.1) where ( ) Ec t0 is the modulus of elasticity of concrete at age t0 . Concrete structures are often subjected to stresses that vary with time from zero at t0 to some value ( ) Δσ c t, t0 at later time t, as depicted in Fig. 2.2( b). This is typical of some cases such as settlement of supports in continuous concrete beams and long term prestress losses in prestressed concrete beams. Since the stress in this case is applied gradually, the creep strain at time t will be less than that in the case when stress is applied at full value at time t0 . To account for this, a dimensionless multiplier χ ( smaller than unity) referred to as the aging coefficient 1,2 is used to express the total strain at time t: ( ) ( ) ( ) [ ( )] 0 0 , 0 1 t, t E t t t t c c c χ ϕ σ ε + Δ = ( 2.2) ( t0 ) t time c stress t 0 t 0 c ( t, t 0 ) stress t time Δ t 0 c ( t0 ) strain c ( t) t time t 0 t ( t) c ( t0 ) c strain time ( a) ( b) Fig. 2.2 Development of concrete strains with time due to: ( a) stress applied at time t0 and sustained to a later time t; and ( b) stress applied gradually from t0 to t. 4 The value of χ ranges from 0.6 to 0.9 and depends on the development of Ec ( t) with time, the creep coefficient ϕ, the time period t − t0 , and the true shape of ( ) Δσ c t, t0 , which is normally not known in advance. For most practical cases, χ can be taken equal to 0.8. It follows from Eq. 2.2 that in the analysis of stresses due to gradually developed forces during a period t0 to t, ( ) Δσ c t, t0 can be related to the change in strains ( ) Δε c t, t0 using the age adjusted elasticity modulus ( ) Ec t, t0 : ( ) ( ) ( ) 0 0 0 1 , , t t Ec t t Ec t + χϕ = ( 2.3) Creep is influenced by the same factors that affect shrinkage in addition to the age of concrete at loading. It is generally accepted that for stress levels below 0.4 fc′ , which is the common range of concrete stresses at service conditions, the creep coefficient ϕ is independent of the applied stress. For most practical situations, ϕ varies between 2 and 4. 2.3 Relaxation of Prestressing Steel Similar to concrete, prestressing steel subjected to stresses more than 50% of its ultimate strength f pu exhibits some creep. In practice, steel used for prestressing is usually subjected to stresses between 0.5 to 0.8 of its ultimate strength. When a prestressing tendon is stretched between two points, it will be subjected to a constant strain. Because of creep, the stress in the tendon decreases ( or relaxes) with time to maintain the state of constant strain. This reduction in stress is known as intrinsic relaxation Δσ pr . It depends on the type of prestressing tendons ( stress relieved or low relaxation), the ratio of the initial stress σ p0 to the yield stress f py and the time t from initial stressing. An equation that is widely used in North America3 for Δσ pr in low relaxation strands is given by 0 0 0.55 40 log( 24 ) p py p pr f t σ σ σ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ = − ( 2.4) 5 In prestressed concrete members, the two ends of the prestressing tendons constantly move toward each other because of the creep and shrinkage effects of concrete, thereby reducing the tensile stress in the tendons. This reduction in tension has a similar effect as if the tendons were subjected to lesser initial stress. Thus, a reduced relaxation value Δσ pr has to be used in the analysis of long term effects in prestressed members: Δσ pr = χ r Δσ pr ( 2.5) where r χ is a dimensionless coefficient less than unity. Ghali and Trevino4 presented a graph to evaluate r χ that depends on, among other factors, the total prestress loss Δσ ps . Since Δσ ps is not known in advance, normally a trial and error procedure is required. A common value for r χ that is used in practice is 0.7. 6 3. Provisions of Bridge Codes for Long term Prestress Losses 3.1 AASHTO LRFD Refined Method5 The total long term prestress loss Δσ ps is expressed as the summation of prestress loss due to creep Δσ ps( cr) , prestress loss due to shrinkage Δσ ps( sh) , and prestress loss due to relaxation Δσ ps( relax) , as follows: Δσ ps( cr) = 12 fcgp − 7Δfcdp ( 3.1) Δσ ps( sh) = ( 13.5 − 0.123RH) ( 3.2) { ( )}Δσ ps( relax) = 0.3 20 − 0.3Δσ ps( fr) − 0.4Δσ ps( es) − 0.2 Δσ ps( sh) + Δσ ps( cr) ( 3.3) where fcgp = concrete stress at center of gravity of prestressing steel at transfer; Δfcdp = change in concrete stress at center of gravity of prestressing steel due to permanent loads applied after transfer; RH = relative humidity in percent; and Δσ ps( fr) and Δσ ps( es) = prestress losses due to friction and elastic shortening, respectively. 3.2 AASHTO LRFD Approximate Method5 For post tensioned concrete bridges with spans up to 160 ft ( 50 m), stressed at concrete age of 10 to 30 days with low relaxation strands and subjected to average exposure conditions, the following equations are given by AASHTO LRFD for box girder bridges: Δσ ps = 17 + 4PPR ( Upper bound) ( 3.4) Δσ ps = 15 + 4PPR ( Average) ( 3.5) PPR is the partial prestress ratio given by: ps py s y ps py A f A f A f PPR + = ( 3.6) where Aps and As = area of prestressing and non prestressed steel, respectively; and 7 f py and f y = yield strength of prestressing and non prestressed steel, respectively. 3.3 CEB FIP Model Code6 The following equation was suggested to estimate the long term prestress losses: ( ) ( ( )) 0 2 0 1 1 1 , , 0.8 t t I A y A A t t f E c c ps c ps ps ps cgp ps cs pr ps α χϕ α ϕ ε σ σ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + + + Δ Δ = ( 3.7) where α ps = Eps Ec = ratio of modulus of elasticity of prestressing steel to that of concrete; Aps and Ac = areas of prestressing steel and net concrete section, respectively; Ic = second moment of area of net concrete section; and yps = y coordinate of prestressing steel measured downwards from centroid of net concrete section. Equation 3.7 was derived assuming a single layer of prestressing steel; the effect of nonprestressed steel was not taken into account. The reduced relaxation coefficient r χ ( see Eq. 2.5) was taken 0.8. 3.4 Canadian Highway Bridge Design Code7 ( CHBDC) For post tensioned concrete bridges with low relaxation strands and As Aps ratio less than unity, CHBDC recommends the use of the following equations: [ ] ( ) Δσ ps cr = − RH 2 α ps fcdg − fcdp ( ) 1.61.37 0.77( 0.01 ) ( 3.8) Δσ ps( sh) = 13.6 − 0.12RH ( 3.9) pu pu pu ps cr ps sh pu p ps relax f f f f 0.002 1.25 3 0 0.55 0.34 ( ) ( ) ) ( ≥ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ + Δ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ = − σ σ σ σ ( 3.10) 8 4. Proposed Method of Analysis 4.1 Assumptions • One layer of prestressing steel • One concrete type for the entire cross section • Any cross sectional shape • Prestressing and dead load are applied at the same time t0 to the concrete section ( this may not reflect normal construction operations and will be researched at a later date) • The wearing surface load is ignored or could be considered at an earlier time t0 • Assume the prestressing is applied at the same time of end of curing of concrete; i. e., shrinkage will start to take place at ts = t0. 4.2 Sign Convention The following sign convention will be used throughout the report. Axial force N is positive when it is tensile. Bending moment M and its associated curvature ψ are positive when they produce tension at the bottom fiber of the cross section. Stress σ and strain ε are positive when they produce tension. Positive vertical distance y from the centroid of the cross section is measured downward. It follows that the concrete shrinkage, ε cs ; the reduced steel relaxation, Δσ pr ; and the total prestress loss, Δσ ps , are always negative quantities. However, for convenience of illustration, the absolute value of Δσ ps will be used in graphs. 4.3 Steps of Analysis The procedure of time dependent analysis of a concrete section can be summarized in four steps, 8 as shown in Fig. 4.1. Step 1: Determine the distribution of instantaneous strain at time t0 due to dead load and prestressing force after immediate losses ( friction, anchorage slip and elastic shortening). 9 The distribution of strain can be defined by the strain at an arbitrary point O, ε O( t0 ) and the curvature ( ) ψ t0 . The transformed section properties at time t0 should be used. Since at this stage the prestressing steel is not yet bonded to concrete, its area should not be included in the analysis. Note that the assumption of introducing the prestressing and dead load at the same instant is fairly acceptable since prestressing causes the member to camber and the gap that forms between the member and formwork is usually sufficient to activated the dead load. Fig. 4.1 Four steps for the analysis of time dependent effects ( after Ghali et al. 2002). Step 2: Determine the hypothetical change in strain distribution in the period t0 to t due to creep and shrinkage. The change in strain is ( ) ϕε O t0 + ε cs and the change in curvature is ( ) ϕψ t0 . Ans 2 Ans1 Aps O εO( t0) ψ( t0) Step 1: Instantaneous strain shrinkage ϕψ( t0) ϕεO( t0) Step 2: Free shrinkage and creep creep εcs ΔM ΔN Δψ( t, t0) ΔεO( t, t0 Step 4: Restraining forces applied in reversed directions Δσ Restraint Step 3: Artificial restraint of concrete deformations ΔM ΔN 10 Step 3: Apply artificial restraining stresses gradually on the cross section during the period t0 to t to counteract the hypothetical strains calculated in Step ( 2). The restraining stress Δσ restrained at any fiber y can be calculated by { [ ( ) ( ) ] } Δσ restrained = − Ec ϕ ε O t0 + ψ t0 y + ε cs ( 4.1) where Ec is given by Eq. 2.3. Note that Δσ restrained is applied only on the net concrete section. The change in concrete strains due to relaxation of prestressing steel can be artificially prevented by application of a force equal to Apsσ pr at the centroid of prestressing steel. Step 4: Integrate the artificial stresses determined in Step 3 to get a normal force ΔN and a moment ΔM at point O. To eliminate the artificial restraint, apply ΔN and ΔM in reversed direction on the age adjusted transformed section to determine the long term changes in strains and curvatures of the cross section. Since the prestressing ducts are shortly grouted after prestress transfer, the properties of the cross section include the area of the prestressing steel at this stage. 4.4 Derivation of Method of Analysis In the four steps presented in the previous section, an arbitrary reference point O was selected to perform all the calculations. This is most suited for computer programming and for structures built in stages; however, the equations become quite involved since point O is not the centroid of the cross section. Since the objective of this report is to present a method that is simple enough for use by practicing engineers, the centroid of the cross section will be determined in each step. 11 ( a) Step 1: Instantaneous strains Determine the instantaneous strain and curvature at time t0 due to the dead load and prestressing forces after immediate losses: O1 ( t0 ) Nequivalent [ Ec ( t0 ) A] ε = ; ψ ( t0 ) = Mequivalent [ Ec ( t0 ) I ] ( 4.2) where O1 = centroid of transformed section at time t0 ; Nequivalent = equivalent normal force due to dead weight and prestressing; Mequivalent = equivalent moment due to dead load and prestressing at centroid of the transformed section at time t0 ; ( ) 1 0 ε O t = axial strain at O1 due to applied loads at time t0 ; ( ) ψ t0 = curvature due to applied loads at time t0 ; A = area of the transformed section at time t0 ; I = second moment of area of the transformed section at time t0 about O1; and ( ) Ec t0 = modulus of elasticity of concrete at time t0 . Since only single layer of prestressing is assumed, the elastic deformation of concrete takes place when the jacking force is applied, and there is automatic compensation for the elastic shortening loss. The steel stress in prestressing steel immediately after transfer is equal to the initial stress minus the immediate losses ( friction plus anchorage slip). 12 ps A / 2 ps y y Δ y 2 Δ 1 A ns 1 A / 2 ps O O O 1 3 2 A ns 2 0 ( t ) 1O O 0 2 0 ( t ) O ( Centroid of net concrete section) O ( Centroid of age adjusted transformed section) O ( Centroid of transformed area at time t ) 3 1 2 0 Fig. 4.2 Typical strain distribution in a bridge girder at transfer. ( b) Step 2: Free creep and shrinkage of concrete Determine the axial strain at the centroid of the net concrete section O2 : ( ) ( ) ( ) 2 0 1 0 0 1 ε O t = ε O t + ψ t Δy ( 4.3) where Δy1 = vertical distance between O1 and O2 . For most practical applications, Δy1 is very small compared to the section depth h and can be neglected: ε O2 ≈ ε O1 . The hypothetical change in strain and curvature would be ( ) ϕε O2 t0 + ε cs and ( ) ϕψ t0 . ( c) Step 3: Calculation of artificial force necessary to prevent creep and shrinkage Calculate artificial forces ΔN and ΔM at the centroid of the net concrete section O2 necessary to prevent free creep and shrinkage: 2 ( 0 ) ΔNcreep = − Ec ϕ Ac ε O t ; ( ) ΔMcreep = − Ec ϕ Icψ t0 ( 4.4) ΔNshrinkage = − Ec ε cs Ac ( 4.5) [ ( ) ] ΔN = − EcAc ϕ ε O2 t0 + ε cs ( 4.6) ( ) ΔM = − Ec Icϕ ψ t0 ( 4.7) where Ac = area of the net concrete section; 13 Ic = second moment of area of the net concrete section about O2 ; Ec = age adjusted elasticity modulus of concrete = Ec ( t0 ) ( 1+ χϕ ); χ = aging coefficient; ϕ = creep coefficient between t0 and t; and ε cs = shrinkage strain between t0 and t. ( d) Step 4: Application of the artificial forces in reversed direction Transfer ΔN and ΔM from O2 to the centroid of the age adjusted transformed section O3 : ΔN* = ΔN ( 4.8) ( ) ΔM* = ΔM − ΔN Δy2 − Δy1 ( 4.9) Δy2 = vertical distance between O1 and O3 ( usually positive value for a section at midspan). Usually Δy1 is very small compared to Δy2 : Δy = Δy2 − Δy1 = Δy2 ΔM* = ΔM − ΔN Δy ( 4.10) Apply ΔN* and ΔM* in reversed direction on the age adjusted transformed section: O N ( Ec A) Δε = − Δ * ( 4.11) M ( Ec I ) Δψ = − Δ * ( 4.12) where A = area of the age adjusted transformed section; I = second moment of area of the age adjusted transformed section; Δε O = change in the axial strain between t0 and t at O3; and Δψ = change in curvature between t0 and t Substituting from Eqs. 4.6 through 4.10, Δε O and Δψ can be expressed as: Δε O = kA Δε free ( 4.13) Δψ = kI Δψ free − kh Δε free / h ( 4.14) where 14 A k Ac A = ; I k Ic I = ; I A h k c y h Δ = ( 4.15) ( ) Δε free = ϕ ε O2 t0 + ε cs ; ( ) Δψ free = ϕ ψ t0 ( 4.16) The long term prestress loss Δσ ps between t0 and t can be given by [ ] Δσ ps = Eps Δε O+ Δψ yps ( 4.17) Δσ ps = Eps{ kA Δε free+ yps [ kI Δψ free − khΔε free / h]} ( 4.18) where yps = y coordinate of prestressing steel with respect to the centroid of the ageadjusted transformed section O3 ( Fig. 4.2). 4.5 Prestress Loss Due to Relaxation Following the same procedure to evaluate prestressing losses due to creep and shrinkage, the prestress loss due to relaxation can be determined. The artificial forces to be applied at the centroid of the age adjusted transformed section O3 to prevent relaxation of prestressing steel: { ΔN} relax = Aps Δσ pr ; { ΔM} relax = Aps yps Δσ pr ( 4.19) Apply the artificial forces in reversed direction on the age adjusted transformed section to evaluate the change in axial strain ( ) Δε O relax and curvature ( ) Δψ relax due to relaxation: E A A c ps pr O σ ε − Δ Δ = ; E I A y c ps σ pr ps ψ − Δ Δ = ( 4.20) The long term prestress loss due to relaxation ( ) Δσ ps relax can be computed as ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ = Δ − + I A y A A E E ps ps ps c ps ps( relax ) pr 2 σ σ 1 ( 4.21) Equation 4.21 can be rewritten as ( )⎥⎦ ⎤ ⎢⎣ ⎡ Δ = Δ − Aps + y c ps ps( relax ) pr k k E E σ σ 1 ( 4.22) where 15 A A k ps Aps = ; I A y k ps ps y 2 = ( 4.23) For all practical applications ( see survey in Sec. 4.6), the term ( )⎥⎦ ⎤ ⎢⎣ ⎡ − Aps + y c ps k k E E 1 was found to vary within a narrow range between 0.98 and 0.69. Therefore, an average value of 0.85 can be assumed without sacrificing the accuracy of the method: Δσ ps( relax ) = 0.85Δσ pr ( 4.24) 4.6 Design Aids To evaluate the long term prestress loss Δσ ps , the terms Δε free and Δψ free ( Eq. 4.16) can be easily determined: ( ) 2 0 ε O t and ( ) ψ t0 can be calculated from the initial loading conditions; ϕ and ε cs can be estimated from any design standards, such as ACI 2099 and CEB FIP MC 90.6 The coefficients kA , kI , and kh can be calculated from the geometric dimensions, prestressing and non prestressed steel ratios and the creep coefficient of the cross section or by using design aids, as will be illustrated in the following. A = ps Σ h B B 2w B ns A 2 A t ns1 wB 3 hb wB 4 h t Bw1 Fig. 4.3 Geometric dimensions and reinforcement in a typical bridge cross section. Fig. 4.3 shows a typical cross section of a post tensioned box girder bridge. The number of cells in the figure is chosen arbitrary. The geometric dimensions are defined in Fig. 4.3. Note that the widths of the inclined external webs ( Bw1 and Bw4 ) have to be measured parallel to the horizontal. To be able to determine the practical range of variation of these variables, a survey of bridges currently under construction or soon to be 16 constructed in California was conducted. Table 4.1 summarizes the results of the survey, where: ΣBw = Bw1 + Bw2 + ... = summation of web widths of the cross section; ρ ns1 = Ans1 Bt ht = ratio of non prestressed steel in top slab; ρ ns2 = Ans2 B hb = ratio of non prestressed steel in bottom slab; and ρ ps = Aps h ΣBw = ratio of prestressing steel with respect to web areas. Table 4.1 Survey of bridges under construction in the State of California Variable Range Selected values in parametric studies No. of cells 2 ~ 10 Any number of cells Span length 26 ~ 80 m N/ A ΣBw B 0.1 ~ 0.3 0.1, 0.2, 0.3 ht h 0.05 ~ 0.15 0.05, 0.10, 0.15 hb h 0.05 ~ 0.20 0.05, 0.10, 0.15 Bt B 1.0 ~ 2.0 1.0, 1.5, 2.0 ρ ns1 0.5% ~ 3.0% 0.2, 1.5, 3.0% ρ ns2 0.2% ~ 1.6% 0.2, 1.5, 3.0% ρ ps 0.8% ~ 1.0% 0.8, 1.2% RH 40% ~ 90% 40% – 90% V/ S 5 ~ 7 in. ( 130 – 180 mm) 5, 7 in. ( 130, 180 mm) χϕ 1 ~ 3 1, 2, 3 ε sc 200 ~ 600 × 10 6 500 × 10 6 The National Oceanic and Atmospheric Administration ( NOAA) provides, through its web site, statistical data about the monthly and average annual relative humidity RH 17 values ( morning and afternoon) for major cities and resorts in the United Sates. 10 For the State of California, the lowest average annual RH was reported in Bakersfield ( 39% in the afternoon); Santa Maria has the highest RH ( 87% in the morning). Therefore, a range for RH between 40% to 90% was considered. It should be mentioned that RH values as low as 20% were reported for the community of Bishop but was excluded from the study for scarce bridge construction in this area. In order to determine the common range of V/ S ratios for post tensioned bridges in California, typical bridge cross sections for spans between 26 and 80 m were assumed. The number of cells varied between 2 and 10; the overhang slab was either taken equal to 4 ft ( 1200 mm) or 6.5 ft ( 2000 mm). The bridge dimensions for each span range were taken similar to respective bridges currently under construction. As per AASHTO LRFD recommendations for poorly ventilated enclosed cells ( Article 5.4.2.3.2), only 50% of the interior perimeter is used in calculating the surface area S. The V/ S ratio was found to vary between 5 in. ( 130 mm) and 7 in. ( 180 mm). The RH and V/ S ratios were used to determine the upper and lower bound values of the creep coefficient ϕ ( and accordingly χϕ ) and the shrinkage coefficient ε sc listed in Table 4.1 using varies the empirical models included in ACI 209,9 CEB FIP MC 90,6 AASHTO LRFD, 5 and NCHRP Report 496.11 A spreadsheet was developed to calculate the variation of the coefficients kA , kI , and kh with χϕ for each of the selected geometric dimensions and steel ratios listed in Table 4.1. The following two assumptions were made: ( 1) Ans1 = Ans2 and their centroids are located at mid depth of the top and bottom flanges, respectively; and ( 2) depth of prestressing steel d ps = 0.8h ( for section at mid span) and = 0.2h ( for section at support). The coefficients kA , kI , and kh are presented in graphs in Appendix A. It should be noted that for some graphs, for clarity of presentation, the upper and lower bound values of a specific variable were only used instead of the entire range listed in Table 4.1; linear interpolation can be used for intermediate values not shown in the graphs. 18 4.7 Effect of Continuity Consider a two span continuous beam, as shown in Fig. 4.4. The variation of the tendon profile is parabolic in each span. Other assumptions are as listed in Sec. 4.1. Solve the statically indeterminate beam by any method of structural analysis ( such as force method) to determine the moment diagram at time t0 due to dead load and prestressing ( after immediate losses). 1 l 2l Prestressing tendon Fig. 4.4 Two span continuous prestressed beam. Perform the time dependant sectional analysis as shown previously in Sec. 4.4 for each of the three sections shown in Fig. 4.5 and determine ( Δψ ) i for each section, where i = A, B and C. 2l / 2 l / 2 2 l / 2 1 l / 2 1 A B C Fig. 4.5 Locations of integration points ( sections) in a two span beam. Use the force method to determine the change in internal forces and displacements in the continuous beam. The released structure in Fig. 4.6 with the shown coordinate system shown can be used. Assume the change in angular discontinuity at middle support between t0 and t is ΔD1 and the unknown change in the connecting moment is ΔF1. The change in angular discontinuity ΔD1 can be evaluated as the summation of the two end rotations of each of the simple spans l1 and l2 . Using the method of elastic weights and assuming parabolic variation of curvature in each span, ΔD1 can be expressed as12 [( ) ( ) ] [( ) ( ) ] A B C B ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 1 2 1 ( 4.25) 19 l1 l 2 Δ D1 Fig. 4.6 Released structure and coordinate system for a two span beam. Due to unit load of the connecting moment ΔF1 = 1 that is to be applied gradually on the released structure from zero at time t0 to unity at time t ( Fig. 4.7), determine the change in curvature at each section ( Δψ u1 ) i : ( ) ( Δψ u1) i = − ( ΔF1) i Ec I i ( 4.26) A B C 0.5 Δ( F ) = A 1 1 ( F ) = F = 1 Δ Δ 1.0 B 0.5 1 C Δ( F ) = Fig. 4.7 Moment diagram due to unit value of connecting moment. Evaluate the age adjusted flexibility coefficient f 11 : [ ] [ ( )] u A u B u C u B f l l 1 1 2 1 1 1 11 ( 2( ) 6 2( ) ( ) 6 = Δψ + Δψ + Δψ + Δψ ( 4.27) The change in connecting moment ΔF1 can be computed by solving the compatibility equation 11 1 1 0 f ΔF + ΔD = : 11 1 1 f F D − Δ Δ = ( 4.28) The prestress loss at each section due to continuity at each section ( ) Δσ ps( cont) i can be given by ( ) i ps c ps ps cont i y I F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 σ ( ) ( 4.29) 20 where ( ΔF1) i is the change in moment at each section. For instance, ( F1) A ( F1) B F1 2 Δ = Δ = Δ . 21 5. Comparisons with Bridge Design Specifications A brief summary of the current design equations for long term prestress losses was given in Section 3. Most specifications give separate equations for each prestress loss component due to creep, shrinkage, and relaxation. Therefore, the predictions of the proposed method will be compared with the design equations for prestress loss due to each component individually; the comparison for total prestress loss is presented afterwards. For low relaxation strands, the long term prestress loss due to relaxation is usually a small quantity of the total prestress loss and, therefore, will only be included ( using Eq. 4.22) in the comparison for total prestress losses. In Eq. 4.22, Δσ pr was evaluated from Eq. 2.7 with pr σ Δ and r χ taken equal to 3 ksi and 0.8, respectively. In the proposed method, the prestress loss due to creep Δσ ps( cr) and due to shrinkage Δσ ps( sh) can be given, respectively, by substituting ε cs and ϕ equal to zero in Eq. 4.18: ( ) ϕ ε σ ε ψ ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⎥⎦ ⎤ ⎢⎣ Δ = + ⎡ − h E k y k t k O ps ( cr) ps A O ps I 0 h ( 5.1) cs h ps sh ps A ps h σ E k y k ε ⎭ ⎬ ⎫⎩ ⎨ ⎧ Δ ( ) = − ( 5.2) In all of the comparisons presented in this chapter, the following geometric properties of the bridge cross section are assumed: ΣBw B = 0.2; Bt B = 1.5; and ht h = hb h = 0.1. The depth of the prestressing steel d ps is assumed 0.8h and centroids of the nonprestressed steel in top and bottom slabs are assumed at mid slab depth. The coefficients kA , kI and kh can be evaluated from either Eq. 4.15 or from the graphs in Appendix A. 5.1 AASHTO LRFD Refined and Approximate Methods The AASHTO LRFD refined method for prestress loss due to creep ( Eq. 3.1) is a function of the concrete stress at the center of gravity of prestressing tendons at transfer fcgp and the elastic stress due to additional permanent loads applied after transfer Δfcdp . For post tensioned bridges, almost all the permanent loads ( the dead weight of the bridge 22 and the prestressing) are introduced at transfer. The wearing surface may never be applied, or if applied that would after long time when most of the long term deformations have taken place. Therefore, the term Δfcdp is taken equal to zero and Eq. 3.1 reduces to Δσ ps ( cr) = 12 fcgp ( 5.3) To be able to compare Eq. 5.3 with the proposed method, two stress profiles across the depth of the section at transfer have been assumed, as shown in Fig. 5.1. At transfer, the AASHTO LRFD limits the compression stress at bottom fiber to 0.55 fci and permits no tensile stresses at top fiber. These stress limits were used for stress profile ( 1) ( Fig. 5.1( a)) assuming a specified concrete strength fc′ at 28 days of 4.35 ksi ( 30 MPa). The concrete strength at transfer fci was taken 0.7 fc′ . In the stress profile ( 2) shown in Fig. 5.1( b), the concrete stress at the center of gravity of prestressing steel was kept the same ( 1.33 ksi), but a compression stress of 0.46 ksi ( 0.15 fci ) was assumed at the top fiber. It is believed that stress profiles ( 1) and ( 2) represent common boundary limits for stress states at transfer. Strain profiles ( 1) and ( 2) are obtained from their respective stress profiles by dividing by the concrete modulus of elasticity at transfer. transformed section at time Centroid of net concrete section or Location of prestressing steel h h 1.33 ksi 1.33 ksi = f Strain Profile ( 2) h 0.8 O ( t ) 0 Strain Profile ( 1) 1.67 ksi ( 0.55 ) ci f O ( t ) 0 1.55 ksi cgp Stress Profile ( 2) Stress Profile ( 1) 0.8 h 0.46 ksi ( 0.15 ) 0t ci f Fig. 5.1 Assumed stress and strain profiles at time of transfer. ( a) ( b) 23 The predictions of the proposed method and AASHTO LRFD refined method for longterm prestress losses due to creep are compared in Figs. 5.2( a) and ( b) for prestressing steel ratios of 0.8% and 1.2%, respectively. The independent variable was chosen to be χϕ . The aging coefficient χ is assigned a constant value of 0.8; therefore, variation in χϕ essentially reflects variation in ϕ. The effect of the non prestressed steel is studied by assuming ratios of ρ ns , equal to 0.2%, 1.5%, and 3.0%, for each strain profile. As shown in Fig. 5.2, the AASHTO prediction is a straight horizontal line since it is only a function of the concrete stress at the center of gravity of prestressing steel ( Eq. 5.3). Compared with the proposed method, the AASHTO prediction changes from underestimating to overestimating creep losses as the creep coefficient increases. The AASHTO equation does not take into account the effect of non prestressed steel. While this could be acceptable for pretensioned girders since they contain little or no such reinforcement, this cannot be neglected in post tensioned bridges. As can be seen in Fig. 5.2, the effect of non prestressed steel reduces the absolute value of prestress loss. Among other factors, the long term prestress loss due to creep is dependent on the strain profile of the concrete cross section at time t0 ( application of post tensioning and dead weight). The strain profile can be determined by the strain at an arbitrary reference point ( ) ε O t0 and the slope of the strain profile ( ) ψ t0 . The AASHTO LRFD refined method ( Eq. 5.3) is a function of the concrete stress ( and hence the concrete strain) at the centroid of prestressing steel and therefore recognizes only the effect of the first parameter. As can be seen from both parts in Fig. 5.2, changing the strain profile results in insignificant variation in the prestress loss. It appears that increasing the strain at the net concrete section ( ) ε O t0 in strain profile ( 2) is offset by its reduced slope ( ) ψ t0 ( see Eq. 5.1). Increasing the ratio of prestressed steel ρ ps results in very modest decrease in prestress loss, as can be seen by comparing Fig. 5.2( a) and ( b). 24 0 10 20 30 40 1 2 3 χϕ Prestress Loss, Δσps( cr) ( ksi) AASHTO Refined Method ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Strain profile 2 0 10 20 30 40 1 2 3 χϕ Prestress Loss, Δσps( cr) ( ksi) AASHTO Refined Method ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method Strain profile 2 Fig. 5.2 Comparison between proposed method and AASHTO LRFD refined method for long term prestress losses due to creep: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ( b) ( a) 25 The current AASHTO LRFD equation ( Eq. 3.2) for prestress loss due to shrinkage is only a function of the relative humidity, RH. Although relative humidity is one of the major factors that affect concrete shrinkage, it is not the sole factor ( see Section 2.1). As mentioned in Section 4.6, post tensioned bridges in California have a V/ S ratio that ranges from 5 to 7 in. For moist cured concrete at a relative humidity of 40%, the AASHTO LRFD shrinkage model predicts an ultimate shrinkage strains of 500 × 10 6 and 330 × 10 6, respectively, for bridge cross sections having V/ S = 5 in. and 7 in. ( Note that this difference margin could even be higher when comparing with pretensioned bridges, which normally have V/ S ratio of 3 in. to 4 in.) In other words, the AASHTO equation predicts the same amount of shrinkage losses for two bridge cross sections subjected to different shrinkage strains. As shown in Eq. 5.2, the prestress loss due to shrinkage Δσ ps( sh) varies directly with the shrinkage strain ε cs . It is evident that an equation for prestress loss due to shrinkage should be a function of shrinkage strain ε cs rather than relative humidity RH. Figure 5.3 compares the predictions of the proposed method with the current AASHTO equation ( Eq. 3.2) for prestress losses due to shrinkage. In applying Eq. 5.2, kA and kh depend upon χϕ . Given the narrow range of variation of ϕ according to the AASHTO creep model ( see Fig. 5.2), a constant value of 1.5 was assumed for χϕ . This corresponds to a creep coefficient ϕ of 1.9, which is an approximate mean value for creep coefficients predicted by AASHTO. The shrinkage strain ε cs in Eq. 5.2 was taken according the AASHTO shrinkage model. The apparent kink in the proposed method curves at RH = 80% is due to the fact that AASHTO shrinkage model uses different coefficients for RH higher than 80%. For most of the range studied for RH, the AASHTO prediction underestimates shrinkage losses for members with V/ S = 5 in., whereas changes from overestimating to underestimating the losses as RH increases for members with V/ S = 7 in. The increase in the ratio of non prestressed steel ρ ns reduces the amount of prestress loss, but this reduction is more pronounced in the case of V/ S = 5 in. As in the case with losses due to creep, the effect of prestressed steel ratio ρ ps is insignificant. 26 0 2 4 6 8 10 12 14 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps( sh) ( ksi) AASHTO Refined Method ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method { V/ S ={ 5 in. V/ S = 7 in. 0 2 4 6 8 10 12 14 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps( sh) ( ksi) AASHTO Refined Method ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method { { V/ S = 5 in. V/ S = 7 in. Fig. 5.3 Comparison between proposed method and AASHTO LRFD refined method for long term prestress losses due to shrinkage: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ( a) ( b) 27 The total long term prestress losses are those due to the combined effects of creep, shrinkage of concrete and relaxation of prestressing steel. To evaluate the total long term prestress losses using the proposed method, Eq. 4.18 ( or adding up Eqs. 5.1 and 5.2) was used for the prestress losses due to creep and shrinkage; Eq. 4.22 was used for the prestress loss due to steel relaxation. The creep coefficient ϕ was varied to give χϕ values between 1 and 3 and the concrete shrinkage ε sc was assumed a constant value 500 × 10 6. Equations 3.1, 3.2, and 3.3 were used, respectively, to calculate the prestress losses due to creep, shrinkage, and relaxation in the AASHTO LRFD refined method. In Eq. 3.3, the terms Δσ ps( fr) and Δσ ps( es) ( prestress losses due to friction and elastic shortening, respectively) were not taken into account because: ( 1) Δσ ps( fr) depends on the profile of prestressing tendons and the duct material, factors that are not considered in the present study and are believed to have little or no impact on long term prestress losses; and ( 2) There are no elastic losses ( Δσ ps( es) = 0) for post tensioned girders with one layer of prestressing, as assumed in the present study. In the AASHTO LRFD approximate method ( Eqs. 3.4 and 3.5), the partial prestress ratio PPR ( Eq. 3.6) was calculated assuming yield strength of prestressing steel f py = 243 ksi ( 1675 MPa) and yield strength of non prestressed steel f y = 58 ksi ( 400 MPa). The AASHTO LRFD refined and approximate methods are compared with the proposed method in Fig. 5.4( a) and ( b). Regardless the ratio of non prestressed steel ρ ns , the average and upper bound approximate methods consistently underestimate the total prestress losses. The predictions of the AASHTO refined method change from underestimating to overestimating the total prestress losses as the ratio χϕ increases. The effect of the shape of strain profile at time of post tensioning ( Fig. 5.2) and prestresed steel ratio ρ ps ( Fig. 5.2 through 5.4) are shown to be insignificant. Therefore, these variables will be excluded in further comparisons with design specifications in this section. 28 10 20 30 40 50 1 2 3 χϕ Prestress Loss, Δσps ( ksi) AASHTO Refined Method AASHTO ( Approximate) Upper Bound AASHTO ( Approximate) Average ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% { Proposed method 10 20 30 40 50 1 2 3 χϕ Prestress Loss, Δσps ( ksi) AASHTO Refined Method AASHTO ( Approximate) Upper Bound AASHTO ( Approximate) Average ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Fig. 5.4 Comparison between proposed method and AASHTO LRFD refined and approximate methods for total prestress losses: ( a) ρ ps = 0.8%; and ( b) ρ ps = 1.2%. ( b) ( a) 29 5.2 CEB FIP Model Code The CEB FIP predictions for long term prestress losses due to the individual effects of creep and shrinkage can be given by Eqs. 5.4 and 5.5, respectively, considering only their respective terms in the numerator of Eq. 3.7: ( ) ( ( )) 0 2 0 ( ) 1 1 1 , , t t I A y A A t t f c c ps c ps ps ps cgp ps cr α χϕ α ϕ σ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + Δ = ( 5.4) ( ( )) 0 ( ) 2 1 1 1 t, t I A y A A E c c ps c ps ps ps cs ps sh α χϕ ε σ + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + + Δ = ( 5.5) The CEB FIP MC predictions for creep losses ( Eq. 5.4) are evaluated against the results of the proposed method ( Eq. 5.1) in Fig. 5.5. As shown in the figure, the CEB FIP overestimates the prestress losses since it does not take the effect of non prestressed steel into account. As the non prestressed steel ratio ρ ns increases, the difference between the proposed method and the CEB FIP becomes greater, as shown by the reduced slopes of the prediction curves of the proposed method compared with the CEB FIP curves. On the other hand, the difference between the two methods increases with the increase in the creep coefficient ϕ. A comparison between the proposed method ( Eq. 5.2) and the CEB FIP MC method ( Eq. 5.5) for prestress losses due to shrinkage strain of 500 × 10 6 is shown in Fig. 5.6. Since both equations are linear functions in shrinkage, the conclusions outlined here is valid for any value of shrinkage strain. As expected, creep alleviates shrinkage effects and therefore shrinkage losses decrease with the increase in ϕ. The total prestress losses using the CEB FIP method ( Eq. 3.7) and the proposed method ( Eq. 4.18 plus Eq. 4.22) are compared in Fig. 5.7. The concrete shrinkage ε cs was assumed 500 × 10 6. The general trend of the curves in Fig. 5.7 is quite similar to those in Fig. 5.5. The CEB FIP consistently overestimates the total prestress losses, with increasing divergence from the proposed method with the increase in ϕ and ρ ns values. 30 0 10 20 30 40 1 2 3 χϕ Prestress Loss, Δσps( cr) ( ksi) CEB FIP MC ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Fig. 5.5 Comparison between proposed method and CEB FIP method for long term prestress losses due to creep ( ρ ps = 0.8%). 0 5 10 15 20 1 2 3 χϕ Prestress Loss, Δσps( sh) ( ksi) CEB FIP MC ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method Fig. 5.6 Comparison between proposed method and CEB FIP method for long term prestress losses due to shrinkage ( ρ ps = 0.8%). 31 10 20 30 40 50 1 2 3 χϕ Prestress Loss, Δσps ( ksi) CEB FIP MC ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Fig. 5.7 Comparison between proposed method and CEB FIP method for total long term prestress losses ( ρ ps = 0.8%). 5.3 Canadian Highway Bridge Design Code ( CHBDC) The comparisons between the CHBDC and the proposed method with will be performed with respect to the relative humidity RH since it is the main variable in the CHBDC equations ( Eq. 3.8 through 3.10). For each value of RH, the creep and shrinkage coefficients ( to be used in the proposed method) were calculated using the empirical models of the CHBDC and assuming a V/ S ratio of 5 in. It should be noted that an increase in RH essentially reduces the creep and shrinkage values. The prestress losses due to creep according to both methods are shown in Fig. 5.8. The CHBDC equation considerably underestimates prestress losses due to creep for all ratios of non prestressed steel ρ ns ; the difference decreases with the increase in relative humidity RH. Figure 5.9 shows the CHBDC and the proposed method predictions for shrinkage losses. In case of ρ ps = 0.2%, the CHBDC equation compares very well with the proposed 32 method. For ρ ns = 1.5 and 3.0%, the CHBDC equation underestimates the shrinkage losses for RH values less than 70%; for higher values of RH, the CHBDC predictions compare well with the proposed method for all values of ρ ns . The total prestress losses using the proposed method ( summation of Eq. 4.18 and 4.22) and CHBDC method ( summation of Eq. 3.8 through 3.10) are compared in Fig. 5.10. In Eq. 3.10, f pu was taken 270 ksi and the ratio of steel stress at transfer to ultimate strength σ p0 f pu was assumed 0.7. The CHBDC predictions significantly underestimate the total losses, except for sections with ρ ns = 0.2% and RH greater than 75%. Similar to the observations made for creep losses, the difference between the two methods decreases with increasing RH values. 0 10 20 30 40 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps( cr) ( ksi) CHBDC 2000 ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method Fig. 5.8 Comparison between proposed method and CHBDC method for long term prestress losses due to creep ( ρ ps = 0.8%). 33 0 2 4 6 8 10 12 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps( sh) ( ksi) CHBDC 2000 ρ ns = 0.2% ρ ns = 1.5% Proposed { ρ ns = 3.0% method Fig. 5.9 Comparison between proposed method and CHBDC method for long term prestress losses due to shrinkage ( ρ ps = 0.8%). 0 10 20 30 40 50 60 30 40 50 60 70 80 90 100 Relative Humidity, RH (%) Prestress Loss, Δσps ( ksi) CHBDC 2000 ρ ns = 0.2% ρ ns = 1.5% ρ ns = 3.0% Proposed { method Fig. 5.10 Comparison between proposed method and CHBDC method for total long term prestress losses ( ρ ps = 0.8%). 34 6. Examples To further illustrate the use of the proposed method, two examples of cast in place posttensioned bridges currently under construction in San Diego County will be analyzed for long term prestress losses; the results from the proposed model will be compared with the current specifications of bridge codes. Since the construction drawings for these bridges were produced in SI units, the input data and results in this section, unlike the rest of the report, will be given in SI units only. The following parameters are assumed in the analysis of both bridges: ϕ = 2.5; χ = 0.8; ε cs = − 500 × 10 6; ( ) Ec t0 = 22.5 GPa; Eps = 195 GPa; Ens = 200 GPa. To account for immediate losses, the curvature friction coefficient μ and anchor set are assumed 0.2 and 10 mm, respectively. Since only one layer of prestressing is assumed, there are no immediate losses due to elastic shortening. 6.1 Lake Hodges Bridge Fig. 6.1 shows a half elevation and a half cross section of the Lake Hodges Bridge. Sections A to E shown in Fig. 6.1( a) are analyzed. The concrete dimensions of the bridge at the analyzed sections are shown in Fig. 6.1( b) and Table 6.1. The non prestressed steel ratios at top and bottom slabs, ρ ns1 and ρ ns2 , respectively, and the prestressed steel ratio ρ ps , are listed in Table 6.2. The variation of the prestressing force P after transfer, as a result of immediate losses, and its location d ps are also given in Table 6.2; posttensioning is performed from both ends. A B 40.7 m C D 45.8 m E 21.35 m Fig. 6.1( a) ( a) 35 C. L. 305 Typ Bt = 12480 mm ht hb h= 1850 mm B = 11260 mm Ans 2 Ans 1 Fig. 6.1 Lake Hodges Bridge: ( a) Half elevation; and ( b) Half cross section. Table 6.1 Lake Hodges Bridge: Concrete Dimensions Section ht hb Bt B ΣBw B ht h hb h A 215 180 1.11 0.12 0.12 0.1 B 215 305 1.11 0.12 0.12 0.16 C 215 180 1.11 0.12 0.12 0.1 D 215 305 1.11 0.12 0.12 0.16 E 215 180 1.11 0.12 0.12 0.1 Table 6.2 Lake Hodges Bridge: Reinforcement and Prestressing Section Ans1 ( mm2) ρ ns1 (%) Ans2 ( mm2) ρ ns2 (%) Aps ( mm2) ρ ps (%) d ps ( mm) d ps h P ( kN) A 32290 0.6 22060 0.54 42280 1.06 1510 0.82 55576 B 67410 1.25 46111 0.67 42280 1.16 355 0.19 55732 C 32290 0.6 22060 0.54 42280 1.06 1510 0.82 53472 D 67410 1.25 46111 0.67 42280 1.16 355 0.19 51303 E 32290 0.6 22060 0.54 42280 1.06 1510 0.82 49079 ( b) 36 The coefficients kA , kI , and kh ( Eq. 4.15) are evaluated for each section and listed in Table 6.3. The computer program CPF13 is used to analyze the bridge due to its own weight and prestressing. The output results from the program were used to determine the strain at the centroid of net concrete section ( ) ε O t0 and the slope of the strain diagram ( ) ψ t0 . The prestress loss due to creep and shrinkage Δσ ps is calculated using Eqs. 4.16 and 4.18. The prestress loss due to continuity Δσ ps( cont ) is evaluated using Eqs. B. 17 through B. 23. As can be seen from Table 6.3, Δσ ps( cont ) is a very small amount compared to Δσ ps . The total pestress loss Δσ ps( total ) is given is the right column of the table. The prestress loss for Lake Hodges Bridge using various code predictions are listed and compared to those of the proposed method in Table 6.4. While the AASHTO approximate method predictions are close to those of the proposed method, the AASHTO detailed method underestimates the total losses by an average of 24%. On the other hand, the CEB FIP and the CHBDC predictions are, respectively, 13% higher and 11% less than the predictions of the proposed method. It should be noted that the comparison results in Table 6.4 are intended only to this particular bridge along with the creep and shrinkage values considered and should not be taken as a general conclusion. Even without changing the creep and shrinkage values, these ratios could change with changing the bridge cross section, as will be seen in the next comparison. Table 6.3 Lake Hodges Bridge: Analysis results Section kA kI kh ( ) ε O t0 ( x 10 6) ( ) ψ t0 ( x 10 6 / m) Δσ ps ( MPa) Δσ ps( cont ) ( MPa) Δσ ps( total ) ( MPa) A 0.84 0.83 0.13  180 34 139 1 140 B 0.79 0.77  0.15  149  65 112  1 111 C 0.84 0.83 0.13  173 14 142 0 142 D 0.79 0.77  0.15  137  37 114 1 115 E 0.84 0.83 0.13  159  6 142  2 140 37 Table 6.4 Lake Hodges Bridge: Comparison with design specifications AASHTO Approximate AASHTO Detailed CEB FIP CHBDC Section Δσ ps Ratio* Δσ ps Ratio* Δσ ps Ratio* Δσ ps Ratio* A 128 0.91 102 0.73 151 1.08 121 0.86 B 123 1.11 89 0.80 134 1.21 102 0.92 C 128 0.90 103 0.73 154 1.09 123 0.87 D 123 1.08 90 0.79 136 1.19 104 0.91 E 128 0.90 103 0.73 154 1.08 123 0.87 Average 0.98 0.76 1.13 0.89 * Ratio of prestress loss using design specification to that of the proposed method. 6.2 Duenda Road Over crossing Fig. 6.2 shows an elevation and a cross section of the Duenda Road Over crossing. Sections A to C shown in Fig. 6.2( a) are analyzed. The concrete dimensions of the bridge at the analyzed sections are shown in Fig. 6.1( b) and Table 6.5. Non prestressed and prestressing steel data are given in Table 6.6. Post tensioning is performed from the longspan end. The analysis results and comparisons with design specifications are reported in Tables 6.7 and 6.8, respectively. The reason for choosing this bridge for analysis is to determine whether a significant difference in the length of two adjacent spans in a continuous bridge could have any effect on the prestress losses due to continuity. As shown in Table 6.7, the continuity effect is quite insignificant and could be ignored. This could be explained by referring to Eq. 4.17, which shows that the prestress loss depends on the change in axial strain at the centroid of the net concrete section Δε O and the change in curvature Δψ . For most practical applications, the change in axial strain is the most dominant factor. The analysis for continuity effects only calculates the additional change in Δψ . In addition, in the calculation of angular discontinuity ΔD1 at one end of intermediate support ( see for instance Eq. 4.25 for two spans) ( ) Δψ A and ( ) Δψ B are of 38 opposite signs therefore reduce ΔD1 , which in turn reduces the long term change in continuity moment ΔF1 . 58.8 m 38.2 m A B C h= 2350 mm 305 Typ Bt = 12510 mm B = 7880 mm hb Ans 1 ht Ans 2 Fig. 6.2 Duenda Road Over crossing: ( a) Elevation; and ( b) Cross section. Table 6.5 Duenda Road Over crossing: Concrete Dimensions Section ht hb Bt B ΣBw B ht h hb h A 220 185 1.51 0.16 0.09 0.08 B 220 305 1.51 0.16 0.09 0.13 C 220 255 1.51 0.16 0.09 0.11 ( a) ( b) 39 Table 6.6 Duenda Road Over crossing: Reinforcement and Prestressing Section Ans1 ( mm2) ρ ns1 (%) Ans2 ( mm2) ρ ns2 (%) Aps ( mm2) ρ ps (%) d ps ( mm) d ps h P ( kN) A 11148 0.43 10001 0.69 36000 1.19 1890 0.80 44788 B 27837 1.07 14910 0.57 36000 1.19 460 0.20 44411 C 24401 1.21 14910 0.74 36000 1.19 1145 0.49 43855 Table 6.7 Duenda Road Over crossing: Analysis results Section kA kI kh ( ) ε O t0 ( x 10 6) ( ) ψ t0 ( x 10 6 / m) Δσ ps ( MPa) Δσ ps( cont ) ( MPa) Δσ ps( total ) ( MPa) A 0.81 0.80 0.27  343  45 162 1 163 B 0.78 0.79  0.20  122  123 136  2 134 C 0.78 0.83  0.005  335 63 175 0 175 Table 6.8 Duenda Road Over crossing: Comparison with design specifications AASHTO Approximate AASHTO Detailed CEB FIP CHBDC Section Δσ ps Ratio* Δσ ps Ratio* Δσ ps Ratio* Δσ ps Ratio* A 129 0.79 129 0.79 173 1.06 160 0.98 B 129 0.96 107 0.80 154 1.15 129 0.96 C 129 0.74 130 0.74 199 1.14 162 0.93 Average 0.83 0.78 1.12 0.96 * Ratio of prestress loss using design specification to that of the proposed method. 40 7. Conclusions Based on the analytical studies presented in the present section, the following conclusions can be made: • The long term behavior of concrete bridges is a rather involved procedure that depends on many parameters. The prediction of long term prestress losses from equations that are function of only one or two parameters, as in the case of all the equations of bridge codes, cannot produce accurate results for all cases. • Although relative humidity is one of the major factors that affect the shrinkage and creep strains, it is not the only one. Equations for prestress losses that are functions of only the relative humidity can lead to misleading results. It is recommended that the prediction equations be functions of the creep and shrinkage coefficients as determined from codes of practice. • Accounting for the effect of non prestressed steel is very essential to produce reliable results for prestressing losses. Neglecting this effect, as in the case of the CEB FIP method, can greatly overestimate the prestress losses. Taking this effect into account in an empirical fashion, as in the case of the other prediction equations, can produce predictions that do not follow the actual trend of prestress losses. • The AASHTO LRFD upper bound approximate method is in fact a lower bound. The approximate average method lies outside the range of prestress loss predictions. • The CHBDC gives reasonable and better predictions for prestress losses compared with the AASHTO LRFD and CEB FIP methods. 41 References 1. Trost, H., “ Auswirkungen des Superpositionspringzips auf Kriech und Relaxationsproblems bei Beton und Apannbeton,” Beton und Stahlbetonbau, V. 62, No. 10, 1967, pp. 230 238; No. 11, 1967, pp. 261 269 ( in German). 2. Bazant, Z. P., “ Prediction of Concrete Creep Effects Using Age Adjusted Effective Modulus,” ACI Journal, V. 69, No. 4, 1972, pp. 212 217. 3. Magura, D. D., Sozen, M. A., and Siess, C. P., “ A Study of Stress Relaxation in Prestressing Reinforcements,” PCI Journal, V. 9, No. 2, 1964, pp. 13 57. 4. Ghali, A., and Trevino, J., “ Relaxation of Steel in Prestressed Concrete,” PCI Journal, V. 30, No. 5, 1985, pp. 82 94. 5. American Association of State Highway and Transportation Officials, “ AASHTOLRFD Bridge Design Specifications,” Third Edition, Washington, DC, 2004. 6. Comité Euro International du Beton − Fédération Internationale de la Précontrainte, “ Model Code for Concrete Structures,” CEB FIP MC 90, London, UK, 1993. 7. Canadian Highway Bridge Design Code, CAN/ CSA S6 00, Rexdale, Canada, 2000. 8. Ghali, A., Favre, R., and Elbadry, M. M., “ Concrete Structures: Stresses and Deformations,” Third Edition, Spon Press, London & New York, 2002, 584 pp. 9. ACI Committee 209, “ Prediction of Creep, Shrinkage, and Temperature Effects in Concrete Structures,” Committee Report 209R 92, American Concrete Institute, Detroit, MI, 1992, 47 pp. 10. http:// www. ncdc. noaa. gov/ oa/ climate/ online/ ccd/ avgrh. html 11. Tadros, M. K., Al Omaishi, N., Serguirant, S. J., and Gallt, J. G., “ Prestress Losses in Pretensioned High Strength Concrete Bridge Girders,” NCHRP Report 496, Transportation Research Board, Washington, DC, 2003, 63 pp. 12. Ghali, A., Neville, A. M., and Brown, T. G., “ Structural Analysis: A Unified Classical and Matrix Approach,” Fifth Edition, Spon Press, London & New York, 2004, 844 pp. 42 13. Elbadry, M. M., and Ghali, A., “ User’s Manual and Computer Program CPF: Cracked Plane Frames in Prestressed Concrete,” Research Report CE 85 2, Department of Civil Engineering, University of Calagry, Calgary, Alberta, Canada, 1990, 82 pp. 43 Appendix A: Coefficients kA , kI and kh Figures A. 1 to A. 18 give the coefficients kA , kI and kh included in Eq. 4.18 to calculate the long term prestress loss Δσ ps . The symbols in the following figures are defined below ( see Fig. 4.3): Bt and B = widths of top and bottom flanges, respectively; ht and hb = depths of top and bottom flanges, respectively; h = total depth of concrete section; Ans1 and Ans2 = areas of non prestressed steel in top and bottom flanges, respectively; Aps = area of prestressing steel; ΣBw = Bw1 + Bw2 + ... = summation of web widths of the cross section; ρ ns1 = Ans1 Bt ht = ratio of non prestressed steel in top flange; ρ ns2 = Ans2 B hb = ratio of non prestressed steel in bottom flange; and ρ ps = Aps h ΣBw = ratio of prestressing steel with respect to web areas. In the graphs below, the modulus of elasticity of prestressing steel Eps and nonprestressed steel Ens were assumed 27.5 ksi ( 190 GPa) and 29 ksi ( 200 GPa), respectively. The modulus of elasticity of concrete Ec at time of post tensioning was assumed 3.6 ksi ( 25 GPa); the aging coefficient χ was assumed 0.8. Two cases of top and bottom thicknesses were assumed: Case 1 ht h = 0.05 and hb h = 0.05; and Case 2 ht h = 0.15 and hb h = 0.15. The following should be noted when using these graphs: • The value of kA is the same for sections with d ps = 0.8h or 0.2h; therefore, no distinction was indicated on the graphs. • The value of kI is the same for sections with d ps = 0.8h or 0.2h and having Bt B = 1. For sections having Bt B = 2, two curves ( in dashed lines) are shown for kI . 44 • The value of kh for sections with d ps = 0.2h and having Bt B = 1 is the same ( but with opposite sign) for sections with d ps = 0.8h. For sections having Bt B = 2, two curves ( in dashed lines) are shown for kh . 45 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 1 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. 46 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 2 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. 47 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = Fig. A. 3 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 0.8%. 48 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 4 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. 50 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 6 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 0.8%. 51 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 7 kA , kI and kh for the case ΣBw B = 0.1, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. 52 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.2 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.2 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 8 kA , kI and kh for the case ΣBw B = 0.2, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. 53 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 9 kA , kI and kh for the case ΣBw B = 0.3, ρns1 = ρns2 = 3.0%, ρ ps = 0.8%. 54 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 10 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%. 55 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.2 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.2 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 11 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%. 56 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.8  0.6  0.4  0.2 0.0 0.2 0.4 0.6 0.8 1 2 3 4 χϕ kh Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.3 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 12 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 0.2%, ρ ps = 1.2%. 57 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 13 kA , kI and kh for the case ΣBw B = 0.1, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%. 58 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 14 kA , kI and kh for the case ΣBw B = 0.2, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%. 59 0.6 0.7 0.8 0.9 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.5 0.6 0.7 0.8 0.9 1 2 3 4 χϕ kI Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.6  0.4  0.2 0.0 0.2 0.4 0.6 1 2 3 4 χϕ kh Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 15 kA , kI and kh for the case ΣBw B = 0.3, ρ ns1 = ρ ns2 = 1.5%, ρ ps = 1.2%. 60 0.2 0.4 0.6 0.8 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Σ B w / B = 0.1 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.1 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 16 kA , kI and kh for the case ΣBw B = 0.1, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%. 61 0.2 0.4 0.6 0.8 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.2 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 17 kA , kI and kh for the case ΣBw B = 0.2, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%. 62 0.2 0.4 0.6 0.8 1.0 1 2 3 4 χϕ kA Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 0.4 0.5 0.6 0.7 0.8 1 2 3 4 χϕ kI Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h  0.4  0.2 0.0 0.2 0.4 1 2 3 4 χϕ kh Σ B w / B = 0.3 Case 1, B t / B = 1.0 Case 1, B t / B = 2.0 Case 2, B t / B = 1.0 Case 2, B t / B = 2.0 Case 1, B t / B = 2.0 Case 2, B t / B = 2.0 y ps = 0.2h Fig. A. 18 kA , kI and kh for the case ΣBw B = 0.3, ρns1 = ρns2 = 3.0%, ρ ps = 1.2%. 63 Appendix B: Prestress Loss Due to Continuity The approach to estimate prestress losses due to continuity in two unequal span continuous beam was outlined in Section 4.6. In the following, the approach will be extended to symmetric beams with two to five spans. The coordinate system and locations of integration points ( sections) are illustrated in Figs. B. 1( a) to ( d). The sections are located either at intermediate supports or at mid spans. In case of four or five spans, the equations are to be solved for two unknows: the connecting moments over intermediate supports ΔF1 and ΔF2 . The prestress loss due to continuity at each section is evaluated for the combined effect of the values of ΔF1 and ΔF2 at this section; see Eqs. B. 16 and B. 23. For definition of symbols used in the equations, refer to Sec. 4.6. l 1 l 1 A B A D 1 l 1 l 2 A B C D 1 l 1 D 1 B A l 1 l 2 A B C D 1 l 2 D 2 D C l 1 B A D 1 l 1 l 2 A B C D 1 l 3 D 2 D E l 1 D C D 2 B A D 1 l 1 Fig. B. 1 Coordinate system and locations of integration points ( sections) for continuous beams: ( a) two spans; ( b) three spans; ( c) four spans; and ( d) five spans. ( a) ( b) ( c) 64 ( a) Two Spans: [( ) ( ) ] A B ΔD = l 2 Δψ + Δψ 3 1 1 ( B. 1) [( ) ( )] u A u B f l 1 1 1 11 2 3 = Δψ + Δψ ( B. 2) 11 1 1 f F D − Δ Δ = ( B. 3) ( ) i ps c ps ps cont i y I F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 σ ( ) ( B. 4) ( b) Three Spans: [( ) ( ) ] [( ) ( ) ] A B C B ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 1 2 1 ( B. 5) [( ) ( )] [( ) ( )] u A u B u C u B f l l 1 1 2 1 1 1 11 2 6 2 6 = Δψ + Δψ + Δψ + Δψ ( B. 6) 11 1 1 f F D − Δ Δ = ( B. 7) ( ) i ps c ps ps cont i y I F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 σ ( ) ( B. 8) ( c) Four Spans: [( ) ( ) ] [( ) ( ) ] A B C B ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 1 2 1 ( B. 9) [( ) ( ) ] C D ΔD = l 2 Δψ + Δψ 3 2 2 ( B. 10) [( ) ( )] [( ) ( )] u A u B u C u B f l l 1 1 2 1 1 1 11 2 6 2 6 = Δψ + Δψ + Δψ + Δψ ( B. 11) [( ) ( ) ] u C u D f l 2 2 2 22 2 3 = Δψ + Δψ ( B. 12) [ ( ) ] u C f l 1 2 21 2 3 = Δψ ; [ ( ) ] u C f l 2 2 12 2 6 = Δψ ( B. 13) 65 ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Δ Δ = − ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Δ Δ ⎥⎦ ⎤ ⎢⎣ ⎡ 2 1 2 1 21 22 11 12 D D F F f f f f ( B. 14) ( ) ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − Δ − Δ ⎥⎦ ⎤ ⎢⎣ ⎡ − − − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Δ Δ 2 1 21 11 22 12 2 11 22 12 21 1 1 D D f f f f F f f f f F ( B. 15) ( ) i ps c ps ps cont i y I F F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ + Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 2 σ ( ) ( B. 16) ( d) Five Spans: [( ) ( ) ] [( ) ( ) ] A B C B ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 1 2 1 ( B. 17) [( ) ( ) ] [( ) ( ) ] C D C E ΔD = l Δψ + Δψ + l 2 Δψ + Δψ 6 2 6 2 3 2 ( B. 18) [( ) ( )] [( ) ( )] u A u B u C u B f l l 1 1 2 1 1 1 11 2 6 2 6 = Δψ + Δψ + Δψ + Δψ ( B. 19) [( ) ( ) ] [( ) ( )] u C u D u C u E f l l 2 2 3 2 2 2 22 2 6 2 6 = Δψ + Δψ + Δψ + Δψ ( B. 20) [ ( ) ] u C f l 1 2 21 2 6 = Δψ ; [ ( ) ] u C f l 2 2 12 2 6 = Δψ ( B. 21) ( ) ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − Δ − Δ ⎥⎦ ⎤ ⎢⎣ ⎡ − − − = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ Δ Δ 2 1 21 11 22 12 2 11 22 12 21 1 1 D D f f f f F f f f f F ( B. 22) ( ) i ps c ps ps cont i y I F F E E ⎟⎠ ⎞ ⎜⎝ ⎛ Δ + Δ ⎟ ⎟⎠ ⎞ ⎜ ⎜⎝ ⎛ Δ = 1 2 σ ( ) ( B. 23) 



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