【正文】 parameters for all layers. Step 4: Elimination of artificial restraint. The artificial forces ΔN and ΔM can be applied in reversed direction on the ageadjusted transformed section to give the true change in strain at O, ΔεO, and in curvature, Δψ, such that (14a) (14b) where is the second moment of about its centroid and is the area of ageadjusted transformed section defined as (15) where Ef and Ep are the moduli of elasticity for the FRP reinforcement and tendons, respectively, and the is as defined in Eq. (11). Substituting Eqs. (12) and (13) into Eqs. (14a), (14b) and (15) gives (16) and (17) 11 where (18) The timedependent change in strain in prestressing tendons Δεp can then be evaluated using Eq. (19) and the timedependent change in stress in prestressing tendons (described by Eq. (20)) is the sum of EpΔεp and the reduced relaxation. Δεp=ΔεO+ypΔψ (19) (20) Substitution of Eqs. (16) and (17) into Eq. (20) gives an expression for the longterm prestress loss, Δσp, due to creep, shrinkage, and relaxation as (21) It should be noted that the last term in Eq. (21), , is zero in the case of prestressed members using CFRP tendons. (23) (24) 4. Application to continuous girders Prestressing of continuous beams or frames produces statically indeterminate bending moments (referred to as secondary moments). As mentioned previously, ε1(t0) and ψ(t0) (Eqs. (7), (8) and (9)) represent the strain parameters at a section due to dead load plus the primary and secondary moments due to prestressing. The 12 timedependent change in prestress force in the tendon produces changes in these secondary moments, which are not included in Eq. (21). This section considers the effect of the timedependent change in secondary moments on the prestress loss. Step 1: Considering a twospan continuous beam, as shown in Fig. 4(a) where the variation of the tendon profile is parabolic in each span, the statically indeterminate beam can be solved by any method of structural analysis (such as the force method) to determine the moment diagram at time t0 due to dead load and prestressing. (14K) Fig. 4. Twospan continuous prestressed girder. (a) Dimensions and cable profile。 FRP。 Shrinkage Nomenclature A area of cross section d vertical distance measured from top fiber of cross section 2 E modulus of elasticity ageadjusted elasticity modulus of concrete fpu ultimate strength of prestressing tendon h total thickness of concrete cross section I second moment of area O centroid of ageadjusted transformed section t final time (end of service life of concrete member) t0 concrete age at prestressing y coordinate of any fiber measured downward from O χ aging coefficient χr reduced relaxation coefficient α ratio of modulus of elasticity of FRP or steel to that of concrete Δεc(t,t0) change in concrete strain between time t0 and t ΔεO change in axial strain at the centroid of ageadjusted transformed section O Δσc(t,t0) stress applied gradually from time t0 to its full amount at time t Δσpr intrinsic relaxation reduced relaxation Δσp total longterm prestress loss Δψ change in curvature εcs shrinkage strain of concrete between t0 and t εc(t0) instantaneous strain at time t0 (t, t0) creep coefficient between t0 and t σc(t0) stress applied at time t0 and sustained to a later time t σp0 initial stress of prestressing tendon ρ reinforcement ratio ψ curvature Ω the ratio of the difference between the total prestress loss and intrinsic relaxation to the initial stress Subscripts 3 1 transformed section at t0 c concrete cc concrete section f FRP reinforcement or flange p prestressing FRP tendon ps prestressing steel tendon s steel reinforcement Article Outline Nomenclature 1. Introduction 2. Relaxation of FRP prestressing tendons 3. Proposed method of analysis . Initial steps . Timedependent change in concrete stress . Longterm deflection 4. Application to continuous girders 5. Development of design aids 6. Illustrative example 7. Summary Acknowledgements References 1. Introduction The use of fiber reinforced polymer (FRP) tendons as prestressing reinforcements have been proposed in the past decade and a few concrete bridges have already been constructed utilizing fiber reinforced polymer (FRP) tendons. Compared to conventional steel prestressing tendons, FRP tendons have many advantages, including their noncorrosive and nonconductive properties, lightweight, and high tensile strength. Most of the research conducted on concrete girders prestressed with 4 FRP tendons has focused on the shortterm behavior of prestressed members。這種方法滿足了平衡性和兼容性的要求，避免使用一些經(jīng)驗 公式 ?！被炷两Y(jié)構(gòu)中大多數(shù)的 FRP 筋的研究和應(yīng)用要么是碳纖維增強聚合體或 者是 芳族聚酸胺纖維 增強的聚合體。本篇論文的目的是通過對用 FRP 筋制作的預(yù)應(yīng)力混凝土構(gòu)件中 隨時間而定的應(yīng)變和應(yīng)力 的評估這樣一個簡單的分析法來強調(diào)錯誤的第二來源。由此得出結(jié)論，收縮時應(yīng)變 ε 為負值。當(dāng)鋼筋受到的應(yīng)力低于屈服應(yīng)力的 50%時，不會呈現(xiàn)出可感知的松弛，對 AFRP筋的測試表明在很低的應(yīng)力作用下它們會產(chǎn)生松弛。因此，松弛的減少量應(yīng)該采用預(yù)應(yīng)力構(gòu)件長期效應(yīng)的分析值，因此 (3)，其中 χ r是無量綱系數(shù)決不一致。這種方法將會得出一個一次方程，容易被實踐工程師運用，而不是冗長的矩陣分析法只能用于特殊用途的計算機程序。當(dāng) y = ycc 時，關(guān)系如圖3所示，因此 (Δ ε cc)free= ε cc(t0)+ε cs， (7)這兒的 ycc是混凝土凈截面質(zhì)心處的y坐標(biāo)， 是 t0到 t 時間內(nèi)的徐變系數(shù)， ε cs是在相同時間內(nèi)的收縮， ε cc(t0)是在 混凝土凈截面質(zhì)心處的應(yīng)變，它們的關(guān)系如下 ε cc(t0)=ε 1(t0)+(yccy1)ψ (t0) (8)，其中 y1是在 t0時刻換算面積處的質(zhì)心， ψ (t0)是在 t0時的曲率。 (16) (17) (18) 預(yù)應(yīng)力鋼筋中應(yīng)變隨時間的變化能按（ 19）式計算。 步驟 1：考慮一個兩跨連續(xù)梁，如圖 4（ a）所示，每跨腱變化的輪廓都是拋物線的，超靜定梁可以通過結(jié)構(gòu)分析的任何方法來解決由于恒載和預(yù)應(yīng)力引起的在 t0時刻產(chǎn)生的彎矩。（ b） 連接處彎矩