【正文】 etensioning) at y = ycc as shown in Fig. 3, such that (Δεcc)free= εcc(t0)+εcs (7) where ycc is the y coordinate of the centroid of the concrete section, is the creep coefficient for the period t0 to t, and εcs is the shrinkage in the same period and εcc(t0) is the strain at the centroid of the concrete section given by 9 εcc(t0)=ε1(t0)+(yccy1)ψ(t0) (8) where y1 is the centroid of the transformed area at t0, and ψ(t0) is the curvature (slope of the strain diagram) at t0. Also free curvature is Δψfree= ψ(t0) (9) (15K) Fig. 3. Typical prestressed concrete section and the strain diagram immediately after transfer. Step 3: Artificial restraining forces. The free strain calculated in Step 2 can be artificially prevented by a gradual application of restraining stress, whose value at any fiber y is given by (10) where is the ageadjusted modulus of concrete [5] and [6], used to account for creep effects of stresses applied gradually to concrete and is defined as (11) The artificial restraining forces, ΔN at the reference point O (which is the centroid of the ageadjusted transformed section), and ΔM, that can prevent strain changes due to creep, shrinkage and relaxation can be defined as (12) and 10 (13) where Ic, yp, and are the second moment of Ac about its centroid, y coordinate of the centroid of the FRP tendons, and the reduced relaxation stress between times t0 and t. It should be noted that if the section contains more than one layer of prestresssing tendons, the terms containing Ap or ypAp should be substituted by the sum of the appropriate 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。目前已有一種簡單的方法，計算添加碳纖維增強的聚合體或者是 芳族聚酸胺纖維 增強的聚合體的連續(xù)的預(yù)應(yīng)力混凝土構(gòu)件的長期的預(yù)應(yīng)力損失和混凝土應(yīng)力的變化。同那些用預(yù)應(yīng)力鋼筋制作的梁相比，混凝土的應(yīng)力變化和變形要么變小，要么變大，這取決于所用的 FRP 筋的類型和所考慮的 橫截面 初始應(yīng)力的分布。最近 ACI 委員會在關(guān)于用 FRP 筋的預(yù)應(yīng)力混凝土結(jié)構(gòu)的報告中指出“關(guān)于預(yù)應(yīng)力的長期損失和隨時間變化的彎曲度和變形的研究是必要的 ? 。 混凝土的徐變和收縮以及預(yù)應(yīng)力筋的松弛引起混凝土結(jié)構(gòu)長期的變形。 在預(yù)測長期的預(yù)應(yīng)力損失發(fā)生錯誤可能是因為以下幾個方面的原因：（ 1）在評估材料長期的特性上不準確（如：混凝土的徐變和收縮以及預(yù)應(yīng)力筋的松弛）；（ 2）使用的分析方法不正確。 為了避免 這篇論文產(chǎn)生混淆，采用協(xié)定的一致的符號。向下的變形為正。 2. FRP 預(yù)應(yīng)力筋的松弛 與混凝土和鋼筋相似， AFRP 預(yù)應(yīng)力筋當遭受到持續(xù)的應(yīng)變時會顯示出徐變。這種應(yīng)力的減少被認為是固有松弛 Δ σ pr。在測試中 σ p1/σ p0的比值在 和 之間變化，平均值是 。 這種應(yīng)力的減少于鋼筋受到較小的初始應(yīng)力有相似的效果。 Ω 是總的預(yù)應(yīng)力損失與固有松弛的差和初始應(yīng)力的比，表達式為 (6)。考慮到由簡單的混凝土組成的任意截面，在 t0處受到預(yù)應(yīng)力和永久荷載這種程序能夠進一步發(fā)展。在任何的纖維層，由于永久荷載和預(yù)應(yīng)力的效應(yīng)下，能計算出在時間 t0處的應(yīng)變和曲率。在 t0到 t的時間內(nèi)由于徐變和收縮引起的混凝土應(yīng)變的分布式通過混凝土凈截面區(qū)域質(zhì)心處的值 (Δ ε cc)free來表示， Ac代表總面積減去 FRP 筋的面積，在后張拉的情況下， Af是 總面積減去 預(yù)張拉管道的面積或者是減去 FRP 筋的面積，是在先張拉的情況下。步驟 2里計算出的自由徐變是可以通過逐步的控制應(yīng)力來人為的預(yù)防，在任意纖維層 y 處 (10)，其中是經(jīng)調(diào)整后的混凝土模量，用來說明逐步施加到混凝土上的應(yīng)力效應(yīng)，被定義為 (11)，在參考點處人為控制的力可以阻止由于徐變，收縮，松弛引起的應(yīng)變改變， Δ N和 Δ M 表達式分別是 (12) 和 (13)， Ic, yp, and 分別是面積的二階矩， FRP 筋質(zhì)心處的 y坐標，在 t0和 t時間內(nèi)由于松弛減少的應(yīng)力。其中 Ef和 Ep分別是 FRP 筋和鋼筋的彈性模量， 得表達式為（ 11）。（ 20）給出了由于徐變，收縮，和松弛引起的長期的預(yù)應(yīng)力損失 Δ σ p，(21)。 25 這部分考慮了預(yù)應(yīng)力損失中次彎矩隨時間變化的效應(yīng)。圖 5所示的用坐標系表示的基本結(jié)構(gòu)可以運用。 Δ D1可以表示成(25) (10K) 圖 （ a） Released structure and coordinate system。 幾何系數(shù) kA, kI, kcc和 kp取決于截面的幾何形狀和材料的參數(shù) Ef/Ec(t0), Ep/Ec(t0)， χ 。表中沒有的數(shù)據(jù)可以用線性內(nèi)插法得到。在大多數(shù)研制試驗中用的預(yù)應(yīng) 力梁，輔助設(shè)計使得計算方法進一步簡化。 參考書目 [1] ACI Committee 440. FRP 筋預(yù)應(yīng)力混凝土結(jié)構(gòu) 4R04， 美國混凝土協(xié)會， Farmington Hills, MI, 2020. [2] H. Saadatmanesh and . Tannous,《芳綸纖維增強塑料筋的長期特征》 , ACI Mater J 96 (1999) (3), pp. 297– 305. [3] A. Ghali and J. Trevino,《預(yù)應(yīng)力混凝土中鋼筋的松弛》 , PCI J 30 (1985) (5), pp. 82– 94.