【正文】 在大多數(shù)研制試驗(yàn)中用的預(yù)應(yīng) 力梁，輔助設(shè)計(jì)使得計(jì)算方法進(jìn)一步簡化。 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 。圖 5所示的用坐標(biāo)系表示的基本結(jié)構(gòu)可以運(yùn)用。（ 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)。步驟 2里計(jì)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(diǎn)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(dá)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(biāo)，在 t0和 t時(shí)間內(nèi)由于松弛減少的應(yīng)力。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(jì)算出在時(shí)間 t0處的應(yīng)變和曲率。 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(dá)式為 (6)。在測試中 σ p1/σ p0的比值在 和 之間變化，平均值是 。 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(dāng)遭受到持續(xù)的應(yīng)變時(shí)會(huì)顯示出徐變。 為了避免 這篇論文產(chǎn)生混淆，采用協(xié)定的一致的符號(hào)。 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。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。C, relationships for a and b were proposed [2] as (2) In a prestressed concrete member, the two ends of the prestressing tendon constantly move toward each other because of creep and shrinkage of concrete, thereby reducing the tensile stress in the tendon. This reduction in tension has a similar effect to that when the tendon is subjected to a lesser initial stress. Thus, a reduced relaxation value, , should be used in the analysis of longterm effects in prestressed members, such that (3) where χr is a dimensionless coefficient less than unity. Following an approach previously suggested by Ghali and Trevino [3] to evaluate χr for prestressing steel tendons, χr for AFRP tendons can be calculated as (log t in Eq. (1) is taken equal to 5 for 100,000 h): (4) where (5) 7 and ζ is a dimensionless time function defining the shape of the tendon stress–time curve. The value of ζ increases from 0 to 1 as time changes from initial prestress time t0 to final time t. Ω is the ratio of the difference between the total prestress loss Δσps(t) and intrinsic relaxation Δσpr(t) to the initial stress σp0, expressed as (6) Fig. 1 shows the variation of χr with Ω for σp0/fpu = , , and , which represents the mon values of initial prestressing ratios [1]. As will be shown in a later section, Ω typically varies between and and a value of χr = can be assumed for practical purposes. (20K) Fig. 1. Reduced relaxation coefficient χr for AFRP. 3. Proposed method of analysis The analysis follows the four generic steps proposed by Ghali et al. [4] and depicted schematically in Fig. 2. The procedure can be developed considering an arbitrary section consisting of a simple type of concrete, subjected at time t0 to both prestressing and dead loads. The method will result in a simple equation that is easy to use by practicing engineers instead of lengthy matrix analysis that could only be used in specialpurpose puter programs. In addition to the initial strain profile of the cross section, the equation is only a function of four dimensionless coefficients that can be easily calculated (or interpolated from graphs) and the creep coefficient and shrinkage. 8 (56K) Fig. 2. Four steps of analysis of timedependent effects (after Ghali et al. [4]). . Initial steps Step 1: Instantaneous strains. At any fiber, the strain and the curvature at time t0 due to the dead load and prestressing effects (primary + secondary) can be calculated. Alternatively, at this stage, the designer may have determined the stress distribution at t0 to verify that the allowable stresses are not exceeded. In this case, the strain diagram at t0 can be obtained by dividing the stress values by the modulus of elasticity of concrete at t0, Ec(t0). Step 2: Free creep and shrinkage of concrete. The distribution of hypothetical free change in concrete strain due to creep and shrinkage in the period t0 to t is defined by its value (Δεcc)free at the centroid of the area of the concrete section, Ac (defined as the gross area minus the area of the FRP reinforcement, Af, minus the area of the prestressing duct in the case of posttensioning, or minus the area of the FRP tendons, Ap, in case of pretensioning) 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