【正文】 ngth 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。 Longterm。 FRP。 Relaxation。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 pr