## 3 Long-Term Losses of Prestress  
The prestress losses in prestressing tendons can largely be divided into short-term losses (also called instantaneous losses or immediate losses) and long-term losses (also called time-dependent losses). Figure 2 shows the types of the prestress losses and the corresponding PF. Details on each prestress loss can be found in the literature [32]. Although the short-term losses are beyond the scope of this study, the friction coefficients related to friction loss were derived in an innovative way using the Smart Strand in two previous studies [2,3]. The present study focuses on the long-term losses. The creep loss occurs because a concrete member subjected to compression by prestressing tendons is shortened for a long time due to the creep. Then, the prestressing tendon is also shortened with the prestress loss. The creep loss can be calculated by Equation (5) based on the composite action between concrete and a tendon, as proposed by [PERSON] et al. [33] and adopted in ACI 423.10R [34]. \[\Delta f_{pCR}=nC_{t}f_{c}, \tag{5}\]  
where \(\Delta f_{pCR}\): creep loss, \(n\): modular ratio (=\(E_{p}/E_{c}\)), \(E_{p}\): modulus of elasticity of a strand (200,000 MPa), \(E_{c}\): modulus of elasticity of concrete, which was calculated as a function of specified concrete compressive strength (\(f_{c}\)') according to ACI 318 [35] in this study, \(C_{t}\): creep coefficient, and \(f_{c}\): compressive stress of concrete at the location of a strand caused by prestressing, self-weight, and superimposed permanent dead loads, where \(f_{c}\) can be calculated by Equation (6). Note that 0.9\(P_{i}\) was used instead of \(P_{i}\) to approximately account in advance for the reduction of \(P_{i}\) during long-term losses [32]. \[f_{c}=\left(\frac{0.9P_{i}}{A_{c}}+\frac{0.9P_{i}e_{p}}{I_{c}}e_{sm}\right)- \frac{M_{d}}{I_{c}}e_{sm}, \tag{6}\]  
where \(P_{i}\): total initial PF (refer to Figure 2), \(A_{c}\): area of the concrete section, \(I_{c}\): second moment of area of the concrete section, \(e_{p}\): eccentricity of the tendon centroid with respect to the concrete centroid, \(e_{sm}\)  
Figure 2: Prestress losses. eccentricity of a specific strand (Smart Strand in this study) of which the stress needs to be obtained, and \(M_{d}\): bending moment due to self-weight and superimposed permanent dead loads. Additionally, \(e_{sm}\) was used in Equation (6) instead of \(e_{p}\) because the theoretical prestress loss at a Smart Strand, not the average prestress loss of all tendons, needs to be compared with the value measured at the Smart Strand for a reasonable comparison. Concrete creep depends on time, concrete compressive strength, member shape, relative humidity, loading age, type of cement, curing condition, temperature, and concrete stress. Equation (7) in ACI 209R [36], originally proposed by [PERSON] and [PERSON] [37], has been frequently used to formulate the time-dependency of creep. \[C_{t}=\frac{t^{0.6}}{10+t^{0.6}}C_{u}, \tag{7}\]  
where \(t\): time after prestressing (days) and \(C_{u}\): ultimate creep coefficient. Equation (7) was suggested, with \(C_{u}=2.35\), for a prestressing age of 7 days for moist-cured concrete and of 1-3 days for steam-cured concrete. Several correction factors are applied to account for other conditions and various effects on the creep [36]. The value of \(C_{u}\) varies depending on the provisions; for example, ACI 423.10R [34] suggested 1.6 for post-tensioned members. The final creep loss can be obtained by using \(C_{u}\) instead of \(C_{t}\) in Equation (5). The shrinkage of concrete due to drying shrinkage and autogenous shrinkage can also cause prestress loss by the reduction of the length of the prestressing tendon. The shrinkage loss can be expressed by Equation (8). \[\Delta f_{p5H}=E_{p}(\varepsilon_{sh})_{t}, \tag{8}\]  
where \(\Delta f_{p5H}\): shrinkage loss, \(E_{p}\): modulus of elasticity of the strand, and \((\varepsilon_{sh})_{t}\): shrinkage strain.