Analysis of Short-Term Prestress Losses in Post-tensioned Structures Using Smart Strands

A 20-m-long full-scale post-tensioned specimen was fabricated to investigate the short- and long-term characteristics of PF using Smart Strands, as shown in Fig. 4. Three ducts with different curvatures were arranged as denoted by T1, T2, and T3 in Fig. 4. Twelve 15.2-mm-diameter strands, with ultimate tensile strength (\(f_{pu}\)) of 1860 MPa, were inserted into an 85-mm-diameter galvanized metal duct and tensioned up to \(0.7f_{pu}\) at one end of the specimen using a multi-strand jack, in the order of T1, T2, and T3, when the concrete compressive strength reached 30 MPa with the mix proportion shown in Table 1. The Smart Strands were selectively inserted into each duct together with regular strands as shown in the anchor heads of Fig. 5 with the identification of each hole. Three types of Smart Strands were fabricated with three, five, and seven FBGs, respectively, which are indicated as a number in the Smart Strand in Fig. 5. Five and seven FBGs were equally spaced along a Smart Strand with the margin of 1 m at both ends, whereas three FBGs were located at 1, 10, and 16 m from the jacking end.

To extend the applicability of Smart Strands to real infrastructures, two different types of post-tensioned PSC girder bridges (bridges A and B) were employed. Both bridges have eight 60-m-long girders, but their specified concrete compressive strengths are 60 MPa and 80 MPa in bridges A and B, respectively. These strengths were used for the theoretical calculation of elastic shortening loss, because tensioning was performed after a sufficient duration to attain them. Figs. 6 and 7 show the main dimensions and configuration of the galvanized metal ducts in A and B’s girder, respectively. All the strands used in these girders have 15.2-mm diameter and \(f_{pu}\) = 1860 MPa.

Total four Smart Strands were fabricated with seven FBGs. Two different girders (girders 1 and 2) were chosen in each bridge and one Smart Strand was assigned to each girder. As also shown in Figs. 6 and 7, one Smart Strand has equally spaced FBGs, whereas the remaining three Smart Strands have the FBGs at the center and mainly near both ends to investigate the effect of anchorage slip on the PF distribution. The Smart Strands were inserted into the uppermost duct in the girder, where a larger friction loss would be induced by the larger curvature than that of the other ducts. The Smart Strand was added to the original regular strands, which numbered 21 and 11, and the inner diameters of the corresponding ducts were 105 and 75 mm for bridges A and B, respectively. The strands in each duct were tensioned at both ends using two multi-strand jacks in both bridges, where the Smart Strand was tensioned approximately up to \(0.67f_{pu}\). The Smart Strand experienced the elastic shortening loss caused by all the strands included in the other ducts, because the uppermost duct with the Smart Strand was tensioned first in both bridges.

The PF distributions as affected by short-term prestress losses were represented for the Smart Strand T1–H7 (see Figs. 4 and 5 for the location) in Table 2 and Fig. 8. The PF distribution was obtained by connecting the PF values measured at each FBG that correspond to the markers in Fig. 8, where the strands were tensioned at the left end. The PF of a strand can vary slightly from strand-to-strand even in a duct due to the different degree of interlocking and slip at a hole of the anchor head. This aspect was analyzed by Cho et al. (2015) using elasto-magnetic sensors. This explains why the PF of T1–H7 did not attain 180 kN, which corresponds to \(0.7f_{pu}\), as initially intended after tensioning even at the left jacking end.

Fig. 8 apparently shows how the PF distributions are affected by each short-term prestress loss. The PF gradually decreases from the left end to the right end due to the friction loss during tensioning. However, this decreasing trend does not match the linear decrease resulting from the theoretical mathematical formula of Eq. (3), possibly due to the unexpected local wobble of a duct and inevitable entanglement of some strands while inserting into a duct and jacking. Then, the PF distribution near the anchorage zone decreases due to the anchorage seating loss, as depicted in Fig. 3. The PF distribution further decreases according to subsequent tensioning of the remaining strands included in the other ducts, in the order of T2 and T3.

The measured PF distributions were also compared with those theoretically estimated by Eqs. (3)–(8) at 2-m intervals in Fig. 8. The friction coefficients used for the theoretical PF distribution, which were derived from the previously proposed methodologies (Jeon et al., 2015; Kim et al., 2019), are \(\mu\) = 0.21/rad and \(k\) = 0.0034/m. For a more reasonable comparison, these friction coefficients based on the measured PF distribution were adopted instead of those provided in the existing provisions. With an assumed anchorage slip of 6 mm, \(l_{set}\) in Fig. 3 was approximately 13 and 11 m in measurement and theory, respectively. The overall trend of PF distributions in each stage of short-term prestress losses was similar for the measured and theoretical values, although they showed some differences in absolute values. The final measured PF after short-term prestress losses was only 4.6% less than the theoretical value at mid-span. These results in Fig. 8 confirmed the usefulness of the existing predictive equations for the short-term prestress losses in estimating the PF distribution for design purposes.

Figs. 9 and 10 show the PF distributions of girder 1 in bridge A and girder 2 in bridge B, respectively, during the history of short-term prestress losses for both the measurement and theory. They were obtained based on the PF values at FBGs indicated by the markers in Figs. 9 and 10. The PF distributions of the measured data during tensioning, which correspond to the graphs for the friction loss in Figs. 9 and 10, do not exhibit a perfectly symmetrical shape, despite the tensioning at both ends, possibly for the reasons mentioned in Sect. 5.1. In particular, the PF distribution near the left end in Fig. 9 was very irregular. The friction coefficients used to produce theoretical PF distributions are: \(\mu\) = 0.266/rad and \(k\) = 0.0015/m in Fig. 9 and \(\mu\) = 0.105/rad and \(k\) = 0.0012/m in Fig. 10, which were also obtained based on previous studies (Jeon et al., 2015; Kim et al., 2019).

Even the theoretical PF distribution during tensioning in Fig. 10 does not show any linearity, which is explained as follows: in the parabolic shape of a tendon, the increase of \(\alpha\) and \(l\) shown in Eq. (3) above is almost proportional to the horizontal distance along a member, which results in the linear PF distribution according to the assumption of Eq. (3), as evident in Figs. 8 and 9. However, the tendon profile of bridge B differs from the typical parabolic shape, as shown in Fig. 7, resulting in the nonlinear PF distribution in Fig. 10. This causes an inevitable discrepancy in estimating the anchorage slip loss, because the assumptions to draw Eqs. (4)–(6) are based on the linearity, as shown in Fig. 3. In this case, it is still recommended to apply the same equations for practical design purposes, using the averaged linear slope in the PF distribution during tensioning, as done in Fig. 10.

The anchorage slip was assumed to be 6 mm and the resulting \(l_{set}\) was approximately 20 m in both bridges. However, the trend of the measured anchorage slip differed from the idealized shape in Fig. 3 in that the slip was not limited to the end jacking regions but had an influence even on the mid-span, although the slip was mainly concentrated on the end regions. Furthermore, the PF distributions at the end regions before and after the anchorage seating did not exhibit a perfect mirror image with the reversed slopes, as shown in Fig. 3. This suggests that the actual PF distribution in real PSC structures can be more complicated than that realized in ideal conditions of test specimens.

The actual elastic shortening loss tended to be larger than the estimation in both bridges: 16.0 and 11.0 kN for measurement and theory, respectively, at the mid-span of bridge A, and 17.5 and 5.1 kN, respectively, at the mid-span of bridge B. As a result, the measured final PFs after short-term losses were smaller than the theoretical ones in both bridges: the differences were 4.8 and 11.8% for bridges A and B, respectively. In Sect. 6.3, we will investigate whether the theoretical elastic shortening loss can be further adjusted to approach the measured value by applying more realistic values for the variables in Eqs. (7) and (8).

Despite these sources of differences, the design equations for the short-term prestress losses remain plausible for predicting the PF distribution in actual structures. However, the theoretical PF distribution was sensitive to the assumed values for the friction coefficients and slip displacement, which highlights the importance of reasonable design parameters. If the actual short-term PF differs greatly from the theoretical estimation, the concrete stresses can exceed the allowable stresses, possibly resulting in cracks, crushing, and abnormal camber or deflection. Therefore, it is desirable to ensure a sufficient margin in stresses when designing a PSC structure to account for such discrepancies.

As shown in Sect. 5, the friction loss occurs prior to other losses during tensioning and has a significant effect on the subsequent PF distributions. Although detailed analyses on the friction loss and methodologies to determine reasonable friction coefficients are beyond the scope of this study, they have been dealt with in previous studies (Jeon et al., 2015; Kim et al., 2019). The friction coefficients recommended in various provisions show a wide range of variation, e.g.: \(\mu\) = 0.15∼0.25, 0.14∼0.22, and 0.19/rad; \(k\) = 0.00066, 0.0010∼0.0023, and 0.00095∼0.0019/m in AASHTO LRFD bridge design specifications (AASHTO, 2020), Post-tensioning manual (PTI, 2006), and Eurocode 2 (CEN, 2004), respectively. In this study, however, more realistic friction coefficients were obtained using the proposed methodologies of previous studies (Jeon et al., 2015; Kim et al., 2019) and applied to the analyses for a more reasonable comparison with the measured PF. As was noted in Sect. 5, the friction coefficients varied depending on the characteristics and conditions of each structure.

Although the amount of draw-in slip is usually assumed in the design by referring to the information provided by a manufacturer of the anchorage device, it strongly affects the PF distribution. For example, if the slip is increased from 3 to 6 mm, \(l_{set}\) and anchorage slip loss increase by as much as 41%, according to Eqs. (4) and (5). Therefore, the manufacturer often suggests a conservatively large amount of slip of 6 mm, as evident in some relevant documents (VSL International Ltd., 2015). Therefore, the 6-mm slip was assumed in the calculations of the theoretical PF in Figs. 8, 9, and 10.

In this study, the PF distributions measured using the Smart Strands were utilized to validate the amount of slip. The principle underlying this procedure is that Eq. (6) can be applied to any shape, not only a triangle. That is, Eq. (6) can be extended to the form of Eq. (9), where Fig. 11 shows an example of the in-between shaded region for the PF distribution of T1–H7 in Fig. 8 that corresponds to the numerator of the following equation:

where \(P_{{\text{before}}}\) and \(P_{{\text{after}}}\) indicate the PF before and after the anchorage seating, respectively, and all other notations have the same meanings as Eq. (6). The area of the shaded region, which can be idealized as a polygon, can be readily obtained by dividing the region into several sub-regions, as illustrated in Fig. 11, and summing up the area of each sub-region. The shape of the region which is not covered by the measurement of FBGs can be appropriately assumed referring to the overall shape.

On the other hand, when the strand is tensioned at both ends, only half the area with respect to mid-span is considered to calculate the slip occurring at each end. For example, Fig. 12 represents the shaded region for the PF distribution in girder 2 of bridge B shown in Fig. 10. The two curves do not meet at any point in this case as the slip affects overall length of the strand.

Tables 3 and 4 show the amount of slip estimated in each Smart Strand using the above-mentioned procedure in the full-scale specimen and bridges, respectively. In Table 3, only the Smart Strands with five or seven FBGs were used for the calculation, because those with three FBGs could not produce sufficiently accurate PF distribution to determine the slip (see Fig. 5). However, T1–H11 and T3–H11 were unavailable, because the entire PF distribution could not be obtained due to the damage in some FBGs. The anchorage slip in the strands of T3 affected the entire length due to the absence of curvature friction. The Smart Strand in girder 2 of bridge A was unavailable in Table 4, because it was damaged after tensioning.

The average draw-in slip was 5.6 mm both in the full-scale specimen and actual bridges, which validated the general assumption of 6 mm in slip. However, the coefficient of variation that is defined by the standard deviation divided by average was much higher in the bridges than in the specimen, possibly because the full-scale specimen could be tested in more controlled and favorable circumstances than at a construction site of actual bridges. Tables 3 and 4 also suggest that the assumption of slip less than 4 mm is so unconservative that it may lead to overestimation of PF.

Finally, Eqs. (4) and (5) based on the assumption of the linear slope of the PF distributions affected by friction, as shown in Fig. 3, can be extended to consider the more realistic slope together with Eq. (9). However, this procedure would involve more complicated equations than Eqs. (4) and (5), and, therefore, is not recommended from a practical point of view.

The measured elastic shortening loss largely differed from the theoretical one in Figs. 8, 9, and 10. Therefore, we examined whether this discrepancy could be reduced by adopting certain strategies. The assumptions implied in Eqs. (7) and (8) are as follows. First, Eq. (8) assumes a perfect bond between the strand and surrounding concrete, because the modular ratio (\(n\)) is used based on strain compatibility. However, regardless of the subsequent grouting into the duct to realize bonded tendon, the tendon remains unbonded until all the tensioning operation is terminated. In the unbonded condition, the tendon strain does not coincide with the strain of the surrounding concrete but is averaged along the length of the tendon. Although this assumption is inevitable to derive a practical equation for elastic shortening loss, it may induce some errors. Second, although \(P_{i}\) is used in Eq. (7), \(P_{i}\) cannot be determined prior to the calculation of short-term losses. Therefore, \(P_{i} \approx 0.9P_{j}\) is usually assumed to simplify the calculation. However, in reality, \(P_{i}\) does not maintain a constant value of \(0.9P_{j}\) but varies along the span, as evident in Figs. 8, 9, and 10. This is another inevitable assumption that cannot easily be rectified. Third, although a constant section was assumed in calculating \(M_{d}\) in Eq. (7), the section is usually gradually enlarged near the end region, as shown in Figs. 6 and 7, to accommodate anchorage devices and stress concentration at the end of post-tensioned structures. This would have minor effect on the distribution of \(M_{d}\). Fourth, the gross section was used in the calculation of \(A_{c}\), \(I_{c}\), and \(M_{d}\) in Eq. (7). That is, it was assumed that the entire section is filled with concrete for convenience in calculation. In reality, however, the net section with the area of ducts deducted is a more accurate expression during tensioning and anchoring. On the other hand, although the concept of transformed section can also be used to account for the contribution of reinforcements, the effect on \(A_{c}\) and \(I_{c}\) would be negligible. Finally, the accuracy of \(E_{c}\) included in \(n\) of Eq. (8) can be an issue, because \(E_{c}\) varies widely.

In this study, we attempted to improve two of these aforementioned assumptions. First, the improvement made by adopting the net section is discussed. Table 5 compares the gross section with the net section, where the outer diameter of each corrugated duct was conservatively used to calculate the deducted area. In fact, consideration of the duct area affects all the variables included in Eq. (7). It also slightly changes the location of the centroidal axis along the member length, affecting \(e_{p}\), \(e_{sm}\), and \(I_{c}\) at each point, because the locations of ducts vary in longitudinal direction. However, the variables in Eq. (7) are not greatly affected by considering the net section, because the change in area only ranges from 1.6 to 5.3% in Table 5. Because the re-evaluation of \(e_{p}\), \(e_{sm}\), and \(I_{c}\) at every point in consideration involves cumbersome tasks, only the effect of change in \(A_{c}\) and \(M_{d}\) on the elastic shortening loss is analyzed. Table 6 shows the magnitude of the elastic shortening loss at mid-span in Figs. 8, 9, and 10 as affected by the deduction of duct area. The elastic shortening loss of the full-scale specimen was 6.7 kN for T1–H7 in Fig. 8. The elastic shortening loss of the twelve Smart Strands inserted into T1 ranged from 5.3 to 7.5 kN and averaged 6.2 kN. This revealed that the elastic shortening loss increased by 2.7∼17.6% and approached the measured values by employing the net section instead of the gross section in original calculation. Nevertheless, the difference between the measurement and theory did not greatly diminish and the measured elastic shortening loss was still much larger than theoretical one.

Second, the effect of the modulus of elasticity on the elastic shortening loss is investigated. When compared to \(E_{p}\) which is considered a constant value of 200,000 MPa in this study, \(E_{c}\) varies depending on various factors, including compressive strength, unit weight, and moisture condition of concrete, various properties and volumetric proportion of aggregate (ACI Committee 318, 2019; Mehta & Monteiro, 2006; Neville, 1996). Many provisions provide the formula for \(E_{c}\) as a function of dominant factors, such as compressive strength and unit weight (ACI Committee 318, 2019; CEN, 2004; KCI, 2012). Table 7 presents the modulus of elasticity calculated according to some representative provisions, where the specified compressive strength included in the formula indicates the strength attained when tensioning. In the original calculation of this study, \(E_{c}\) was obtained from KCI (2012). The modulus of elasticity of the full-scale specimen ranged from 93.5 to 119.2% when the original \(E_{c}\) is considered 100%, whereas that of the bridges A and B ranged from 106.0 to 113.8%. As shown in Eq. (8), the modulus of elasticity of concrete is inversely proportional to the elastic shortening loss, resulting in Table 8. That is, using \(E_{c}\) based on other provisions increased the difference between the theoretical elastic shortening loss and the measured one in most cases. However, the theoretical values that can be adjusted by modifying \(E_{c}\) remained insignificant when compared to the measured elastic shortening loss.

Based on the above analyses, we concluded that accurate prediction of the elastic shortening loss is difficult, because almost all the variables included in the predictive equations are affected by the accuracy in calculation and provisions. Furthermore, the predictive equations are founded on several intrinsic assumptions. The amount of the elastic shortening loss can vary to some extent by applying some improvement or the alternatives to each variable. A combination of two or more strategies would increase the variability in the elastic shortening loss. However, it was difficult to determine which strategy is effective, because the difference from the measured elastic shortening loss remained large in this study and the trends of the measured and theoretical PF distributions are arbitrarily dependent on a structure. Further study is required to accumulate more measurement data to clarify the tendency of elastic shortening loss.