Estimation of Friction Coefficient Using Smart Strand

While the prestressing tendons are tensioned using a jack, the loss of prestress occurs along the tendons due to the friction between the tendons and the sheath in a post-tensioning system. Figure 1 demonstrates two major types of friction. Curvature friction is induced at the curved section of a sheath, where the tendons come into contact with the sheath during tensioning. On the other hand, wobble friction occurs even in a straight sheath due to the unintended deformation of the sheath during handling or casting of concrete, although the sheath is supported at a certain interval before casting.

The predictive equation of the prestressing force as affected by the friction can be derived as shown in Eq. (1) (Nilson 1987).where P _x1 and P _x2 are the prestressing forces at the points of x1 and x2 on a tendon, respectively, with x2 closer to the tensioning point, μ is the curvature friction coefficient, k is the wobble friction coefficient, α is the variation of angle between x1 and x2, and l is the distance between x1 and x2.

The friction coefficients specified in various codes, specifications, and manuals, etc. are summarized and compared in Table 1. It is noted that some provisions provide different friction coefficients according to the type and surface condition of the prestressing steel and sheath. Although Table 1 represents the case of strands in a galvanized metal sheath that is most frequently used worldwide, it still shows a wide range of variation of the values from provision to provision, and even in a single provision. For the wobble friction coefficient, the lowest value is 0.00066 as specified in several American provisions (AASHTO 2002; AASHTO 2014; Caltrans 2005; PCI 2011), whereas the highest value is 0.0066 as presented in the Korean design codes (KCI 2012; KRTA 2010) and the previous ACI 318 code (ACI 2008). The difference is as much as ten times. On the other hand, most of the curvature friction coefficients fall into the range of 0.15–0.25, with the recommended value of around 0.20, except for the exceptionally high value specified in the Japanese specifications (JRA 2012; JSCE 2007).

Therefore, a number of attempts have been made to develop more reasonable friction coefficients. However, in several studies, one of the two types of friction coefficients was assumed, while the other friction coefficient was evaluated (Kitani and Shimizu 2009; Moon and Lee 1997); this involves an intrinsic inaccuracy that is strongly affected by the initial assumption of the value of a friction coefficient. The errors that may be induced by this type of methodology were analyzed in some studies (Park and Gil 2004; Park and Kang 2003). Some studies referred to the strains measured by the conventional electrical resistance strain gauges attached to the surface of a strand (Jeung et al. 2000). It is generally accepted, however, that the reliability of the strains obtained by this method is somewhat questionable due to a number of sources of uncertainty and inaccuracy. Gupta (2005) developed a technique to measure the prestressing force at any point of a strand using a tension tester based on the relationship between the lateral deflection and tension of the strand. However, it may be regarded that this method uses an indirect measurement of tension, which possibly involves some errors. Therefore, a more reliable methodology is required to derive realistic friction coefficients in terms of acquirement of the actual strain distribution of a strand and evaluation of the friction coefficients using the measured data.

Test variables were established to include various cases of PSC beams or girders constructed in practice. Test variables of the full-scale test of a PSC beam for evaluating friction coefficients are the diameter, curvature, and filling ratio of sheaths as shown in Table 2. Seven, twelve and nineteen strands are inserted into the sheath with 66, 85, and 100 mm diameters, respectively, in usual cases. However, the effect of the filling ratio on the friction coefficients was also taken into consideration by intentionally reducing the number of strands. In addition to the parabolically curved sheaths, straight sheaths were also arranged to separate the curvature effect from the wobble effect. The curvature shown in Table 2 does not refer to a mathematical curvature, but is defined by the sag of a curved sheath divided by the specimen length. Minor lateral curvature, which is inevitable when arranging many sheaths at mid-span, was ignored. The degree of the curvature of sheaths was determined by taking into account the ordinary curvature in PSC girder bridges, including box girders, and the intentionally increased curvature. Similar to an ordinary bonded post-tensioning system, the strands were not lubricated.

Figure 3 shows the 20 m long full-scale test specimen, where the height varies from 2.0 to 2.5 m to realize the largest curvature with an economical use of concrete. The specified compressive strength of the specimen is 40 MPa, which is the same as that of the standard PSC girders used in Korea. Also, self-consolidating concrete with a slump flow of 600 mm was used to accommodate the casting work and to ensure complete compaction of concrete, even in the case of congested sheaths and reinforcements.

The tendons were tensioned at only one end with the opposite end remaining as the dead end in order to increase friction loss and thus to purposely highlight the friction effect, although in practice, the tendons can also be tensioned at both ends. The jacking force per strand was increased by 20–30 MPa until it reached 180–200 kN, which nearly corresponds to the allowable tensile stress during jacking (0.80f _pu or 0.94f _py , where f _pu : Ultimate strength, and f _py : Yield strength) of the prestressing steel with an ultimate strength of 1860 MPa (ACI 2014; KCI 2012). A total of 13 Smart Strands were fabricated, where six strands have seven Bragg gratings (G1–G7 from dead end to live end) and the remaining seven strands have five Bragg gratings (G1–G5 from dead end to live end) that are equally spaced along the length. Among the total number of strands in one sheath, Smart Strands occupied some portion and normal strands occupied the remaining portion. The locations of the Smart Strands in the holes of the anchor head were determined so that various interactions related to the friction (between strands and between the strands and a sheath) can be accounted for. Figure 4 shows the arrangement of Smart Strands and normal strands in the case of the anchor head with 19 holes, corresponding to the sheath with a diameter of 100 mm and curvature of 0.0785 in Fig. 3. The figure also shows the sequence of work performed to investigate the effect of the filling ratio on friction coefficients.

In addition to the FBG sensors in the Smart Strands, extra measures were taken to complement or compare with the data obtained by the FBG sensors. First, a load cell was installed at the dead end to measure the total prestressing force at that location. Second, electro-magnetic (EM) sensors were installed immediately inside the jack, as shown in Fig. 5a, to measure the individual jacking force of the strands that reflects the jack loss (Cho et al. 2015). Third, electrical resistance gauges were attached to the strand surface corresponding to the selective locations of the FBG sensors. The dead end, i.e. the passive side of the strands, was realized by applying compressed grips as shown in Fig. 5b instead of normal wedges, to ensure reuse of the Smart Strands by minimizing the damage that may possibly be caused by the wedges. The optical fibers were connected to lead wires and a data logger by using a fusion splice.

Figure 6 shows an example of the relationship between jacking force and strains obtained in one of the 19 strands (strand A in Fig. 4) inserted in a sheath with a diameter of 100 mm and curvature of 0.0785 from in Fig. 3. Strand A is an example of a Smart Strand with five Bragg gratings, while seven gratings were also applied for some other Smart Strands. Because the strands were tensioned up to less than the allowable stress, the strains are within an elastic range and thus are almost linearly proportional to the jacking force. It can also be identified in Fig. 6 that the Smart Strands can be utilized to check that the jacking force is correctly transmitted through the tendons.

The distribution of the prestressing force obtained at the same Smart Strand as that shown in Fig. 6 is presented in Fig. 7, where the values measured at the load cell, EM sensor, and jack are also indicated. The strains measured at the Bragg gratings of a Smart Strand can be converted to the prestress at each grating by multiplying the modulus of elasticity of the Smart Strand, and can further be converted to the prestressing force by multiplying the cross sectional area of the Smart Strand. However, the computation of the prestressing force from the strain of the core wire of a Smart Strand can further be refined by accounting for the twist of helical wires and the difference of the cross sectional area and material property between the helical wires and the core wire as proposed in Eq. (2) (Cho et al. 2013; KICT 2013) in an analytical way. In a usual approximate calculation for a normal strand, E _p  = 2 × 10⁵ MPa and A _p  = 138.7 mm² are used to produce E _p A _p  = 27,740 kN. On the other hand, the refined calculation for a Smart Strand using Eq. (2) results in (E _p A _p ) _smart  = 26,007 kN.where P is the prestressing force at the location of a Bragg grating, ε _p is the strain measured at a Bragg grating of a Smart Strand, f _c and f _h are the correction factors for core wire and helical wires, respectively, E _p,c and E _p,h are the modulus of elasticity of core wire and helical wires, respectively, A _p,c and A _p,h are the cross sectional area of core wire and helical wires, respectively, and (E _p A _p ) _smart is the equivalent E _p A _p for a Smart Strand.

It should be noted from Fig. 7 that the prestressing force of each strand can be separately determined in the Smart Strands and EM sensors in each load level (Lv. 1–Lv. 9), while only the averaged prestressing force in each strand can be obtained in the jack and load cell system by dividing the total force by the number of strands. This implies another advantage of the Smart Strands system; it can individually predict the distribution of the prestressing force of a specific strand. Therefore, the difference of the friction coefficients of each strand can be evaluated in the Smart Strands system, which can be used to investigate the variation of friction coefficients depending on the location of a strand inside the sheath. This shows a clear contrast to the conventional method where only the average friction coefficients can be derived inside a sheath using the elongation and jacking force measured at a jack, and the force measured at a load cell installed at the dead end, if available (Jeon et al. 2009). Also, it can be seen that the difference in prestressing forces between the EM sensor and the jack refers to the amount of jack loss that occurs due to the friction inside the jack.

Friction coefficients can be determined by applying the basic equation shown in Eq. (1) and the distribution of prestressing force as presented, for example, in Fig. 7. In this study, the friction coefficients were evaluated in two steps for sheaths with a specific diameter. First, the wobble friction coefficient was evaluated in the straight sheath. Since the variation of angle (α) does not exist in the straight sheath, the wobble friction coefficient (k) can be obtained from two prestressing forces (P _x1 and P _x2) that were arbitrarily selected in a Smart Strand, judging from the form of Eq. (1), with the term of μα removed. The curvature friction coefficient (μ) can then be evaluated from Eq. (1) by applying two prestressing forces on a curved Smart Strand within a curved sheath with the wobble friction coefficient maintained as the previously obtained value for the straight sheath of the same diameter.

As can be expected, the friction coefficients obtained in such a way vary depending on the two prestressing forces chosen. Therefore, a statistical approach is required to derive friction coefficients that are more reliable. During the statistical process, some of the friction coefficients may exhibit exceptionally high or low values when compared to the ordinary range of the coefficients shown in Table 1. This behavior can be attributed to the abnormal distribution of prestressing force that can occur in a local region due to an excessive twist of strands while inserting or jacking, or the inevitable irregularity of alignment of a sheath caused by insufficient support combined with the casting pressure of concrete. Therefore, data filtering has been performed for a minority of these exceptional values based on the upper or lower limits of the friction coefficients shown in Table 1. The filtering was performed in two different ways and the results are compared. The first case is based on the two Korean design codes; Structural Concrete Design Code (KCI 2012) and Design Code for Highway Bridges (KRTA 2010). Therefore, the wobble and curvature friction coefficients that were calculated outside the range of 0.0015–0.0066/m and 0.15–0.25/radian, respectively, have been excluded from the statistics. It can be identified that the values of friction coefficients of ACI 318-08 (ACI 2008) are almost identical to those of the Korean design codes. In the second case, the entire provisions in Table 1 were accounted for and, as a result, the effective range was extended to 0.00066–0.0066/m and 0.14–0.30/radian for the wobble and curvature friction coefficients, respectively.

When Eq. (1) is used to evaluate the friction coefficients, any two arbitrary prestressing forces measured at different points can be adopted, regardless of where they are measured among the Smart Strand, load cell, EM sensor, and jack. In this respect, two different approaches were employed in this study. First, two prestressing forces corresponding to the jack and one of the gratings in a Smart Strand were referred to. As mentioned previously, however, the prestressing force measured at the jack is only an average value and does not represent the exact prestressing force of the strand under consideration. Furthermore, although friction loss may also occur inside the jack and at the anchorage devices, the jacking force does not include these losses. These are the sources that may lower the accuracy of the resulting friction coefficients. In order to cope with these problems, in the second method, two prestressing forces obtained purely in two gratings of a Smart Strand were employed.

In the above evaluation process of friction coefficients, the accurate calculation of the distance and the variation of angle between two points is needed. The distance between two arbitrary points of Bragg grating can be easily calculated, since the gratings are embedded at equal spaces along the strands. The variation of angle can be calculated by measuring the angle formed between two tangent lines drawn from the two grating points. In order to perform this calculation, the following mathematical equation, Eq. (3) (Kreyszig 2011), for calculating the length of a curve is required to inversely obtain the horizontal distance, i.e. x coordinate, corresponding to the grating point. The curved form of a sheath is assumed to be a parabola, as has frequently been assumed in design practice.where l is the length of the partial curve of f(x) between x = a and x = b, and f(x) is the shape of a sheath assumed as a parabola.

The results of the statistical analyses are presented in Figs. 8 and 9 for wobble and curvature friction coefficients, respectively. In these figures, ‘jack-grating’ implies that the jacking force and prestressing force at a grating were used for analysis, while ‘grating-grating’ indicates that two prestressing forces obtained at two gratings were adopted. Also, ‘Korean provisions’ and ‘entire provisions’ imply that the statistical data were filtered based on the limit of the two Korean design codes (KCI 2012; KRTA 2010) and entire provisions shown in Table 1, respectively.

The average wobble and curvature friction coefficients of the cases shown in Figs. 8 and 9 were evaluated as 0.0038/m and 0.21/radian, respectively. Therefore, the wobble friction coefficient was slightly smaller than the average value of 0.0041/m in the Korean design codes (KCI 2012; KRTA 2010), while the curvature friction coefficient was a little larger than the average value of 0.2/radian in the Korean design codes. In general, however, the evaluated values were close to the average values specified in the Korean design codes. Also, it can be observed that, in each pair of the wobble and curvature friction coefficients, if the wobble friction coefficient is increased, the corresponding curvature friction coefficient decreases, and vice versa. This can be expected as a matter of course because the two coefficients are interrelated in Eq. (1). The difference of the values in each group, i.e. jack-grating or grating–grating, is due to the difference of the range used for data filtering.

In most of the previous studies using the strands with FBG sensors, only the distribution of prestressing force considering prestress losses was estimated, and the friction coefficients were not derived (Kim et al. 2012; Xuan et al. 2009; Zhou et al. 2009). At most, the distribution of prestressing force obtained by assuming different friction coefficients was compared with the measured data (Kim et al. 2012). The research significance of this study can be found in a direct evaluation of the friction coefficients by utilizing the advanced sensing technology using FBG embedded in a strand. When the scope is extended to the previous studies for proposing friction coefficients, regardless of which method is adopted, the coefficients show a wide range of variation depending on the methodology used (Gupta 2005; Jeon et al. 2009; Kim et al. 2012; Kitani and Shimizu 2009; Moon and Lee 1997) and consistent coefficients have not yet been established. Another factor for this large variation may be the difference in material and workmanship in each study. For example, the degree of wobble friction sensitively varies according to the supporting interval, stiffness, and surface condition of a sheath and to the workmanship dedicated to maintain the original shape of a sheath during the installation of the sheath and the casting of concrete.

The confidence level of each friction coefficient was also investigated as shown in Figs. 8 and 9. The 95 % confidence interval was calculated using the corresponding mathematical equation (Kreyszig 2011) for each method, by assuming normal distribution of the data. Although each method has a narrower band of the confidence interval, the 95 % confidence interval marked with dotted lines in Figs. 8 and 9 only presents the absolute lower and upper limits that can cover all cases with sufficient reliability. Through this type of statistical method, the wide range of the friction coefficients specified in a specific provision can be reduced to enhance the accuracy and reliability. For example, while the range of the vertical axes shown in Figs. 8 and 9 corresponds to that of ACI 318-08 (ACI 2008) and Korean design codes (KCI 2012; KRTA 2010), the range can be narrowed to 0.0021–0.0058/m and 0.178–0.244/radian for wobble and curvature friction coefficients, respectively, by applying a 95 % confidence level. This means that the range was reduced by 27 and 34 % for the wobble and curvature friction coefficients in this study, respectively, which may accommodate the choice of friction coefficients for field engineers and designers.