### 5.1 Methodology

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.

$$ l = \int_{a}^{b} {\sqrt {1 + \left( {f^{\prime}(x)} \right)^{2} } dx} $$

(3)

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.

### 5.2 Analysis Results

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.

### 5.3 Discussion

As mentioned earlier, a number of conventional electrical resistance strain gauges were also implemented in this study for comparison with the Smart Strand system. As has been frequently reported, however, a large number of electrical resistance strain gauges were made unavailable due to the damage of sensors and lead wires during insertion of strands into a sheath and during tensioning. The strains measured by Smart Strands were compared with those measured by the remaining electrical resistance gauges attached to the surface of helical wires, corresponding to the same grating points. The data obtained using electrical resistance gauges showed a great amount of difference from those of Smart Strands, since the electrical resistance gauges may be sensitively affected by such factors as the alignment of the gauges on the helical wires and the workmanship related to the proper bonding of the gauges. In the overall trend, the strains of the conventional gauges were smaller than those of Smart Strands by 20–30 % as representatively shown in Fig. 10. The difference between the strains can be attributed to the difference in the length between the core wire and helical wire, and the slope of the helical wire with respect to the core wire (8.2°), etc. For example, the longer length of the helical wire wound around the core wire can result in a smaller strain. The difference in the actual stresses between the core wire and helical wire has also been investigated by Cho et al. (2013). Therefore, the strains measured using electrical resistance gauges should be interpreted with special care, especially when applied to the strands, although they are still widely used.

In this study, the effect of curvature, diameter, and filling ratio of a sheath, and the effect of the location of a strand in a sheath on the friction coefficients have also been investigated. However, these topics will be dealt with in another paper since they involved extensive analyses. This study presented the general average friction coefficients in terms of wobble and curvature, taking into consideration all the test variables, since the friction coefficients are specified without any limited condition in most provisions as shown in Table 1.