# Estimation of Tendon Force Distribution in Prestressed Concrete Girders Using Smart Strand
## 3 Numerical Verification
### With Variation  
Figure 5 corresponds to the ideal case where the prestress force measured by the smart strand coincides with the average prestress force of the tendon. However, [PERSON] et al. (2010) reported that, in reality, the prestress forces developed in the strands of the tendon differ to each other due to the braiding of the strands and that the prestress force distribution of the strands inside the tendon follows a normal distribution \(N\Big{(}f_{mean},\;\sigma_{f}^{2}\Big{)}\) where \(f_{mean}\) is the average prestress force of the tendon and \(\sigma_{f}\) is the standard deviation among the prestress forces.  
\[\sigma_{f}=0.019f_{mean}+2.286\;[\mathrm{kN}] \tag{11}\]  
Consequently, the strain measured in the smart strand will also exhibit a pattern different to the ideal case. Need is thus to verify if the proposed method would provide an accurate estimation of the tendon force in such a non-ideal case. To that goal, assume the following expression for the strain measured by the smart strand.  
\[\epsilon_{1j,exp}^{1p}=\frac{F_{1}^{1p}\exp\bigl{(}-\mu\alpha_{1j}-\kappa l_{ 1j}\bigr{)}}{E_{p}A_{p}}+var \tag{12}\]  
where \(var\) follows the normal distribution \(N\bigl{(}\epsilon_{mean},\;\sigma_{\epsilon}^{2}\bigr{)}\). The strain \(\sigma_{\epsilon}\) can be obtained by dividing Equation (11) by the elastic modulus and area of the strand.  
Optimization is then performed by applying the variation expressed by Equation (12) in the strain measured by the smart strand for the analysis case A. This optimization was carried out on 10 variations and the results are arranged in Table 2. The average values for each of the optimization variables show minimal error compared to the original system. In addition, the normalized error of the smart strand exhibits an average of about 2.9% and indicates the feasibility of the proposed method.  
Figure 6 plots the strains measured by the smart strand together with the corresponding optimized strains for the variations with the smallest error, largest error and median error. Even if the measured strain profiles are very different to the ideal case, the tendon strain distribution resulting from the optimization appears to be very close to the ideal case. For the worst level of correspondence in Figure 6c, the error reaches only 3.6% at the location with the largest difference. Consequently, the method proposed in this study appears to be sufficiently applicable even in the case where the strain measured by the smart strand exhibits a different distribution profile due to the braiding of the strands.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Variation**} & **Curvature Friction** & **Wobble Friction** & **Working Stroke of** & **Elastic Modulus** & **Normalized** \\  & **Coefficient (/rad)** & **Coefficient (/m)** & **Anchorage (mm)** & **(MPa)** & **Strain Error (\%)** \\ \hline
1 & 0.21 & 0.0020 & 6.0 & 33,943.9 & 1.0 \\
2 & 0.19 & 0.0020 & 6.5 & 33,926.2 & 1.2 \\
3 & 0.20 & 0.0020 & 6.6 & 33,896.1 & 1.6 \\
4 & 0.17 & 0.0020 & 6.3 & 34,001.2 & 1.7 \\
5 & 0.16 & 0.0019 & 7.2 & 33,922.2 & 2.6 \\
6 & 0.22 & 0.0021 & 6.7 & 33,836.8 & 2.7 \\
7 & 0.16 & 0.0019 & 6.1 & 34,057.1 & 2.8 \\
8 & 0.26 & 0.0021 & 6.1 & 33,842.9 & 4.0 \\
9 & 0.24 & 0.0021 & 6.7 & 33,784.2 & 4.1 \\
10 & 0.12 & 0.0018 & 5.2 & 34,291.5 & 7.3 \\ Average & 0.19 & 0.002 & 6.3 & 33,950.2 & 2.9 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Optimization results for variations where the strain measured by the smart strand shows definite fluctuation.  
Figure 6: Comparison of prestress force distribution in cases where the strain measured by the smart strand shows definite fluctuation: (**a**) Variation #1: the smallest error; (**b**) Variation #7: the median error; (**c**) Variation #10: the largest error.