# Estimation of Tendon Force Distribution in Prestressed Concrete Girders Using Smart Strand
## 2 Formulations  
The formulation is done in the case where the smart strand is installed in the first tendon to be prestressed in the PSC girder with multiple tendons. This condition that the tendon installed with the smart strand must be prestressed first is essential for knowing the force distribution in all the tendons equipping the PSC girder. The present formulation can be applied through slight adjustment to the case where several smart strands are installed. The prestress forces developed by the individual strands in a tendon differ to each other at the ends of the girder due to the braiding of the strands and the difference in their initial lengths [10; 11; 12]. This situation leads the prestress forces of the individual strands to also differ to each other inside the tendon itself. Consequently, the prestress force distribution in an individual strand resembles the prestress force distribution curve considering the friction but does not coincide exactly with it. However, the force distribution of the tendon consisting of bundles of these strands  
Figure 1: Shape and composition of conventional and smart strands: (**a**) Conventional strand; (**b**) Smart strand. Figure 2: Prestressed concrete (PSC) girder with multiple tendons. fits with the prestress force distribution curve considering the friction by the force equilibrium. Therefore, the prestress force of the jacked tendon can be expressed as follows [7; 8]. \[F_{ij}^{kp}=F_{i}^{kp}\exp\left(-\mu\alpha_{ij}-\kappa l_{ij}\right) \tag{1}\]  
where \(\mu\) and \(\kappa\) are respectively the curvature friction coefficient and the wobble friction coefficient; \(\alpha_{ij}\) and \(\kappa_{ij}\) are respectively the changes in curvature and arclength in the \(j\)th section of the \(i\)th tendon; and \(F_{i}^{kp}\) and \(F_{ij}^{kp}\) are respectively the prestress force in the anchor of the \(i\)th tendon and the prestress force in the \(j\)th section at the \(k\)th jacking stage. The values of \(\alpha_{ij}\) and \(\kappa_{ij}\) can be obtained from the profile of the tendon. The value of \(F_{i}^{kp}\) is also known since it is the jacking force applied during prestressing. Accordingly, the unknowns in the prestress force distribution of the tendon are \(\mu\) and \(\kappa\) of which values, once determined, will allow the computation of the prestress force (\(F_{ij}^{kp}\)) in each section of the tendon. The wedge is inserted after jacking. In this process, the anchorage slip occurs between the wedge and anchorage head. This anchorage slip provokes the change in the prestress force of the tendon at the jacked end, which can be expressed as follows [7; 8]. \[F_{ij}^{ks}=F_{i}^{ks}\left(F_{i}^{kp},\ \mu,\ \kappa,\ \Delta\right) \tag{2}\]  
where \(\Delta\) is the anchorage slip. Since the tendon including the smart strand is prestressed first in this study, \(k=1\) and \(i=1\). Equations (1) and (2) can thus be rewritten as follows for the tendon with the smart strand. \[F_{1j}^{1p}=F_{1}^{1p}\exp\left(-\mu\alpha_{1j}-\kappa l_{1j}\right) \tag{3}\]  
\[F_{1j}^{1s}=F_{1}^{1s}\left(F_{1}^{1p},\ \mu,\ \kappa,\ \Delta\right) \tag{4}\]  
Dividing the prestress forces obtained from Equations (3) and (4) by the area (\(A_{p1}\)) and the elastic modulus (\(E_{p1}\)) of the tendon gives the strains (\(\epsilon_{1j}^{1p},\ \epsilon_{1j}^{1s}\)). The elastic moduli of the conventional and smart strands composing the tendon are presented in the studies of [PERSON] [13] and [PERSON] et al. [14]. After prestressing of the first tendon including the smart strand, the strain developed in each section of the concrete girder experiences changes due to the prestressing of the subsequent tendon. This also leads the prestress force of the previously prestressed tendon to change. Such changes in the strain must satisfy the following force equilibrium in each section.