# Estimation of Tendon Force Distribution in Prestressed Concrete Girders Using Smart Strand
## 2 Formulations
\[\sum\Delta F=F_{ij}^{t}\cos\theta_{ij}+\sum_{i=1}^{k-1}E_{pi}A_{pi}\Delta \epsilon_{ij}^{t}\cos\theta_{ij}+\int_{Aj}E_{c}\Delta\epsilon_{c,j}^{t}dA=0 \tag{5}\]  
\[\sum\Delta M_{z}=F_{ij}^{t}\cos\theta_{ij}y_{ij}+\sum_{i=1}^{k-1}E_{pi}A_{pi} \Delta\epsilon_{ij}^{t}\cos\theta_{ij}y_{ij}+\int_{Aj}E_{c}\Delta\epsilon_{c, j}^{t}y_{j}dA=0 \tag{6}\]  
\[\sum\Delta M_{y}=F_{ij}^{t}\cos\theta_{ij}z_{ij}+\sum_{i=1}^{k-1}E_{pi}A_{pi} \Delta\epsilon_{ij}^{t}\cos\theta_{ij}z_{ij}+\int_{Aj}E_{c}\Delta\epsilon_{c, j}^{t}z_{j}dA=0 \tag{7}\]  
where \(t\) denotes the prestressing stage (\(t=2p,\ 2s,\ 3p,\ 3s,\ldots\)); \(F_{ij}^{t}\) is the prestress force of the tendon prestressed in each stage and can be obtained by means of Equations (1) and (2); \(\Delta\epsilon_{c,j}^{t}\left(=\epsilon_{c,j}^{t}-\epsilon_{c,j}^{t-1}\right)\) is the change in the strain of concrete occurring in the current stage; and, \(\Delta\epsilon_{ij}^{t}\left(=\epsilon_{ij}^{t}-\epsilon_{ij}^{t-1}\right)\) is the strain change experienced in the previously prestressed tendon at the current stage. Assuming that a plane section of the concrete structure before deformation remains plane after deformation, the strain change (\(\Delta\epsilon_{c,j}^{t}\)) per prestressing stage in the concrete section can be expressed as follows. \[\Delta e^{t}_{c,j}=\Delta e^{t}_{co,j}-y_{j}\Delta x^{t}_{cz,j}+z_{j}\Delta x^{t}_{ cy,j} \tag{8}\]  
where \(\Delta e^{t}_{co,j}\), \(\Delta x^{t}_{cz,j}\) and \(\Delta x^{t}_{cy,j}\) respectively denote the strain change in the neutral axis of the \(j\)th section at the \(t\)th prestressing stage, and the curvature changes with respect to the \(z\)-axis and \(y\)-axis (Figure 3); \(y_{j}\) and \(z_{j}\) respectively represent the distances from the neutral axis of the section in the \(y\)-direction and \(z\)-direction. The integrated terms related to concrete in Equations (5)-(7) are arranged in Appendix A. Moreover, if we assume that the prestressed tendon and concrete behave monolithically, the strain increment of the prestressed tendon can be expressed as follows using the strain of concrete. Figure 2:0  
The unknowns in Equations (3)-(7) can be subdivided into the variables that keep constant values at each prestressing stage and those whose values vary. The variables that remain constant at each prestressing stage are \(\mu,\ \kappa,\ \Delta\) for the tendon and \(E_{c}\) for concrete. The variables with varying values at each prestressing stage are \(\Delta e^{t}_{co,j}\), \(\Delta x^{t}_{cz,j}\), and \(\Delta x^{t}_{cy,j}\) for the deformation of concrete in each section. Among them, the unknowns representing the deformation of concrete can be obtained by the equilibrium of the forces given in Equations (5)-(7). Consequently, the problem of evaluating the tendon prestress force in the PSC girder is transformed to a problem searching for the curvature friction coefficient and wobble friction coefficient of the tendon, the anchorage slip, and the elastic modulus of concrete. Solving Equations (5)-(7) for each section and at every prestressing stage provides then the strain state of the tendons. Among these so-obtained strains, an optimization problem can be formulated so as to minimize the difference between the strain increment \(\left(\Delta e^{t}_{1j}\right)\) of the tendon including the smart strand and the strain increment \(\left(\Delta e^{t}_{1j,exp}\right)\) measured by the smart strand. Equation (10) formulates the optimization problem which, once solved, will allow the remaining four other unknowns to be obtained. Figure 2:1  
Each prestressing stage is subdivided into a substage \((kp)\) in which the tendon is tensioned by the hydraulic jack, and another substage \((ks)\) in which the wedge is fixed in the anchor head. The anchorage slip cannot be obtained in substage-\(kp\) because this substage is related only to the curvature friction coefficient, the wobble friction coefficient and the elastic modulus.