## 2 Theoretical Derivation  
In this study, a static equilibrium over a cross-section of the prestressed segmental beam is employed to obtain the neutral axis location with strain measurement data. The strain data-based methods were used for obtaining the neutral-axis location due to the maturity of the well-established relationship between strain and stress [21]. This cross-section analysis uses strain values on a specific cross-section of a segmental beam to compute the neutral axis location by adopting the assumption of the linear strain distribution. Figure 1 shows two parallel strain gauges installed at the web in the longitudinal direction, with one placed at the top denoted as \(\varepsilon_{\text{st}}\) and one at the bottom denoted as \(\varepsilon_{\text{sb}}\), with a distance of h between them. The distance between the bottom strain gauge and the bottom of the beam section is denoted as m. The neutral axis location \(\text{X}_{0}\) can be calculated based on the linear strain distribution as [24]:  
\[\text{X}_{0}=\frac{\varepsilon_{\text{sb}}\text{h}}{(\varepsilon_{\text{sb}}+ \varepsilon_{\text{st}})}+\text{m} \tag{1}\]  
By assuming linear strain distributions, which implies the cross-section remains plane during deformation, and the beam is considered to be composed of discrete linear fibers [20], the longitudinal strain \(\varepsilon_{\text{I}}\) at any point on the cross-section of the beam can be calculated based on the distance hi from the neutral axis location. The strains at the top and bottom of the cross-section of the beam are denoted as \(\varepsilon_{\text{t}}\) and \(\varepsilon_{\text{b}}\), respectively, as shown in Figure 1. The relationship between I and \(\varepsilon_{\text{sb}}\), and the ratio of the top strain gauge \(\varepsilon_{\text{st}}\) and bottom strain gauge \(\varepsilon_{\text{sb}}\) can be expressed as:  
\[\frac{\varepsilon_{\text{i}}}{\varepsilon_{\text{sb}}}=\frac{\text{h}_{\text {i}}}{\text{h}_{\text{sb}}},\ \frac{\varepsilon_{\text{st}}}{\varepsilon_{\text{sb}}}=\frac{\text{h}_{\text {st}}}{\text{h}_{\text{sb}}} \tag{2}\]  
where \(\text{I}_{\text{i}}\), \(\text{h}_{\text{sb}}\) and \(\text{h}_{\text{st}}\), as indicated in Figure 1, represent the distances from the point of interest, bottom strain gauge, and top strain gauge to the neutral axis, respectively. During the post-tensioning process, only the stress in the prestress tendons causes the deformation of the entire precast segmental beam, since no external loading force is applied on the beam. Accordingly, the static equilibrium of a specific cross-section of the  
Figure 1: Strain distribution along the cross-section of the beam. segmental beam with the post-tensioning force is obtained, as shown in Equation (3). The schematic of the force equilibrium of the beam is illustrated in Figure 2.  
\[\int_{\text{A}}\sigma\text{dA}=\text{F} \tag{3}\]  
During the post-tensioning process, concrete is in the linear elastic range, wherein stress \(\sigma\) is proportional to strain \(\varepsilon\). Following [PERSON]'s law, it has \(\sigma=\varepsilon\ \times\text{E}_{\text{C}}\), where \(\text{E}_{\text{C}}\) denotes the elastic modulus of concrete. The concrete stresses at the top and bottom locations at the cross-section of the beam are denoted as \(\sigma_{\text{b}}\) and \(\sigma_{\text{t}}\), respectively, as shown in Figure 2. Since tensile reinforcements, which discontinue between segments, have only small deformation, and remain in the elastic range \((\sigma_{\text{s}}<f_{\text{y}})\), the stress of the reinforcements can be calculated from the corresponding strain by \(\sigma_{\text{s}}=\varepsilon_{\text{s}}\times\text{E}_{\text{s}}\), where \(\text{E}_{\text{s}}\) is the elastic modulus of steel.