## 2 The SF-iFEM Methodology
### Calculation of Element Measured Strain Based on the Equivalent Neutral Layer  
The measured strain vector \(\varepsilon^{e}\) of the divided element is composed of the measured strain \(\varepsilon^{e}\) of tension and compression and the measured strain \(K^{e}\) of bending [20; 21], as expressed in Equation (5).  
\[\varepsilon^{e}=\varepsilon^{e}+Z_{0}K^{e},\ \varepsilon^{e}=\frac{1}{2}\left[ \left\{\begin{array}{l}\varepsilon^{\pm}_{X}\\ \varepsilon^{+}_{Y}\\ \gamma^{+}_{XY}\end{array}\right\}+\left\{\begin{array}{l}\varepsilon^{-}_ {X}\\ \varepsilon^{-}_{Y}\\ \gamma^{-}_{XY}\end{array}\right\}\right]\ K^{e}=\frac{1}{2z_{0}}\left[ \left\{\begin{array}{l}\varepsilon^{\pm}_{X}\\ \varepsilon^{+}_{Y}\\ \gamma^{+}_{XY}\end{array}\right\}-\left\{\begin{array}{l}\varepsilon^{-}_ {X}\\ \varepsilon^{-}_{Y}\\ \gamma^{-}_{XY}\end{array}\right\}\right] \tag{5}\]  
where \(\varepsilon_{X}\) represents the measured strain in the \(X\) direction of the panel, \(\varepsilon_{Y}\) represents the measured strain in the \(Y\) direction of the upper surface of the panel, \(\gamma_{XY}\) represents the measured shear strain in the \(XY\) plane of the upper surface of the panel, the superscript '\(-\)' represents the strain of the inner side of the structure, and the superscript '\(+\)' represents the strain of the outer side of the structure.  
In the context of conventional plate and shell structures, when subjected to bending deformation, the strain within the plane located at \(1/2\) of the plate thickness from the wall surface is uniformly zero, leading to the designation of this plane as the neutral layer [22]. Consequently, the placement of sensors on either the inner or outer surface of the measured wall panel structure allows for the acquisition of the necessary inner and outer surface strains, which are crucial for the iFEM to reconstruct the strain field.  
For the stiffened ship panel, when subjected to bending deformation, the influence of the stiffeners results in the neutral layer generally not being located halfway through the plate thickness, as illustrated in Figure 2. Considering the monotonic functional relationship among the strain values along the \(Z\)-axis in the cross-section of the ship stiffened panels [23], two strain-sensing points, S1 and S2, are strategically placed on the lateral T-section sidewall and the top surface, respectively. Based on the strains measured at S1 and S2, along with their corresponding spatial coordinates, the equivalent neutral layer position of the ship stiffened panels can be calculated, as expressed in Equation (6).  
\[h_{N}=h_{1}-\frac{h_{2}-h_{1}}{\varepsilon_{2}-\varepsilon_{1}}\varepsilon_{1} \tag{6}\]  
where \(h_{N}\) is the equivalent neutral layer position in the stiffened ship panel, \(h_{1}\) is the \(Z\) coordinate of the strain sensor S1, \(h_{2}\) is the \(Z\) coordinate of the strain sensor S2, \(\varepsilon_{1}\) is the strain data measured by the strain sensor S1, and \(\varepsilon_{2}\) is the strain data measured by the strain sensor S2.  
Figure 2: Sensor layout based on equivalent neutral layer calculation for stiffened ship panel.  
Based on the strain data from any point on the inner surface of the stiffened ship panel, coupled with the position of the equivalent neutral layer, the strain on the outer surface of the stiffened panel can be computed, as expressed in Equation (7). Consequently, in practical engineering applications, there is no need to deploy strain sensors on the outer surface of the ship stiffened panel that comes into contact with seawater. This approach contributes to a reduction in the complexity of the monitoring network.  
\[\varepsilon_{outer}=\frac{\varepsilon_{inner}}{h-h_{N}}h_{N} \tag{7}\]  
where \(\varepsilon_{inner}\) is the strain data measured by the strain sensor S3 arranged on the inner side of the panel, \(\varepsilon_{outer}\) is the strain of the outer side of the stiffened panel, and \(h\) is the thickness of the panel.