## 2 Theoretical Background  
The moment-area method [23] is commonly used to describe the relationship between the moment and deflection of structures. The moment and deflection curves of a simply supported beam subjected to an external load are illustrated in Figure 1. The span of the beam is denoted as \(L\). Take point A as the reference point; tangent deflection \(\Delta\) is the distance between the deflection curve at point B and the tangent line of deflection curve at point A (shown in Figure 1). According to the principle of the moment-area method, \(\Delta\) can be expressed as:  
\[\Delta=\int_{A}^{B}X_{1}(M(z)/EI)\,\mathrm{d}z \tag{1}\]  
in which \(M(z)\) represents moment at \(z\) and \(EI\) represents bending stiffness of the beam. \(X_{1}\) represents the distance between point B and differential \(\mathrm{d}z\). The deflection of point B with the coordinate of \(x\) can be calculated by the moment distribution of the beam. From Figure 1, the deflection of point B can be expressed as:  
\[d(x)=\theta_{A}x-\Delta=\theta_{A}x-\int_{A}^{B}{(x-z)(M(z)/EI)\,\mathrm{d}z} \tag{2}\]  
in which \(\theta_{A}\) represents rotating angle of point A. Similarly, the distance between the deflection curve at point C and the tangent line of deflection curve at point A is equal to \(\theta_{A}L\). Based on Equation (1), \(\theta_{A}\) can be calculated:  
\[\theta_{A}=\frac{\int_{A}^{C}{(L-z)(M(z)/EI)\,\mathrm{d}z}}{L} \tag{3}\]  
Normally, Equation (2) is applicable when the moment distribution of the beam is known. The following section will derive the relationship between strain and deflection when the moment distribution (or external load) is unknown. According to the physical equation, the sectional curvature of a certain point can be expressed as:  
\[\frac{\varepsilon(z)}{h}=\frac{1}{\rho}=\frac{M(z)}{EI} \tag{4}\]  
in which \(\varepsilon(z)\), \(h\) and \(1/\rho\) represent strain at point \(z\), neutral axis height and sectional curvature, respectively. With sectional curvature as the connection between strain and moment, deflection at any point \(x\) can be calculated by measured strain in spite of the unknown load. On the basis of this connection, Equation (2) can be modified as:  
\[d(x)=\theta_{A}x-\int_{A}^{B}{(x-z)(\varepsilon(z)/h)\,\mathrm{d}z} \tag{5}\]  
Generally, the improved moment-area method not only keeps the advantage of the moment-area method, but also requires no information about the position and type of external load. The general equation suitable for estimating both static and dynamic deflection can be achieved as:  
\[d(x,t)=\theta_{A}(t)x-\int_{A}^{B}{(x-z)(\varepsilon(z,t)/h)\,\mathrm{d}z} \tag{6}\]  
in which \(\theta_{A}(t)\) is equal to \(\int_{A}^{C}{(L-z)(\varepsilon(z,t)/h)\,\mathrm{d}z}/L\). \(d(x,t)\) and \(\varepsilon(z,t)\) represent deflection of point \(x\) and strain of point \(z\) at time \(t\), respectively. Figure 1: Principle of moment-area method. Long-gauge fiber Bragg grating (LG-FBG) sensors can measure average strain over a certain length, and possess both static and dynamic measurement capacity [24]. It is assumed that a number of \(m\) long-gauge strain sensors (LGSSs) are installed on the structure. The average strain obtained by \(i\)th LGSS with the gauge length \(L_{i}\) is denoted as \(\bar{\epsilon}_{i}\) (Figure 1(a)). Take a simply supported beam, for example (Figure 1(b)). The beam is free to rotate at support A. The rotating angle at point A is greater than zero and the value of \(\theta_{A}\cdot x\) is greater than \(d(x)\). The deflection of point \(x\) of a simply supported beam at time \(t\) can be expressed as:  
\[ \begin{array}{ll}d(x,t)&=\theta_{A}(t)x-\int_{0}^{x}\frac{\epsilon(z,t)}{h} (x-z)\,\mathrm{d}z\\ &=\frac{x}{L}\sum\limits_{n=1}^{m}\frac{\overline{\tau}_{n}(t)}{h}\int\limits _{L_{n}}(L-z)\mathrm{d}z-\left[\sum\limits_{n=1}^{i-1}\frac{\overline{\tau}_{n }(t)}{h}\int\limits_{L_{n}}(x-z)\mathrm{d}z+\frac{\overline{\tau}_{i}(t)}{h} \int\limits_{i-1}^{x}L_{n}(x-z)\mathrm{d}z\right]\end{array}