## 2 Theory Background and Methodology  
This study investigates the use of strain measurements to estimate displacement and acceleration of structures members and determine their dynamic characteristics, frequency and damping. The monitoring theory and the design of strain sensors were presented in [27]. The main SHM concepts of strain, displacement and acceleration measurements and analyses are presented in [28; 29; 30]. Herein, local response measurements of structures' members are a major advantage of strain measurements [31; 32]. Where, local strains can be estimated through the relationship between strain and displacement [17; 33]. As well as, dynamic acceleration can be detected from dynamic strain measurements through a transfer function. Figure 1 presents the displacement, acceleration and strain responses of point (\(\mathbf{a}\)) using three sensors. This study aims to estimate accurately three responses of point \(\mathbf{a}\) using strain measurements only. The relationship of displacement and strain for a given monitored point \(\mathbf{a}\) along the vertical axis (\(\mathbf{z}\)) of a beam during monitoring time \(\mathbf{t}\), Figure 1, can be simply expressed as follows [17]:  
\[\{\varepsilon_{s}(z,t)\}=[T]\{u_{s}(z,t)\} \tag{1}\]  
where the semi-static strain (\(\varepsilon_{s}\)) and displacement (\(u_{s}\)) vectors are real-time measurements for one monitoring point, and \(\mathbf{T}\) is the transformation matrix between the semi-static strain and displacement responses' vectors. Additionally, the transformation matrix is used to transfer the acceleration to dynamic strain as follows [34]:  
\[\{\varepsilon_{d}(z,t)\}=[D]\{a_{d}(z,t)\} \tag{2}\]  
where, the dynamic strain (\(\varepsilon_{d}\)) and acceleration (\(a_{d}\)) of the monitoring point are measured, and \(\mathbf{D}\) is the transformation matrix between the dynamic strain and acceleration responses' vectors. Thus, the inverse of the transformation matrices was used to calculate the displacement and acceleration of the monitored point from strain measurement. Herein, T and D represent the mode shapes or modal matrices of the vibration mode of point \(\mathbf{a}\). Thus, to estimate the acceleration and displacement from Equations (1) and (2), the modal matrices are obtained analytically or experimentally. [PERSON] [17] obtained the modal matrices experimentally by using displacement frequency response functions; while [PERSON] et al. [21] used the finite element to calculate the modal matrices. In this study, to overcome the complexity of the theoretical and experimental calculations of the frequency responses of structures, the polynomial function was used to transfer the strain measurements to displacement and acceleration responses in the time domain. A polynomial function was used to transform the semi-static displacement and dynamic acceleration responses of strain measurements in the time domain, as follows [35; 36]:  
\[u_{s}(t)=\sum_{n=0}^{N}p_{n}(\varepsilon_{s}(t))^{N-n} \tag{3}\]  
Figure 1: A schematic diagram of beam responses in the z direction. \[a_{d}(t)=\sum_{n=0}^{N}k_{n}(\varepsilon_{d}(t))^{N-n} \tag{4}\]  
where N is the polynomial model order and \(p_{n}\) and \(k_{n}\) are the semi-static and dynamic parameters of the model, respectively. Figure 2 proposes the model design for estimating the semi-static displacement and dynamic accelerations responses from strain measurements. Hence, since dynamic noises and sensors errors affect sensors measurements, filtration methods are applied to remove the unsuitable noises [7; 17]. The moving average (MA) filter is widely used to extract semi-static responses of structures, and the wavelet method is used to improve dynamic signals [7; 26]. Further details on the two filters' theories and their design can be found in [26; 34; 37]. In the present study, the MA and wavelet denoising methods are applied to extract accurate semi-static and dynamic behavior of bridges.