# Temperature Effects on Vibration-Based Damage Detection of a Reinforced Concrete Slab
## 4 Damage Identification in Test Structure
### Damage Identification Considering Temperature Variations  
The linear relationship between the elastic modulus and the first natural frequencies of a simply supported beam can be obtained from Equation (2), which can be written as:  
\[E=\frac{4L^{4}f_{n}^{2}\overline{m}}{n^{4}\pi^{2}I} \tag{27}\]  
where \(L\) is the span length; \(f_{n}\) is the natural frequency of \(n\)-th mode; \(\overline{m}\) is the mass per unit length; \(I\) is the sectional moment of inertia. Consequently, the elastic modulus under the different environmental temperatures can be calculated by the experimental frequencies. The linear relationship between the elastic modulus and temperature can be obtained by least square method, and the linear fitting coefficients R2 of elastic modulus and temperatures is 0.8375, which is shown in Figure 15.  
The linear relationship can be written as:  
\[E=-189.5386\times T+33083.134 \tag{28}\]  
Figure 14: Frequencies of the slab vs. temperature based on ARX model, 10% uniform damage, 5% uniform damage and the 95 percent confidence intervals. Figure 13: Frequencies of the slab vs. temperature based on ARX model, the 95 percent confidence intervals and the finite element model. Then, the linear relationship of temperature vs. elastic modulus is adopted for the finite element model of the slab, and two damage cases are introduced, which are 5% and 10% uniform damage in all elements (20 \({}^{\circ}\)C). Statistical criterion and particle swarm optimization (PSO) are utilized to detect the damage considering the temperature effect. The PSO algorithm randomly generates the initial population named particles, which is first raised by [PERSON] and [PERSON] [43]. The characteristics of velocity and position are assigned to each particle which can be dynamically changed and updated by referring to the best particle in the search space. The algorithm as a powerful optimization tool aims to find the global optimal of an objective function, which is widely use in damage identification [44]. The position \(P\) and velocity \(V\) of each particle are updated by Equations (29) and (30) respectively, which are shown as follows:  
\[P_{i}^{t=t+1}=P_{i}^{t}+V_{i}^{t=t+1} \tag{29}\]  
\[V_{i}^{t=t+1}=\omega V_{i}^{t}+c_{1}\alpha_{1}\cdot(\textit{gbest}-P_{i}^{t=t+ 1})+c_{2}\alpha_{2}\cdot(\textit{pbest}-P_{i}^{t=t+1}) \tag{30}\]  
where \(i\) stands for the \(i\)-th particle; \(t\) is the number of current iterations; \(c_{1}=c_{2}=2\) mean the learning factors; \(\alpha_{1}\) and \(\alpha_{2}\) are random constants in the range of [0, 1]; _gbest_ and _pbest_ are the global optimal particle and optimal position respectively. \(\omega\) is the inertia weight. The objective function can be defined as:  
\[f(x)=10\cdot\sum_{j=1}^{n\text{mod}}\left(\frac{f_{aj}-f_{ej}}{f_{ej}}\right)^ {2}+\sum_{j=1}^{n\text{mod}}\left(\frac{1-\sqrt{MAC_{j}}}{MAC_{j}}\right)^{2} \tag{31}\]  
where \(f_{aj}\) and \(f_{ej}\) are the \(j\)-th analytical and experimental natural frequency respectively; \(MAC_{j}\) is the \(j\)-th Modal Assurance Criterion (Equation (1)), \(nmod\) is the number of modes. The flowchart and identification results are shown in Figures 16 and 17, respectively. Figure 15: Linear relationship between elastic modulus and temperature. From Figure 17 it can be seen that the damage of the RC slab can be precisely located and accurately quantified. In addition, as shown in Table 4, the first analytical frequency of the slab in damaged state does not exceed the confidence interval, so it is evident that the proposed damage identification method is verified and is of good performance.