## 3 Fire Damage Diagnosis System Using Fuzzy Theory
### Membership Function  
In order to use the input variables in the fire damage diagnosis system based on fuzzy theory, they should be fuzzified first, and the triangular and trapezoidal fuzzy sets were used in this study, as shown in Figure 3. The membership function of a triangular fuzzy set can be defined, depending on the range of the input variable (\(x\)), as follows:  
\[triangle(x;a,b,c)=\begin{cases}0&x\leq a\\ \frac{x-a}{b-a}&a\leq x\leq b\\ \frac{c-x}{c-b}&b\leq x\leq c\\ 0&c\leq x\end{cases}, \tag{4}\]  
where \(a\), \(b\) and \(c\) are the constants representing the range of the input variable (\(x\)), as shown in Figure 3a. In addition, the membership function of a trapezoidal fuzzy set can also be defined, as follows:  
\[trapezoid\left(x;a^{\prime},b^{\prime},c^{\prime},d^{\prime}\right)=\begin{cases} 0&x\leq a^{\prime}\\ \frac{x-a^{\prime}}{b-a}&a^{\prime}\leq x\leq b^{\prime}\\ 1&b^{\prime}\leq x\leq c^{\prime}\\ \frac{d^{\prime}-x}{d-c}&c^{\prime}\leq x\leq d^{\prime}\\ 0&c^{\prime}\leq x\end{cases}, \tag{5}\]  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**CPD I** & **Ranges of Grade** & **Grade** & **CPD II** & **Ranges of Grade** & **Grade** \\ \hline \multirow{4}{*}{\begin{tabular}{l} Concrete \\ discoloration \\ \end{tabular} } & Soot & A & & \(TP\leq 300\) & A \\  & Pink & B & Heating & \(300\leq TP<600\) & B \\  & Light gray & C & temperature (\({}^{\circ}\)C) & \(600\leq TP<950\) & C \\  & Light yellow & D & & \(950\leq TP<1200\) & D \\  & Concrete melting & E & & \(1200\leq TP\) & E \\ \hline \multirow{4}{*}{\begin{tabular}{l} Crack width \\ (mm) \\ \end{tabular} } & \(CR\leq 0.1\) & A & & \(CR\leq 0.1\) & A \\  & \(0.1\leq CR<0.2\) & B & & \(0.1\leq CR<0.2\) & B \\  & \(0.2\leq CR<0.3\) & C & Crack width (mm) & \(0.2\leq CR<0.3\) & C \\  & \(0.3\leq CR<0.5\) & D & & \(0.3\leq CR<0.5\) & D \\  & \(0.5\leq CR\) & E & & \(0.5\leq CR\) & E \\ \hline \multirow{4}{*}{\begin{tabular}{l} Spalling \\ \end{tabular} } & No spalling & A & & No spalling & A \\  & \(SP\leq 15\) & B & & \(SP\leq 15\) & B \\  & \(15\leq SP<20\) & C & Spalling (mm) & \(15\leq SP<20\) & C \\  & \(20\leq SP<25\) & D & & \(20\leq SP<25\) & D \\  & \(25\leq SP\) & E & & \(25\leq SP\) & E \\ \hline \multirow{4}{*}{\begin{tabular}{l} Spalling \\ \end{tabular} } & \multirow{4}{*}{\
\begin{tabular}{l} Ratio of design \\ compressive \\ strength of \\ \end{tabular} } & \(RCS\geq 100\) & A \\  & & & (in perfect condition) & \\  & & & \(RCS\geq 100\) & \\  & & & (with slight damage) & B \\  & & & \(85\leq RCS<100\) & C \\  & & & \(70\leq RCS<85\) & D \\  & & & \(70\geq RCS\) & E \\ \hline \multirow{4}{*}{
\begin{tabular}{l} Concrete \\ carbonation depth \\ (mm) \\ \end{tabular} } & \(CA\leq 5\) & A \\  & \(5\leq CA<10\) & B \\  & \(10\leq CA<15\) & C \\  & \(15\leq CA<30\) & D \\  & \(30\leq CA\) & E \\ \hline \hline \end{tabular} _CD_: concrete discoloration; _CR_: crack width; _SP_: spalling; _TP_: heating temperature; _RCS_: ratio of designed compressive strength of concrete to measured strength; _CA_: concrete carbonation depth.  
\end{table}
Table 3: Input variable items for combined cases of primary damages (CPD I and II).  
where \(a^{\prime}\), \(b^{\prime}\), \(c^{\prime}\) and \(d^{\prime}\) are the constants representing the range of the input variable (\(x\)), as shown in Figure 3b. As shown in Figure 4, the fuzzy sets of all of the input variables were carefully constructed based on the detailed guidelines for the Safety Inspection Regulations of Korea Infrastructure Safety and Technology Corporation (KISTEC) [18]. In particular, the concrete discoloration status was divided into four categories: sooty, pink, light gray and concrete melting, as shown in Figure 4a, and the ranges of the heating temperature depending on the discoloration condition were determined based on the Fire Resistance of Reinforced Concrete Structures [1] published by the Korea Concrete Institute (KCI).