# An Integral Fuzzy Model to Evaluate Slab and Beam Bridges with a Preventive Approach
## 3 Methodology
### Second Phase: Fuzzy Evaluation Model  
Based on the elements considered in our catalog presented in Table 3, an integral fuzzy evaluation model was developed using a Mamdani inference system with a centroid defuzzification method. The benefits of using this system include ease of interpretation, intuitive reasoning, and its suitability for processing input information influenced by human judgment [24]. This model is fed with data collected from visual inspections on the deteriorations present at the first hierarchical level of the previously described taxonomy to more accurately identify which property, element, or component exhibits deterioration, allowing for precise and effective maintenance actions. The defuzzification method used for the Mamdani model is described in Equation (1).  
\[x_{Centroid}=\frac{\sum_{i}\mu(x_{i})x_{i}}{\sum_{i}\mu(x_{i})} \tag{1}\]  
where \(\mu(x_{i})\) is the membership value for point \((x_{i})\) in the universe of discourse.  
To establish the severity of each type of deterioration in the properties of the bridge elements within our fuzzy model, three types of membership functions were employed: triangular, trapezoidal, and Gaussian, as these best represent the progression of deterioration in the elements. Figures 4 and 5 show examples of two input variables: (1) access properties and (2) connection properties.  
In Figure 4, the input variable described represents the relationship between the geometric properties of the element. In this case, the relationship between the transverse slope, the longitudinal slope, and the settlement is considered. The membership functions used are triangular, with wide overlaps, reflecting a gradual transition between condition states, without abrupt variations in the level of deterioration.  
In contrast, Figure 5 shows the input variable for the connection condition of steel elements in a bridge, specifically for pedestrian accesses made of steel. This variable is derived from the deterioration conditions inspected in the field, based on the manual prepared in this work. The use of Gaussian functions, as opposed to triangular ones, indicates a low impact of the severity of the deterioration represented by these functions.  
For each deterioration type, we implement membership functions that best fit the observed deterioration behavior. The membership functions used were triangular, trapezoidal, and Gaussian. The fuzzification process was validated with the output surfaces. Examples of the generated surfaces are shown in Section 3.4.  
The integral evaluation model proposed generates two main outputs: (1) the structure condition of the bridge and (2) the urgency of attention. We used triangular membership functions to define the degree of deterioration for both outputs. For example, Figure 6, which corresponds to the evaluation of the overall condition of the bridge, illustrates how these functions represent the progressive deterioration of the structure through broad overlaps between its categories. Similarly, Figure 7 illustrates the assignment of urgency levels for bridge care, indicating that as the bridge falls into a specific category, it requires progressively faster attention as deterioration progresses and the corresponding intervention is not carried out.