## 1 Introduction
In this sense, the Vulnerability Index Method (VIM) constitutes a process that combines the vulnerability classes of the EMS-98 scale and the Italian method [14] for the characterization of the seismic vulnerability of buildings [22; 23]. These methodologies are based on vulnerability indexes that have their roots in a sequence of basic vulnerability indexes obtained according to the structure typology of the building; they are modified with a series of factors related to the structural and constructive characteristics of each building, in a standardized way. Concerning the VIM method, the basic vulnerability indexes _lv_ were developed through the inventory of damage in buildings found in the European countries involved in the project. Thus, the seismic performance of the most habitual structural configurations in urban areas can be characterized by means of the Building Typology Matrix (BTM) [18]. The vulnerability indexes for some building typologies of BTM, consequently included in a previous study [24] as participants of the RISK-UE Workgroup, are shown in Table 1. This proposal standardizes the RISK-UE curves by empirical damage models, consistent with the fuzzy set theory. Similar to the qualitative definitions interpreted through membership functions \(\chi\) proposed by previous studies [23; 25] for the vulnerability categories of the EMS-98 scale, RISK-UE considers a series of membership functions for each structural configuration. A function \(\chi\) defines the belonging of single values of a certain parameter to a specific set: the value of \(\chi\) is 1 when the degree of belonging is reasonable; between 0 and 1 shows that the value of the parameter is infrequent but possible; and is 0 when the parameter is not associated to the set. Thus, VIM sets a value \(lv\)* as the most possible vulnerability index (\(\chi\) = 1), a couple [\(lv^{-}\), \(lv^{+}\)] that generates a commendable interval (\(\chi\) = 0.6), and an interval [\(lv^{\min}\), \(lv^{\max}\)] with the minimum and maximum limits (\(\chi\) = 0.2), characterized as least probable or exceptional. The VIM method adapts this basic index of vulnerability \(lv_{t}\) in line with a series of values of penalty or improvement called modifiers of behavior, according to some structural and urban planning constraints such as the level of design code, the number of floors of the buildings, the geometric and stiffness regularity, or the relative position of the building in the neighborhood. Taking into account different experiences after post-earthquake evaluations [16; 26; 27], these standards are defined in a \(\pm\)0.02 range for an interval between \(-\)0.08 and +0.08 in order to incorporate them into the characterization of the seismic vulnerability. Since the previous described version [18] a series of reviews have been performed in the definition and numerical evaluation of these behavior modifiers [23; 25; 28; 29], calibrating the influence of several dispositions (i.e., sloping ground difference of height between close buildings, irregularity in plant or elevation, etc. ), in the seismic analysis of buildings. Furthermore, the methodology considers an additional factor--the regional modifier--, linked to the specific constructive idiosyncrasy of the study area, the existing seismic design codes, time of construction, or of the judgment of experts from empirical information proceeding from actual observed damage. In order to estimate this modifier, no widely accepted proposal is currently available, with different values in the range of [\(-\)0.088, 0.08] for masonry typologies and [\(-\)0.022, 0.14] for reinforced concrete buildings [23; 25; 29].