Seismic vulnerability in essential buildings through analytical fragility curves Vulnerabılıdad sísmıca en edıfıcacıones esencıales medıante curvas de fragılıdad analítıcas

This article presents a methodology to assess the seismic vulnerability of essential buildings through analytical fragility curves, applied to the administrative building of the Universidad Nacional del Centro del Perú, for various damage states and different levels of seismic demand according to ATC-40 and SEAOC -VISION 2000. It is shown that the fragility curves through the probability of damage matrices allow us to reasonably estimate the probable state of the building after a seismic event, which in the case of the building under analysis shows that it is more vulnerable in the North-South direction (Y axis).


Introduction
Peru is a highly seismic country, as it is part of the circumpacific belt where more than 80 percent of the world's earthquakes occur. These events have shown that buildings with poor seismic performance are prone to significant damage to their structural elements and consequently collapse, resulting in loss of human and economic life. So it is important to conduct seismic vulnerability studies of essential buildings using methods that consider the uncertainty of structural, seismic and geotechnical factors.
The studied building was projected with peruvian code of sismoresistente design E.030 (1997) in Huancayo city, built between 2000 and2002. It has an area of 1944,81 square meters and its main use is administrative. The structure has 10 levels, a cellar intended for the parking area and a semisotano, has a glass facade with aluminium frame whose sloping roof has metal frame by the four fronts from the seventh floor and contains fixed partitions only in the hygienic service environments. In the last level of the building is the elevated tank.
The structural settings involves a portic and shear walls of reinforced concrete join by means of solid slabs and lightened. The building is composed of 22 circular columns (1.00 m and 0.70 m in diameter) and four rectangular columns (0.40 m x0.60 m) from the eighth to the tenth level, also has constant section vaulted beams from the basement to the last level, these are: 90 m,. It also has plates that form the elevator box, two plates that support the staircase, two cutting walls located in common bathroom areas and eight symmetrically arranged plates of different thicknesses (e= 0.30 m and e= 0.40 m).
In the present research the seismic vulnerability is determined by fragility curves for various states of damage and the probability matrix of damage of the administrative building of the National University of Central Peru

Materials and methods
To determine the seismic vulnerability of the building, the structural data of the building were surveyed, the mechanical model of the typology was proposed, loads were measured, soil mechanics were studied, nondestructive tests on the concrete, definition of seismic demand, thresholds and indicators of structural damage.

Study of soil mechanics
Over the study area there is a powerful deposit of coarse sub-rounded granular soil, called conglomerate, which corresponds to an ancient alluvial terrace of the Mantaro River. The mechanical behavior of the foundation soil is summarized in the following (Table 1).

Structural parameters
For the structural evaluation of the building, "in situ tests" were carried out in order to compare the compressive force of concrete ´ according to (Table 2). For this purpose, the Rebound Hammer Test, Schmidt Hammer or Swiss Hammer was used. (ACI, 1995), developed according to ASTM C 805. 2002.

Measurements of gravity loads
The purpose of the load measurements was to estimate the loads acting on the structure and to estimate the seismic weight according to (Table 3).

Structural model of the building
The research project requires a detailed analysis of seismic behavior, so a structural model was defined with the help of Etabs version 19.0.0 software. The structural configuration in the building plan showed a progressive reduction of the constructed area according to (Figure 1).

Seismic demand
The deterministic method was used in the estimation of seismic movement, considering more severe conditions. Due to the lack of historical seismic information, alternative seismic records were used, considering primarily the maximum horizontal acceleration of the terrain (PGA), magnitude and those that produce greater spectral accelerations "Sa" in the fundamental period "T0" with a 5% damping.
Therefore, six records of three seismic events were selected as shown in (Table 4), which according to (Aguilar, 2001) belong to the earthquakes of 1966, 1970 and 1974.

Seismic demand levels
In order to develop the elastic spectra of pseudo accelerations, seismic demand levels were used according to the Applied Technology Council TC-40 (1996) according to Table 5 and the California Association of Structural Engineers SEAOC (1995) shown in Table 6. Table 5. Acceleration associated with E.030 for earthquake levels (ATC-40, 1996).

Spectrum of pseudo seismic demand accelerations
The construction of the spectrum of pseudo seismic demand accelerations according to (Figure 2, was made from the pseudo accelerations of each seismic record obtained with the seismosignal program for a damping of 5% and 10%. For this purpose, the geometric mean, the mean plus a standard deviation and the mean plus two standard deviations of the six seismic records were calculated.

Nonlinear static analysis
The static nonlinear incremental load analysis was performed using the Ebas version 19.0.0 software to estimate the structural response in terms of shear force and displacement at the deck level. For this purpose, the non-linear load-deformation characteristics of the building were considered, subjecting the structure to a monotonic increase of lateral loads

Non-linearity of materials
The constituent models of the materials sought to define the nonlinear behavior of these. Below are the constituent models of concrete in (Figure 3) and (Table 7) and steel shown in (Figure 4).

Capacity curve
The capacity curve of the building under analysis is shown in (Figure 5), obtained by nonlinear static analysis for the two main directions of the building, X and Y. This curve allowed us to assess the behavior of the building when it enters the nonlinear range and determine the maximum response of the structure.

Capacity spectrum
To calculate the thresholds and performance point, the capacity spectrum shown in (Figure 6) was generated from the Pushover curve under the criteria of (FEMA 440, 2005), using the following (Equation 1):

Discrete states of damage
In order to make a qualitative description of the damages produced by earthquake and calculate the fragility curves, discrete damage states according to (Table 8) proposed by (Lagomarsino and Penna, 2003) were used. = 11.09

Thresholds for structural damage
The thresholds for each discrete state of damage were calculated from the mezzanine drift shown in (Table 9). For this purpose we record the evolution of these with respect to the global drift of the roof level of the building, by taking data in each increase of lateral load of the nonlinear static analysis. Where: ̅ , is Median of spectral displacement

Generation of fragility curves
The fragility curves of the administrative building of the National University of Central Peru shown in (Figure 7) were calculated from the spectral floor displacements for each discrete state of damage. Thus obtaining, the probability that the response of the building exceeds a discrete state of damage as a function of the spectral displacement that defines the intensity of the movement of the ground. The following expression of (HAZUS MH 2.1, 2001) was used for this purpose (Equation 2): Where: ̅ , : Median of spectral displacement where the building reaches the damage status threshold, ds; : Standard deviation of the natural logarithm of the displacement spectrum by state of damage ds; : Standard normal cumulative distribution function.
To generate the fragility curves, the median of the spectral displacement (median of the damage thresholds) was calculated and the variability of the damage state by the standard deviation of the natural logarithm of the displacement spectrum by damage state shown in (Table 10), by means of the expressions developed in the RISK-EU project that calculated the variability of the damage state directly with ultimate ductility .

Calculation of probability of damage matrix
The damage probability matrices were obtained by intersecting the performance points with the fragility curves for each damage state according to (Table 16) and (Table 17), making use of spectral displacement. For this purpose, the following equation expression was used (Equation 3):

Capacity point on demand
The capacity point per demand was determined by intersecting the capacity spectrum with the elastic demand spectrum considering 5% damping when intersecting in the linear range of structural capacity. However, in case the intersection occurs in the inelastic range we use the inelastic demand spectrum with 10% damping. Under these considerations we calculate the maximum capacity point for the seismic demand levels of (ATC-40, 1996) shown in (Table 11) and (SEAOC, 1995) shown in (Table 12). Table 11. Performance points of the building under study: (ATC-40, 1996).

Matrix
To illustrate the above, we calculated the probability that the damage in the 12-level building is severe for a spectral displacement of = 5.435 shown in (Figure 14), which represents the maximum displacement expected for the level of Rare earthquake according to. To do this, the probability of severe and complete damage leave for a spectral displacement of = 5.435 , was determined, according to ( Figure  8).

Analysis of results
The probability of occurrence of a state of mild, moderate, severe or complete damage is determined from the fragility curves and capacity point by demand, because the seismic behavior of the building can be quantified by the point of performance. The maximum expected displacement of the building for the seismic levels of the ATC-40 and SEAOC Vision 2000 are obtained by superimposing the spectrum of capacity and demand spectrum, by the method of the capacity diagram -seismic solicitation, proposed by (Chopra and Goel, 1999). For the North-South direction (Y axis), according to (ATC-40, 1996) and with a seismic level of design, it has a probability of 84.62% that the damage is complete, 14.50% that is severe; while according to (SEAOC, 1995), for the seismic level "Rare" equivalent, has a 60.59% chance of damage being complete, 31.14% severe.

Conclusions
Fragility curves obtained for states of mild, moderate, severe damage and collapse show that the administrative building of the National University of Central Peru is more vulnerable in the North-South direction (axis Y) than the East-West direction (axis X) and allow a reasonable estimate of the likely condition of the building after a seismic event.
The contribution of the damage probability matrix is significant, because it allows to predict the level of damage that will reach the building when it suffers earthquake of different levels. Thus, for the East-West direction (X axis) according to the SEAOC and with a seismic level of design, it has a probability of 82.34% that the damage is complete, 16.26% that it is severe; while according to SEAOC