A method of seismic performance evaluation based seismic fragility analysis

For evaluating seismic performance of structures, a method is presented based on increment dynamic analysis and seismic fragility analysis. Firstly, the failure probability equation is deduced, in which the relationships of seismic intensities and response demands can be determined by the increment dynamic analysis. On this basis the probabilities with different damage levels under different seismic intensities are obtained. Secondly, damage parameter R is defined and used to reflect seismic damage level of structures, damage index Re is defined and used to compare with R range and determine the situation of seismic damage. Finally, the procedure is proposed for evaluating structural seismic performance. In order to verify and demonstrate the method, three structural models are designed, and typical earthquake records are selected. The results of example analysis show that this method is convenient for evaluating multi-levels seismic performance of structures.


Introduction
Seismic fragility analysis is the core research module of PBEE a new-generation methodology based on total probability theory [1]. It refers to the probability of structural failure or of reaching a certain performance level under the action of ground motions of different intensities. It is a description of probability distribution of each performance state of a building structure caused by a given disaster, and is characterized by a seismic fragility curve or a seismic fragility matrix.
At present, the studies on seismic fragility analysis and its application in structural seismic evaluation have achieved a certain number of results [2][3][4][5][6][7], but at least a problem still exists that limits their application in engineering. Seismic fragility analysis results are probability and multi-level. Expressing the result as a probability is not easily accepted by engineers, as engineers prefer to use a single value to evaluate the structural damage rather than the failure probability of using multiple failure states, which hinders its application in engineering.
For this reason, a seismic performance evaluation method is proposed in this paper based on increment dynamic analysis (IDA) and seismic fragility analysis, in which a concept of damage index is presented, and is adopted to evaluate the seismic performance of structures.

Seismic fragility analysis and seismic performance evaluation
Under the action of a certain intensity ground motion, the conditional failure probability of the maximum story drift ratio θ m of a structure reaching or exceeding the story drift ratio limit [θ] under a certain extreme state is expressed as Pf. When both θm and [θ] obey the logarithmic normal distribution, the failure probability P f can be calculated by the following equation:  (1) where μ [θ] and σ [θ] are the mean and logarithmic standard deviations of the maximum story drift ratio limit of the structure, respectively, and μ θm and σ θm are the mean and logarithmic standard deviation of the maximum story drift ratio of the structure under different ground motions, respectively.
The θ m of the θ m -PGA capability curve drawn by the IDA method is averaged, and natural logarithms of the vertical and horizontal coordinates are taken to draw a capability curve ln(μ θm )-ln(PGA). It can be seen from previous studies that such curves are basically in line with the linear relationship, so the regression equation is: In this paper, Eq. (3) is used to calculate the failure probability of structures corresponding to different seismic damage grades.
The steps of seismic fragility analysis and seismic performance evaluation based on the simplified IDA method are as follows.
(1) Draw μ θm -PGA capability curves of the structure by using the simplified IDA method and calculate σ 2 θm . (2) Take the natural logarithm of the vertical and horizontal coordinates of the μ θm -PGA capability curve, and substitute it into Eq. (2) to calculate the regression coefficients a and b.
(3) According to different damage levels, substitute PGA, a, b, μ [θ] , σ 2 θm , and σ 2 [θ] into Eq. (3) to calculate the failure probability P f corresponding to different seismic damage levels. Among these, μ [θ] can be determined by the upper limit of θ max corresponding to each damage level in Table 1;σ 2 [θ] is set to 0.399. In the table 1, the 1,2,3 and 4-th seismic performance levels represent structures subjected to small earthquake, medium earthquake, strong earthquake and Super strong earthquake are not bad, repairable, not collapse and near or collapse, respectively. The first three levels are derived from the code, and the 4-th level is aimed at the situation of near collapse or collapse of structures.
(4) Substitute the failure probability value P f,i of the structure at the i-th damage level into Eq. (4), and calculate the probability value P i of the structure at the ith damage level.
where P f,0 =1. (5) Defining R is a damage parameter for different seismic performance levels, and R o is the calculated value of R for different damage levels, which are listed in Table 1.
Substitute the damage probability P and the R o corresponding to each damage level into Eq. (4), and calculate the damage index R e which is the mathematical in which the subscript i represents the corresponding i-th damage level in Table 1. (6) Compare R e with the R range in Table 1 to determine the damage of the structure under different PGA ground motion excitations, and then evaluate whether the structure meets the requirements under different seismic performance levels.

Designed structural models
Three six-story structural models were designed for hard soil sites (I 0 ,I 1 ), intermediate soil sites (II,III), and a soft soil site (IV), and designated as structures 1, 2 and, 3, respectively. The three structural models have a story height of 3.6 m and uniform lateral shear stiffness along the story. The interstory damping uses the Rayleigh damping with a damping ratio of 5%. The basic period is 0.61 s and the fortification intensity is 7. Table 2 lists the story weight, mode of vibration, and interstory shear strength of the three structural models.
The structural models of this calculation example can be simplified to a multi-degree-of-freedom (MDOF) system. The ideal story hysteretic relationship of these three MDOF systems is a bilinear stiffness model, the yield stiffness coefficient is 0.024, and the stiffness degradation coefficient is 0.4.

Damage probability calculation and seismic performance evaluation
The PGA of the selected earthquake acceleration is adjusted from 'small to large'. A dynamic time-history analysis with continuously increasing excitation of the structural models is conducted, that is increment dynamic analysis, and θ m -PGA format capability curves are illustrated in Figure 1.
The damage probability P is calculated by Eq. (3) and Eq. (4), and then the damage index R e is obtained. The damage indexes R e corresponding to different PGA are listed in Table 4. Through table 4 and R ranges in Table 1, it is intuitive to determine whether the structure model meets the seismic performance requirements under different earthquake intensities.
The Code for Seismic Design of Buildings provides ground motion PGA values under small, medium, and strong earthquakes. The structural models of this calculation example are all fortified at 7, so the PGA takes values of 0.036, 0.100, and 0.224 g. The R e values corresponding to the three PGA values are listed in Table 5 to evaluate whether the studied structural models meet seismic performance requirements at 1,2 and 3-th levels. From Table 5, it is shown that: according to the method proposed in this paper, the structural model in accordance with the seismic performance level. In particular, under the action of a large earthquake, the structural damage is small, which does not exceed the damage level corresponding to that of a medium earthquake.  Notes：The symbols ★, ▲, △ represent the structure 1,2 and 3, respectively.

Conclusions
In this paper, a method for evaluating seismic performance of structures is proposed based on seismic fragility analysis, as follows.
(1) The IDA is used to get the relationships between seismic intensities and structural response demands, on this basis, the failure probability equation is given, and then the probabilities with different damage levels under different seismic intensities are obtained.
(2) Damage parameter R is defined and used to reflect seismic damage level of structures, damage index R e is defined and used to compare with R range and determine the situation of seismic damage.
(3) The procedure is given, in which the damage index is calculated to evaluate whether the structure meets the seismic performance requirements at different levels.
In order to validate and demonstrate the proposed method, three structural models are designed according to the hard, Intermediate and soft soil sites, respectively, and 15 typical earthquake records are selected as the horizontal earthquake motion excitation of the structural models. The example results show that this method is clear in concept and simple in operation, and it can be used as a seismic design or checking method of structures.