# Assessment of Seismic Vulnerability of Steel and RC Moment Buildings Using HAZUS and Statistical Methodologies

Authors: Iman Mansouri; Jong Wan Hu; Kazem Shakeri; Shahrokh Shahbazi; Bahareh Nouri
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## Abstract

Designer engineers have always the serious challenge regarding the choice of the kind of structures to use in the areas with significant seismic activities. Development of fragility curve provides an opportunity for designers to select a structure that will have the least fragility. This paper presents an investigation into the seismic vulnerability of both steel and reinforced concrete (RC) moment frames using fragility curves obtained by HAZUS and statistical methodologies. Fragility curves are employed for several probability parameters. Fragility curves are used to assess several probability parameters. Furthermore, it examines whether the probability of the exceedence of the damage limit state is reduced as expected. Nonlinear dynamic analyses of five-, eight-, and twelve-story frames are carried out using Perform 3D. The definition of damage states is based on the descriptions provided by HAZUS, which gives the limit states and the associated interstory drift limits for structures. The fragility curves show that the HAZUS procedure reduces probability of damage, and this reduction is higher for RC frames. Generally, the RC frames have higher fragility compared to steel frames.


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## Body

## 1. Introduction

Collapse fragility curves give information on how the probability of structural damage increases with increasing peak ground acceleration. Collapse capacities are presumed to follow a lognormal distribution. Literature reviews show that utilization of fragility curves began with nuclear facilities; because the damage to these structures can be so devastating, these structures are considered extremely important. Fragility curves were first designed for nuclear power plants in 1980 [2]. Following that, these curves were further developed by Kircher and Martin 1993 [3]; the curves are simple in terms of calculation, and they had been obtained experimentally and with engineering judgment. Moreover, they had considered the vertical and horizontal axes based on emergence probability and the qualitative amount of earth motion, respectively. Recent studies have found the lognormal distribution to provide good results for illustrating the collapse capacity data [4–7]. Several approaches have been suggested for the fragility analysis of structures: empirical, analytical, and hybrid methods [8]. Rossetto and Elnashai [9] used the empirical method by using data gathered on structures damaged during past earthquakes.An investigation on methods to evaluate the seismic collapse of structures by Villaverde [10] and global collapse of frames by Ibarra and Krawinkler [4] as well as the ATC-63 project [11] has reported extensive literature reviews of analytical methods available to evaluate the capacity of structures to withstand seismic collapse. After the Northridge earthquake, more attention was allocated to estimate the amount of structural damage. During a study on California structures, principles of ATC-13 were used to plot the fragility curves in 1994. In that work, fragility curves were plotted for steel moment resisting frames and wooden and reinforced concrete structures. Anagnos et al. [12] undertook further studies based on inserted load distribution in ATC in 1995, and they presented a new model of fragility curves. In this paper, all seismic calculations were performed based on ATC-13, where the probability distribution function had been as assumed to be the normal distribution. Fragility curves were produced for Memphis buildings in 1996, and the displacement among stories was investigated at different levels. Singhal and Kiremidjian estimated fragility curves according to the observations in the one-story buildings. They used the Park-Ang index to assess damage and then expressed the amount by statistical distribution function undergoing various earthquakes [13]. Shinozuka [14] presented an investigative paper about the fragility curve of a one-span bridge that, in comparison with former works, was very precise. Because a nonlinear dynamic technique and statistical data had been used, they presented a method for obtaining fragility curves in bridges. Later, Saxena et al. [15] used this procedure for multispan bridges and reinforced concrete, in which the generation of fragility curves for bridges was presented step by step for the first time, and the probability distribution of the data was assumed to be the normal distribution. Tanaka et al. [16] used the lognormal distribution to calibrate fragility curves in 2000. He classified 3683 bridges into 5 groups, defined the amount of damage for 5 levels, and assessed lognormal distribution parameters.Researchers have also plotted fragility curves for steel, wooden, and concrete structures by using the damage functions and PGV in the Japan Kobe in 2000 [17]. They came to the conclusion that the reinforced concrete and steel structures have the minimum and maximum fragility, respectively, in this city. Furthermore, the production procedure of fragility curves generated using the capacity spectrum method was presented in 2001. In this work, researchers plotted fragility curves for displacement among stories by utilizing the nonlinear static analysis method. Pagni [18] presented a damage model for old reinforced concrete elements in Washington University. He introduced 12 states of damage, which included primary cracking and the common face of beam-column. Moreover, damage states were divided into 2 groups of cracking and crumbling, and the best distribution was selected. Fragility curves were generated for steel moment resisting frame buildings by using Perform software and FEMA codes in 2006. Nonlinear dynamic analysis of 2-, 3-, 4-, 6-, 8-, and 10-story frames was performed. Ozer et al. [19] dealt with developing seismic failure curves for 3-, 5-, 7-, and 9-story buildings that were designed based on the TSC-1998 prevalent code. The structure model was the concrete moment resisting frame which was designed two-dimensionally by using Sap 2000 and IDARC-2D software, which were used for analysis and evaluation. Structure frames were classified into poor (with low quality) and superior (with high quality) groups based on the special features, construction process, and observed performance after large earthquakes in Turkey. Failure curves were plotted in different groups according to the parameters affecting the seismic performance, and the number of stories was considered as an important parameter in a building’s seismic vulnerability. Barkhordary and Tariverdilo [20] studied an evaluation of the effect of column bar patching and bar slipping on the structure’s vulnerability, utilizing concrete moment resisting frame systems in Iran. The structure model was considered two-dimensionally and was analyzed using OpenSees (2006). The seismic reliabilities of two typical stainless steel legged wine storage tanks have been studied by Colombo and Almazán [21]. Seismic fragility of nonlinear connections in large piping systems has been evaluated by Tadinada and Gupta [22]. Seismic fragility of RC structures with shape memory alloys has been investigated by Mirtaheri et al. [23]. Finally, the results of additive dynamic analysis were used for generating the fragility curves. Nazri and Saruddin [24] analyzed steel and concrete moment resisting frames under near and far field earthquakes using IDA, and, according to the fragility curve, they presented the possibility of reaching or exceeding damage limits for each structure at 5 levels of performance introduced in FEMA-273.HAZUS (Hazard US) is a well-known loss estimation methodology that defines five damage states, none, slight, moderate, extensive, and complete, using physical (qualitative) descriptions of damage to building elements. The HAZUS methodology has been applied to various seismic hazard assessment studies by adapting capacity and fragility curves for structures in specific regions [25, 26].In areas with significant seismic activity, choosing the kind of structure has always been a serious challenge for designer engineers. In this regard, development of fragility curves can help designer select a structure that has the least fragility. On the other hand, the question presented is that, among all common methods, which one is more accurate and comprehensive? In this paper, we aim to provide an answer to this and other important questions.


## 2. Damage States

To investigate the vulnerability of structures, it is necessary to consider a damage index. In this study, interstory drift is considered as the damage index. Additionally, a series of failure modes is defined and evaluated in terms of seismic vulnerability. In this paper, the limit states used are based on the HAZUS-MH MR5 guidelines [1]. According to HAZUS, there are 36 building types, and each type is divided into three categories in terms of number of stories: low-rise, mid-rise, and high-rise, which generally correspond to 1–3, 4–7, and 8+ stories, respectively. The limit and intensity of damage to structural components of a building are described by one of four damage states: slight, moderate, extensive, and complete [1]. The values for each damage state are presented in Table 1. Damage occurs when the drift value reaches the limit state.Table 1
Interstory drift values for the damage states [1].
Damage state
Mid-rise RC structure
High-rise RC structure
Mid-rise steel structure
High-rise steel structure
Slight
0.0033
0.0025
0.004
0.003
Moderate
0.0067
0.005
0.008
0.006
Extensive
0.02
0.015
0.02
0.015
Complete
0.0533
0.04
0.0533
0.04


## 3. Building Design and Modeling

According to HAZUS guideline, 4- to 7-story buildings are classified as mid-rise buildings and the buildings with higher 8 stories are classified as tall buildings. Therefore, in this study 5-, 8-, and 12-story buildings are analyzed as a sample of low-rise, border of mid- and high-rise, and high-rise buildings, respectively. The 5-, 8-, and 12-story steel and concrete buildings have the same plan view (Figure1) and seismic design category; they are designed based on the AISC 360-10, ACI 318, and ASCE 7-10 codes. The height of the first story is 2.7 m (to top of beam), and the height of all the other stories is 3.2 m. The dead and live loads are 460 and 200 kg/m2 uniformly distributed over each floor. The specimens are analyzed using Perform 3D v4.0.3, which is a nonlinear dynamic analysis program. Beams, columns, and connection regions are modeled individually, and ground acceleration records were applied to the specimen models. All of the structural members for all the buildings can be found in Figures 21 and 22 and Tables 5, 6, 7, 8, 9, and 10.Figure 1
Typical plan view.


## 4. Ground Motion Database

According to [27], accurate assessment of damage demand can be achieved using 10 to 20 ground motion records. In this study, ten ground motion records were selected for the nonlinear time history analyses. The selected records consist of site classesC andD extracted from the PEER database and scaled from 0.1 g to 1.5 g by steps of 0.1 g. Pertinent information on the ground motion database is presented in Table 2.Table 2
Ground motion database.
Number
Earthquake
Year
M
w
Station
PGA (g)
1
Imperial Valley
1979
6.5
SAOHOP Casa Flores
0.506
2
Imperial Valley
1979
6.5
SAOHOP Casa Flores
0.287
3
Loma Prieta
1989
6.9
Capitol
0.529
4
Loma Prieta
1989
6.9
Capitol
0.443
5
Morgan Hill
1984
6.2
Hallas Valley
0.312
6
Northridge
1994
6.7
Rinaldi Receiving Station
0.838
7
Northridge
1994
6.7
Rinaldi Receiving Station
0.472
8
Northridge
1994
6.7
LA-Hollywood Stro FF
0.358
9
Parkfield
1966
6.1
Cholame #5
0.442
10
Whittier Narrows
1987
6
Glendale-Las Palmas
0.296


## 5. Fragility Analysis

To facilitate the assessment of the seismic vulnerability of the structures, fragility functions defining the probability of meeting or exceeding a specific limit state given an earthquake intensity level were developed using the nonlinear time history analyses described above.The Pacific Earthquake Engineering Center gives a formulation for the fragility analysis and probabilistic seismic assessment framework. This framework proposes a relationship between a certain key decision variable, DV (e.g., annual seismic loss), and a series of conditional probabilities, deaggregating average annual frequency (AAF) of exceeding the decision parameter,λDV, in terms of damage measure, DM, and ground motion intensity measure, IM, as follows: (1)λDV=∬GDV∣DMdgDM∣IMdλIM,in which (DV∣DM) denotes the probability that the DV exceeds specified values given that the engineering DMs are equal to special values, GDM∣IM is the probability that the DMs exceed these values given that the IM is equal special value, and λIM is the AAF of the ground motion IM.The partGDM∣IM in the deaggregated equation is named the seismic fragility function. Seismic fragility relates the probability of achieving or exceeding predefined levels of damage to the severity of ground motion intensity. Analytical fragility curves are derived from the results of numerical models of the buildings subjected to ground motion records. The fragility function is given in (2)PD>Di∣IM=Φln⁡x/Diβd∣IM2+βc2+βm2,where Φ(·) stands for the standard normal cumulative distribution function, Di is the superior bound for each damage level, x is the median value of demand as the function of IM, βd∣IM is the logarithmic standard deviation of the demand conditioned on the IM, βc is the capacity uncertainty, and βm is the modeling uncertainty.The output of the linear regression analysis, in a log-transformed space and with empirically based limit-state models, may be used to develop closed-form expressions for fragility functions, assuming limit-state models also follow a lognormal distribution:(3)P≤D=Фln⁡Sd/Scβsd,in which the left-hand side is the probability of exceeding the damage limit, D is damage, Ф is the standard normal cumulative distribution, Sd is the median estimate of demand (maximum story drift), Sc is the median estimate of capacity, and βsd=βD∣IM2+βc2, where βD∣IM is the dispersion of the demand conditioned on the intensity measure and βc is the dispersion of the capacity.


### 5.1. HAZUS Methodology

The values ofSc and βsd for various building types were taken from the HAZUS guidelines (Table 3).Table 3
Values ofSc and βsd according to [1].
Type
Slight
Moderate
Extensive
Complete
S
c
β
s
d
S
c
β
s
d
S
c
β
s
d
S
c
β
s
d
S1L
1.3
0.80
2.59
0.76
6.48
0.69
17.28
0.72
S1M
2.16
0.65
4.32
0.66
10.80
0.67
28.80
0.74
S1H
3.37
0.64
6.74
0.64
16.85
0.65
44.93
0.67
C1L
0.9
0.81
1.8
0.84
5.40
0.86
14.40
0.81
C1M
1.5
0.68
3
0.67
9.00
0.68
24.00
0.81
C1H
2.16
0.66
4.32
0.64
12.96
0.67
34.56
0.78The parameterSd can be calculated as follows:(4)ln⁡Sd=aln⁡PGA+b,where a and b are regression coefficients which can be obtained using regression analysis of incurred damage versus PGA [1]. For example, Figure 2 shows the regression of the 5-story RC frame.Figure 2
Probabilistic seismic demand (drift) for the 5-story RC frame.After defining and modeling the nonlinear properties of the elements, the scaled ground motion records are applied to structures using Perform 3D software. For each model, the maximum interstory drifts are derived and compared with HAZUS criteria to determine the number meeting or exceeding the limit states. For example, the 5-story RC frame is analyzed 10 times at an acceleration of 0.7 g for 10 ground motion records; the results (Table4) show that, in 10, 8, 3, and 0 cases, the structure has experienced slight, moderate, extensive, and complete damage states, respectively.Table 4
Number of instances reaching or exceeding the limit states for the 5-story RC frame.
PGA
Slight
Moderate
Extensive
Complete
0.1 g
3
0
0
0
0.2 g
6
3
0
0
0.3 g
8
5
0
0
0.4 g
9
6
3
0
0.5 g
10
8
3
0
0.6 g
10
8
3
0
0.7 g
10
8
3
0
0.8 g
10
9
5
0
0.9 g
10
10
5
0
1.0 g
10
10
5
2
1.1 g
10
10
6
2
1.2 g
10
10
7
2
1.3 g
10
10
7
3
1.4 g
10
10
10
3
1.5 g
10
10
10
4Table 5
Sections for the 5-story RC frame.
Story
Column
Beam
C1
C2
C3
C4
Size
Longitudinal bars
Top bars
Bottom bars
1
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
45
×
40
3
ϕ
20
3
ϕ
24
2
ϕ
24
12
ϕ
20
12
ϕ
18
12
ϕ
20
12
ϕ
20
2
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
45
×
40
3
ϕ
20
3
ϕ
24
2
ϕ
24
12
ϕ
20
12
ϕ
18
12
ϕ
20
12
ϕ
18
3
C
40
×
40
C
40
×
40
C
40
×
40
C
40
×
40
B
40
×
35
3
ϕ
20
3
ϕ
24
2
ϕ
24
12
ϕ
18
12
ϕ
18
12
ϕ
18
10
ϕ
18
4
C
40
×
40
C
40
×
40
C
40
×
40
C
40
×
40
B
40
×
35
3
ϕ
20
3
ϕ
24
2
ϕ
24
12
ϕ
18
8
ϕ
18
12
ϕ
18
8
ϕ
18
5
C
40
×
40
C
4
0
×
40
C
40
×
40
C
4
0
×
40
B
40
×
35
3
ϕ
20
3
ϕ
24
2
ϕ
24
8
ϕ
18
8
ϕ
18
8
ϕ
18
8
ϕ
18Table 6
Sections for the 8-story RC frame.
Story
Column
Beam
C1
C2
C3
C4
Size
Longitudinal bars
Top bars
Bottom bars
1
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
45
3
ϕ
20
3
ϕ
24
3
ϕ
24
12
ϕ
22
12
ϕ
20
12
ϕ
22
12
ϕ
20
2
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
45
3
ϕ
20
3
ϕ
24
3
ϕ
24
12
ϕ
18
12
ϕ
20
12
ϕ
18
10
ϕ
20
3
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
45
3
ϕ
20
3
ϕ
24
3
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
4
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
40
3
ϕ
18
3
ϕ
24
3
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
5
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
40
3
ϕ
18
3
ϕ
24
3
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
6
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
40
3
ϕ
18
3
ϕ
24
3
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
7
C
45
×
45
C
45
×
45
C
45
×
45
C
45
×
45
B
45
×
40
3
ϕ
18
3
ϕ
20
2
ϕ
20
8
ϕ
18
8
ϕ
18
8
ϕ
18
8
ϕ
18
8
C
45
×
45
C
45
×
45
C
45
×
45
C
45
×
45
B
45
×
40
3
ϕ
18
3
ϕ
20
2
ϕ
20
8
ϕ
18
8
ϕ
18
8
ϕ
18
8
ϕ
18Table 7
Sections for the 12-story RC frame.
Story
Column
Beam
C1
C2
C3
C4
Size
Longitudinal bars
Top bars
Bottom bars
1
C60 × 60
C60 × 60
C60 × 60
C60 × 60
B60 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
16
ϕ
24
16
ϕ
24
16
ϕ
24
16
ϕ
24
2
C60 × 60
C60 × 60
C60 × 60
C60 × 60
B60 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
12
ϕ
24
12
ϕ
20
12
ϕ
24
12
ϕ
20
3
C60 × 60
C60 × 60
C60 × 60
C60 × 60
B60 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
4
C55 × 55
C55 × 55
C55 × 55
C55 × 55
B55 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
5
C55 × 55
C55 × 55
C55 × 55
C55 × 55
B55 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
6
C55 × 55
C55 × 55
C55 × 55
C55 × 55
B55 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
7
C50 × 50
C50 × 50
C50 × 50
C50 × 50
B50 × 45
3
ϕ
20
3
ϕ
24
3
ϕ
24
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
8
C50 × 50
C50 × 50
C50 × 50
C45 × 45
B50 × 45
3
ϕ
20
3
ϕ
24
3
ϕ
24
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
9
C50 × 50
C50 × 50
C50 × 50
C50 × 50
B50 × 45
3
ϕ
20
3
ϕ
24
3
ϕ
24
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
10
C45 × 45
C45 × 45
C45 × 45
C45 × 45
B45 × 40
3
ϕ
18
3
ϕ
24
2
ϕ
24
10
ϕ
18
12
ϕ
20
10
ϕ
18
12
ϕ
20
11
C45 × 45
C45 × 45
C45 × 45
C45 × 45
B45 × 40
3
ϕ
18
3
ϕ
24
2
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
12
C45 × 45
C45 × 45
C45 × 45
C45 × 45
B45 × 40
3
ϕ
18
3
ϕ
24
2
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18Table 8
Sections for the 5-story steel frame.
Story
Column
Girder
C1
C2
C3
C4
d
h
b
f
t
f
t
w
1
Box
Box
Box
Box
28
25.6
15
1.2
0.8
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
2
Box
Box
Box
Box
28
25.6
15
1.2
0.8
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
3
Box
Box
Box
Box
28
25.6
15
1.2
0.8
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
20
×
20
×
1.2
4
Box
Box
Box
Box
25
22.6
15
1.2
0.6
20
×
20
×
1.2
20
×
20
×
1.2
20
×
20
×
1.5
20
×
20
×
1.2
5
Box
Box
Box
Box
25
22.6
15
1.2
0.6
20
×
20
×
1.2
20
×
20
×
1.2
20
×
20
×
1.5
20
×
20
×
1.2Table 9
Sections for the 8-story steel frame.
Story
Column
Girder
C1
C2
C3
C4
d
h
b
f
t
f
t
w
1
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
2
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
3
Box
Box
Box
Box
28
24
15
2
0.8
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
4
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
5
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.2
20 × 20 × 1
6
Box
Box
Box
Box
25
22
15
1.5
0.6
25 × 25 × 1.5
20 × 20 × 1
25 × 25 × 1.2
20 × 20 × 1
7
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
8
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1Table 10
Sections for the 12-story steel frame.
Story
Column
Girder
C1
C2
C3
C4
d
h
b
f
t
f
t
w
1
Box
Box
Box
Box
30
26
18
2
0.8
35 × 35 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
2
Box
Box
Box
Box
30
26
18
2
0.8
35 × 35 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
3
Box
Box
Box
Box
30
26
18
2
0.8
35 × 35 × 1.5
30 × 30 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
4
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
5
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
6
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
25 × 25 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
7
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
30 × 30 × 1.5
8
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
9
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
20 × 20 × 1
25 × 25 × 1.5
25 × 25 × 1.5
10
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
25 × 25 × 1.5
11
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
12
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1The fragility curves for both steel and RC frames are plotted as shown in Figures3–8. The curves include four damage states based on HAZUS guidelines.Figure 3
Comparison of the fragility of the 12-story steel and RC frame buildings for the slight damage state.Figure 4
Comparison of the fragility of the 12-story steel and RC frame buildings for the moderate damage state.Figure 5
Comparison of the fragility of the 12-story steel and RC frame buildings for the extensive damage state.Figure 6
Comparison of the fragility of the 12-story steel and RC frame buildings for the complete damage state.Figure 7
Comparison of the fragility of the 8-story steel and RC frame buildings for the slight damage state.Figure 8
Comparison of the fragility of the 8-story steel and RC frame buildings for the moderate damage state.The curves shifting left mean that the probability of damage has increased. All curves show that, for slight and moderate states, the slope of curve is initially steep and then decreases. Usually, this slope change occurs at PGA > 0.8 g, because until 0.8 g the probability of exceeding the limit states varies between 0 and 100% while after 0.8 g this probability closely approaches 100%. For the extensive and complete damage states, the slope of the curve is nearly constant. The results indicate that in the steel frames the highest probability of extensive damage is 60%, while this value is 80% for RC frames. The probabilities of complete damage for steel and RC frames are 10–15% and 20–30%, respectively.The fragilities of the steel frames for slight, moderate, extensive, and complete states are compared in Figures9–12.Figure 9
Comparison of the fragility of the 8-story steel and RC frame buildings for the extensive damage state.Figure 10
Comparison of the fragility of the 8-story steel and RC frame buildings for the complete damage state.Figure 11
Comparison of the fragility of the 5-story steel and RC frame buildings for the slight damage state.Figure 12
Comparison of the fragility of the 5-story steel and RC frame buildings for the moderate damage state.Results reveal that, in general, the increase in the number of stories leads to increase in the fragility of both steel and RC structures. This difference is more significant in RC frames. For example, in the moderate damage state, the difference between 5 and 12 steel frames is 15%, while this value for 5 and 12 RC frames is approximately 30%. These comparisons can also be seen in Figures13–16.Figure 13
Comparison of the fragility of the 5-story steel and RC frame buildings for the extensive damage state.Figure 14
Comparison of the fragility of the 5-story steel and RC frame buildings for the complete damage state.Figure 15
The fragility curves for the 5-story steel frame buildings.
(a)
(b)
(c)
(d)Figure 16
The fragility curves for 8-story steel frame.
(a)
(b)
(c)
(d)The fragilities of the 12-story steel and RC frames are compared for slight, moderate, extensive, and complete damage states as shown in Figures3–6.Figures3–6 demonstrate that the fragility of a 12-story RC frame is higher compared to that of a 12-story steel frame. This difference is more significant in higher damage states, that is, at the slight damage state there is minimal difference, and at the complete state, there is maximal difference. These figures also show that the curves of the slight and moderate damage states begin to converge with increasing PGA, while the curves of the extensive and complete states begin to diverge with increasing the PGA.The fragilities of the 8-story steel and RC frames are compared for slight, moderate, extensive, and complete damage states, as shown in Figures7–10.Figures7–10 show that, similar to 12-story models, 8-story RC frames have higher fragility compared with steel frames. These figures also show that the curves of the slight and moderate damage states begin to converge with increasing PGA, while the curves of the extensive and complete states begin to diverge with increasing the PGA.The fragilities of the 5-story steel and RC frames are compared for the slight, moderate, extensive, and complete states, as shown in Figures11–14.Evaluation of Figures11–14 indicates similar results for 12- and 8-story frames. Furthermore, in the 5-story frames at the complete damage state, the fragility curves nearly overlap at low PGA and only diverge slightly at high PGA. The most significant difference is seen at the moderate damage state, and the least significant is at the complete damage state.


### 5.2. Statistical Methodology

As described previously, the fragility curve can be obtained using this equation:(5)1Pf=ϕln⁡Sd/ScβD∣IM2+βc2.In Section 5.1, the required parameters and coefficients have been defined using HAZUS. In this part, these parameters are obtained using nonlinear dynamic analyses and a statistical postprocessing. The ground motion records are scaled from 0.1 g to 1.5 g by a step of 0.1 g.Figure15 shows the fragility curves for a 5-story steel frame using both HAZUS and the statistical procedure. The figure shows that, generally, the statistical method yields higher fragility. The most significant difference is seen at the extensive damage state and the least significant at the complete damage state. The curves for the slight damage state nearly overlap, and at PGA > 0.7 g the probability approaches 100%. At the moderate damage state, the difference between the statistical and HAZUS methodologies is approximately 10%.The comparison between statistical and HAZUS methodologies for an 8-story steel frame is shown in Figure16.The comparison between statistical and HAZUS methodologies for a 12-story steel frame is shown in Figure17.Figure 17
The fragility curves for the 12-story steel frame buildings.
(a)
(b)
(c)
(d)As is evident in Figures15 to 18, the HAZUS method yields a reduced probability of damage. The difference between the curves is higher compared to the 5-story steel frame. Similar to the 5-story steel frame, in the extensive damage state, the most significant difference (30%) is observed.Figure 18
The fragility curves for the 5-story RC frame buildings.
(a)
(b)
(c)
(d)Figure17 shows the fragility curves for a 12-story steel frame. The most significant difference is between the moderate and extensive damage states (20%). This difference is 10% for the slight damage state, and in the complete state the curves are approximately the same.The fragility curves for RC frames are presented in Figure18.Figure18 shows that the HAZUS procedure yields a reduced probability of damage. At three damage states, slight, moderate, and extensive, the difference between the HAZUS and statistical procedures is 25%, but at the complete state, the curves nearly overlap at low PGA and diverge only slightly at large PGA. A comparison between the 5-story steel and RC frames shows that this difference percentage is higher in the RC case. Figure 19 shows that similar conclusions can be drawn for an 8-story RC frame.Figure 19
The fragility curves for the 8-story RC frame buildings.
(a)
(b)
(c)
(d)Seismic vulnerability of a 12-story RC frame has been assessed using two approaches (Figure20). The comparison shows that, similar to previous conclusions, the HAZUS procedure yields reduced damage probability except at PGA = 0.6 g for the extensive damage state. The most significant difference can be seen in the extensive damage state (statistical method yields 20% higher fragility). In other states, the curves are fairly close together.Figure 20
The fragility curves for 12-story RC frame buildings.
(a)
(b)
(c)
(d)Figure 21
Typical plan view of RC columns.Figure 22
Typical plan view of steel columns.Generally, in RC frames, the difference between the statistical and HAZUS procedures is 20%.


## 5.1. HAZUS Methodology

The values ofSc and βsd for various building types were taken from the HAZUS guidelines (Table 3).Table 3
Values ofSc and βsd according to [1].
Type
Slight
Moderate
Extensive
Complete
S
c
β
s
d
S
c
β
s
d
S
c
β
s
d
S
c
β
s
d
S1L
1.3
0.80
2.59
0.76
6.48
0.69
17.28
0.72
S1M
2.16
0.65
4.32
0.66
10.80
0.67
28.80
0.74
S1H
3.37
0.64
6.74
0.64
16.85
0.65
44.93
0.67
C1L
0.9
0.81
1.8
0.84
5.40
0.86
14.40
0.81
C1M
1.5
0.68
3
0.67
9.00
0.68
24.00
0.81
C1H
2.16
0.66
4.32
0.64
12.96
0.67
34.56
0.78The parameterSd can be calculated as follows:(4)ln⁡Sd=aln⁡PGA+b,where a and b are regression coefficients which can be obtained using regression analysis of incurred damage versus PGA [1]. For example, Figure 2 shows the regression of the 5-story RC frame.Figure 2
Probabilistic seismic demand (drift) for the 5-story RC frame.After defining and modeling the nonlinear properties of the elements, the scaled ground motion records are applied to structures using Perform 3D software. For each model, the maximum interstory drifts are derived and compared with HAZUS criteria to determine the number meeting or exceeding the limit states. For example, the 5-story RC frame is analyzed 10 times at an acceleration of 0.7 g for 10 ground motion records; the results (Table4) show that, in 10, 8, 3, and 0 cases, the structure has experienced slight, moderate, extensive, and complete damage states, respectively.Table 4
Number of instances reaching or exceeding the limit states for the 5-story RC frame.
PGA
Slight
Moderate
Extensive
Complete
0.1 g
3
0
0
0
0.2 g
6
3
0
0
0.3 g
8
5
0
0
0.4 g
9
6
3
0
0.5 g
10
8
3
0
0.6 g
10
8
3
0
0.7 g
10
8
3
0
0.8 g
10
9
5
0
0.9 g
10
10
5
0
1.0 g
10
10
5
2
1.1 g
10
10
6
2
1.2 g
10
10
7
2
1.3 g
10
10
7
3
1.4 g
10
10
10
3
1.5 g
10
10
10
4Table 5
Sections for the 5-story RC frame.
Story
Column
Beam
C1
C2
C3
C4
Size
Longitudinal bars
Top bars
Bottom bars
1
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
45
×
40
3
ϕ
20
3
ϕ
24
2
ϕ
24
12
ϕ
20
12
ϕ
18
12
ϕ
20
12
ϕ
20
2
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
45
×
40
3
ϕ
20
3
ϕ
24
2
ϕ
24
12
ϕ
20
12
ϕ
18
12
ϕ
20
12
ϕ
18
3
C
40
×
40
C
40
×
40
C
40
×
40
C
40
×
40
B
40
×
35
3
ϕ
20
3
ϕ
24
2
ϕ
24
12
ϕ
18
12
ϕ
18
12
ϕ
18
10
ϕ
18
4
C
40
×
40
C
40
×
40
C
40
×
40
C
40
×
40
B
40
×
35
3
ϕ
20
3
ϕ
24
2
ϕ
24
12
ϕ
18
8
ϕ
18
12
ϕ
18
8
ϕ
18
5
C
40
×
40
C
4
0
×
40
C
40
×
40
C
4
0
×
40
B
40
×
35
3
ϕ
20
3
ϕ
24
2
ϕ
24
8
ϕ
18
8
ϕ
18
8
ϕ
18
8
ϕ
18Table 6
Sections for the 8-story RC frame.
Story
Column
Beam
C1
C2
C3
C4
Size
Longitudinal bars
Top bars
Bottom bars
1
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
45
3
ϕ
20
3
ϕ
24
3
ϕ
24
12
ϕ
22
12
ϕ
20
12
ϕ
22
12
ϕ
20
2
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
45
3
ϕ
20
3
ϕ
24
3
ϕ
24
12
ϕ
18
12
ϕ
20
12
ϕ
18
10
ϕ
20
3
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
45
3
ϕ
20
3
ϕ
24
3
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
4
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
40
3
ϕ
18
3
ϕ
24
3
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
5
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
40
3
ϕ
18
3
ϕ
24
3
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
6
C
50
×
50
C
50
×
50
C
50
×
50
C
50
×
50
B
50
×
40
3
ϕ
18
3
ϕ
24
3
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
7
C
45
×
45
C
45
×
45
C
45
×
45
C
45
×
45
B
45
×
40
3
ϕ
18
3
ϕ
20
2
ϕ
20
8
ϕ
18
8
ϕ
18
8
ϕ
18
8
ϕ
18
8
C
45
×
45
C
45
×
45
C
45
×
45
C
45
×
45
B
45
×
40
3
ϕ
18
3
ϕ
20
2
ϕ
20
8
ϕ
18
8
ϕ
18
8
ϕ
18
8
ϕ
18Table 7
Sections for the 12-story RC frame.
Story
Column
Beam
C1
C2
C3
C4
Size
Longitudinal bars
Top bars
Bottom bars
1
C60 × 60
C60 × 60
C60 × 60
C60 × 60
B60 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
16
ϕ
24
16
ϕ
24
16
ϕ
24
16
ϕ
24
2
C60 × 60
C60 × 60
C60 × 60
C60 × 60
B60 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
12
ϕ
24
12
ϕ
20
12
ϕ
24
12
ϕ
20
3
C60 × 60
C60 × 60
C60 × 60
C60 × 60
B60 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
4
C55 × 55
C55 × 55
C55 × 55
C55 × 55
B55 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
5
C55 × 55
C55 × 55
C55 × 55
C55 × 55
B55 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
6
C55 × 55
C55 × 55
C55 × 55
C55 × 55
B55 × 50
3
ϕ
20
4
ϕ
26
4
ϕ
26
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
7
C50 × 50
C50 × 50
C50 × 50
C50 × 50
B50 × 45
3
ϕ
20
3
ϕ
24
3
ϕ
24
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
8
C50 × 50
C50 × 50
C50 × 50
C45 × 45
B50 × 45
3
ϕ
20
3
ϕ
24
3
ϕ
24
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
9
C50 × 50
C50 × 50
C50 × 50
C50 × 50
B50 × 45
3
ϕ
20
3
ϕ
24
3
ϕ
24
10
ϕ
22
12
ϕ
20
10
ϕ
22
12
ϕ
20
10
C45 × 45
C45 × 45
C45 × 45
C45 × 45
B45 × 40
3
ϕ
18
3
ϕ
24
2
ϕ
24
10
ϕ
18
12
ϕ
20
10
ϕ
18
12
ϕ
20
11
C45 × 45
C45 × 45
C45 × 45
C45 × 45
B45 × 40
3
ϕ
18
3
ϕ
24
2
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18
12
C45 × 45
C45 × 45
C45 × 45
C45 × 45
B45 × 40
3
ϕ
18
3
ϕ
24
2
ϕ
24
10
ϕ
18
10
ϕ
18
10
ϕ
18
10
ϕ
18Table 8
Sections for the 5-story steel frame.
Story
Column
Girder
C1
C2
C3
C4
d
h
b
f
t
f
t
w
1
Box
Box
Box
Box
28
25.6
15
1.2
0.8
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
2
Box
Box
Box
Box
28
25.6
15
1.2
0.8
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
3
Box
Box
Box
Box
28
25.6
15
1.2
0.8
30
×
30
×
1.5
30
×
30
×
1.5
30
×
30
×
1.5
20
×
20
×
1.2
4
Box
Box
Box
Box
25
22.6
15
1.2
0.6
20
×
20
×
1.2
20
×
20
×
1.2
20
×
20
×
1.5
20
×
20
×
1.2
5
Box
Box
Box
Box
25
22.6
15
1.2
0.6
20
×
20
×
1.2
20
×
20
×
1.2
20
×
20
×
1.5
20
×
20
×
1.2Table 9
Sections for the 8-story steel frame.
Story
Column
Girder
C1
C2
C3
C4
d
h
b
f
t
f
t
w
1
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
2
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
3
Box
Box
Box
Box
28
24
15
2
0.8
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
4
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
5
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.2
20 × 20 × 1
6
Box
Box
Box
Box
25
22
15
1.5
0.6
25 × 25 × 1.5
20 × 20 × 1
25 × 25 × 1.2
20 × 20 × 1
7
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
8
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1Table 10
Sections for the 12-story steel frame.
Story
Column
Girder
C1
C2
C3
C4
d
h
b
f
t
f
t
w
1
Box
Box
Box
Box
30
26
18
2
0.8
35 × 35 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
2
Box
Box
Box
Box
30
26
18
2
0.8
35 × 35 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
3
Box
Box
Box
Box
30
26
18
2
0.8
35 × 35 × 1.5
30 × 30 × 1.5
35 × 35 × 1.5
35 × 35 × 1.5
4
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
5
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
6
Box
Box
Box
Box
28
24
15
2
0.8
30 × 30 × 1.5
25 × 25 × 1.5
30 × 30 × 1.5
30 × 30 × 1.5
7
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
30 × 30 × 1.5
8
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
25 × 25 × 1.5
9
Box
Box
Box
Box
26
22
15
2
0.6
25 × 25 × 1.5
20 × 20 × 1
25 × 25 × 1.5
25 × 25 × 1.5
10
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
25 × 25 × 1.5
11
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
12
Box
Box
Box
Box
25
22
15
1.5
0.6
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1
20 × 20 × 1The fragility curves for both steel and RC frames are plotted as shown in Figures3–8. The curves include four damage states based on HAZUS guidelines.Figure 3
Comparison of the fragility of the 12-story steel and RC frame buildings for the slight damage state.Figure 4
Comparison of the fragility of the 12-story steel and RC frame buildings for the moderate damage state.Figure 5
Comparison of the fragility of the 12-story steel and RC frame buildings for the extensive damage state.Figure 6
Comparison of the fragility of the 12-story steel and RC frame buildings for the complete damage state.Figure 7
Comparison of the fragility of the 8-story steel and RC frame buildings for the slight damage state.Figure 8
Comparison of the fragility of the 8-story steel and RC frame buildings for the moderate damage state.The curves shifting left mean that the probability of damage has increased. All curves show that, for slight and moderate states, the slope of curve is initially steep and then decreases. Usually, this slope change occurs at PGA > 0.8 g, because until 0.8 g the probability of exceeding the limit states varies between 0 and 100% while after 0.8 g this probability closely approaches 100%. For the extensive and complete damage states, the slope of the curve is nearly constant. The results indicate that in the steel frames the highest probability of extensive damage is 60%, while this value is 80% for RC frames. The probabilities of complete damage for steel and RC frames are 10–15% and 20–30%, respectively.The fragilities of the steel frames for slight, moderate, extensive, and complete states are compared in Figures9–12.Figure 9
Comparison of the fragility of the 8-story steel and RC frame buildings for the extensive damage state.Figure 10
Comparison of the fragility of the 8-story steel and RC frame buildings for the complete damage state.Figure 11
Comparison of the fragility of the 5-story steel and RC frame buildings for the slight damage state.Figure 12
Comparison of the fragility of the 5-story steel and RC frame buildings for the moderate damage state.Results reveal that, in general, the increase in the number of stories leads to increase in the fragility of both steel and RC structures. This difference is more significant in RC frames. For example, in the moderate damage state, the difference between 5 and 12 steel frames is 15%, while this value for 5 and 12 RC frames is approximately 30%. These comparisons can also be seen in Figures13–16.Figure 13
Comparison of the fragility of the 5-story steel and RC frame buildings for the extensive damage state.Figure 14
Comparison of the fragility of the 5-story steel and RC frame buildings for the complete damage state.Figure 15
The fragility curves for the 5-story steel frame buildings.
(a)
(b)
(c)
(d)Figure 16
The fragility curves for 8-story steel frame.
(a)
(b)
(c)
(d)The fragilities of the 12-story steel and RC frames are compared for slight, moderate, extensive, and complete damage states as shown in Figures3–6.Figures3–6 demonstrate that the fragility of a 12-story RC frame is higher compared to that of a 12-story steel frame. This difference is more significant in higher damage states, that is, at the slight damage state there is minimal difference, and at the complete state, there is maximal difference. These figures also show that the curves of the slight and moderate damage states begin to converge with increasing PGA, while the curves of the extensive and complete states begin to diverge with increasing the PGA.The fragilities of the 8-story steel and RC frames are compared for slight, moderate, extensive, and complete damage states, as shown in Figures7–10.Figures7–10 show that, similar to 12-story models, 8-story RC frames have higher fragility compared with steel frames. These figures also show that the curves of the slight and moderate damage states begin to converge with increasing PGA, while the curves of the extensive and complete states begin to diverge with increasing the PGA.The fragilities of the 5-story steel and RC frames are compared for the slight, moderate, extensive, and complete states, as shown in Figures11–14.Evaluation of Figures11–14 indicates similar results for 12- and 8-story frames. Furthermore, in the 5-story frames at the complete damage state, the fragility curves nearly overlap at low PGA and only diverge slightly at high PGA. The most significant difference is seen at the moderate damage state, and the least significant is at the complete damage state.


## 5.2. Statistical Methodology

As described previously, the fragility curve can be obtained using this equation:(5)1Pf=ϕln⁡Sd/ScβD∣IM2+βc2.In Section 5.1, the required parameters and coefficients have been defined using HAZUS. In this part, these parameters are obtained using nonlinear dynamic analyses and a statistical postprocessing. The ground motion records are scaled from 0.1 g to 1.5 g by a step of 0.1 g.Figure15 shows the fragility curves for a 5-story steel frame using both HAZUS and the statistical procedure. The figure shows that, generally, the statistical method yields higher fragility. The most significant difference is seen at the extensive damage state and the least significant at the complete damage state. The curves for the slight damage state nearly overlap, and at PGA > 0.7 g the probability approaches 100%. At the moderate damage state, the difference between the statistical and HAZUS methodologies is approximately 10%.The comparison between statistical and HAZUS methodologies for an 8-story steel frame is shown in Figure16.The comparison between statistical and HAZUS methodologies for a 12-story steel frame is shown in Figure17.Figure 17
The fragility curves for the 12-story steel frame buildings.
(a)
(b)
(c)
(d)As is evident in Figures15 to 18, the HAZUS method yields a reduced probability of damage. The difference between the curves is higher compared to the 5-story steel frame. Similar to the 5-story steel frame, in the extensive damage state, the most significant difference (30%) is observed.Figure 18
The fragility curves for the 5-story RC frame buildings.
(a)
(b)
(c)
(d)Figure17 shows the fragility curves for a 12-story steel frame. The most significant difference is between the moderate and extensive damage states (20%). This difference is 10% for the slight damage state, and in the complete state the curves are approximately the same.The fragility curves for RC frames are presented in Figure18.Figure18 shows that the HAZUS procedure yields a reduced probability of damage. At three damage states, slight, moderate, and extensive, the difference between the HAZUS and statistical procedures is 25%, but at the complete state, the curves nearly overlap at low PGA and diverge only slightly at large PGA. A comparison between the 5-story steel and RC frames shows that this difference percentage is higher in the RC case. Figure 19 shows that similar conclusions can be drawn for an 8-story RC frame.Figure 19
The fragility curves for the 8-story RC frame buildings.
(a)
(b)
(c)
(d)Seismic vulnerability of a 12-story RC frame has been assessed using two approaches (Figure20). The comparison shows that, similar to previous conclusions, the HAZUS procedure yields reduced damage probability except at PGA = 0.6 g for the extensive damage state. The most significant difference can be seen in the extensive damage state (statistical method yields 20% higher fragility). In other states, the curves are fairly close together.Figure 20
The fragility curves for 12-story RC frame buildings.
(a)
(b)
(c)
(d)Figure 21
Typical plan view of RC columns.Figure 22
Typical plan view of steel columns.Generally, in RC frames, the difference between the statistical and HAZUS procedures is 20%.


## 6. Conclusions

This paper investigated the seismic vulnerability of steel and RC frames of various heights by the use of the seismic fragility analysis. Two methodologies, HAZUS and a statistical method, have been employed, and the results were compared. The main conclusions are as follows:(i)
Generally, an increase in PGA leads to an increased fragility.(ii)
The slope of the fragility curve is larger at lower PGA and smaller at higher PGA.(iii)
For both steel and RC frames, the increase in number of stories leads to higher fragility (especially in RC frames).(iv)
For RC frames at slight and moderate damage states, the fragility curves of 5- and 8-story frames are very close together, while the 12-story curve shows a significant difference.(v)
In steel frames, the difference in fragility for 8- and 12-story frames is not significant (all 4 damage states are similar), while the 5-story frame shows difference from others.(vi)
Generally, the fragility of RC frames is higher compared to steel frames.(vii)
Comparison between the HAZUS and statistical methodologies shows that the fragility curves obtained from the statistical procedure yield higher fragility; the most difference is approximately 30%.



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*Source: 2698932-2017-09-28.xml*