Applied Mechanics and Materials Vols.

638-640 (2014) pp 1848-1853 Submitted: 2014-07-07

© (2014) Trans Tech Publications, Switzerland Accepted: 2014-07-11
doi:10.4028/www.scientific.net/AMM.638-640.1848 Online: 2014-09-19

SEISMIC FRAGILITY ANALYSIS OF HILL BUILDINGS WITH UNEVEN

GROUND COLUMN HEIGHTS

Linqing Huang 1,a ,Liping Wang 1, b and Chaolie Ning *2,c

1
Architectural and Civil Engineering Institute, Chongqing University of Science Technology,
Chongqing 401331, China
2
Collage of Civil Engineering, Tongji University, Shanghai 200092, China
a b c
hlinqing@126.com, wangliping98@163.com, ningchaolie@126.com

Keywords: Hill building with uneven ground column heights; Seismic fragility; Exceedance
probability; Peak ground acceleration

Abstract. The hill buildings sited on slopes have been widely constructed in mountainous regions. In
order to estimate the seismic vulnerability of the hill buildings with uneven ground column heights
under the effect of potential earthquakes, the exceedance probabilities of the hill buildings sited on
different angle slopes in peak ground acceleration (PGA) are calculated and compared by using the
incremental dynamic analysis method. The fragility curves show the slope angle has considerable
influence on the seismic performance. Specifically, the exceedance probability increases with the
increasing of the slope angle at the same performance level.

Introduction
Since the scarcity of flat ground in hilly areas, the construction of reinforced concrete structures sited
on slopes is very common. As shown in Fig.1, those buildings are designed to accommodate the hilly
terrain by adopting the uneven ground column heights. Therefore, the investigation of seismic
behavior for those hill buildings is of utmost importance [1-4]. Recently, the seismic damage of the
hill buildings has emphasized the need of risk assessment for the existing hill building stock to
estimate the potential damage from earthquakes in the future. Seismic risk analysis is a useful tool to
identify the seismic vulnerability. Because a seismic risk analysis for a given structure is to conclude
to an approximation, the probability of a given level of damage to be sustained by the building type
due to the occurrence of an earthquake, based on a given and probable scenario. Fragility curves
convey the graphical representation of the probability of exceeding a given damage state for a wide
range of motion intensities. Hence, fragility curves can be calculated to estimate the exceedance
probability as a function of ground motion indices. By comparing the fragility curves of the hill
buildings on the different slope angles, the aim of reminding the difference in the seismic damage for
the hill buildings with uneven ground column heights is achieved for structural engineers.

1 2 3
A B

Fig. 1. The schematic diagram of the hill buildings Fig. 2. Hill building with uneven ground column heights

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 Applied Mechanics and Materials Vols. 638-640 1849

Sample Buildings
Sample buildings, with respect to the slope angle of 5o, 10o, 15o, 20o, 25o, 30o, were designed
respectively according to the Chinese Seismic Design Code (CSDE) GB50011-2010 as shown in Fig.
2. The dimension of each building in the strong direction was set with 6 m width and 3 m story height.
The column height in the ground storey of those hill buildings are illustrated in Table 1. The cross
section of all beams is 250×500 mm. The concrete is assumed to have the compressive strength of 30
MPa. Two different types of reinforcement steel, Grade 300 and 400, was considered with the
characteristic yield strength of 300 MPa and 400 MPa respectively. In addition, the connection beam
AB as shown in the Fig.2 ensures the requirement of shear span ratio for exterior bottom columns
when the angle of the slope is 20o, 25o and 30o.
Table 1 Column heights in the ground storey for each sample building (m)

Slope Angle
Number
5o 10o 15o 20o 25o 30o
① 2.7 3.7 4.8 3.8+2.2 4.4+2.8 5.1+3.4
② 2.1 2.7 3.2 3.8 4.4 5.1
③ 1.6 1.6 1.6 1.6 1.6 1.6

Before performing the incremental dynamic analysis (IDA), the first step is to determine the
dynamic response characteristics of the buildings, i.e. natural periods and mode shapes, which
characterize the basic dynamic behavior and indicate the potential responses to the seismic loadings.
These necessary data was provided by Eigen-value analysis, which emphasizes the structures
collective behaviors through the shape of Eigen functions. The first three natural periods of vibration
can be shown in Table 2. It is observed that those hill buildings have little difference in the natural
periods.
Table 2 Natural period of buildings (s)

Slope Angle
Natural Period
5o 10o 15o 20o 25o 30o
1 0.0534 0.0552 0.0577 0.0558 0.0561 0.0564
2 0.0052 0.0054 0.0055 0.0056 0.0056 0.0056
3 0.0015 0.0010 0.0018 0.0019 0.0020 0.0021

Ground motions
Two main uncertainties exist in the fragility analysis. The first one goes to the uncertainty resulted
from the scatter of material properties and the second one is the randomness of earthquakes. In this
paper only the latter uncertainty was considered to make the estimation of fragility curves. The
selection of representative ground motions has been crucially important due to the sensitivity of
structural seismic performance to the characteristics of ground motions. Therefore, representative
ground motions are selected by using the double frequency method [5]. It is generally confirmed that
10 - 20 ground motions are sufficient enough in the investigation for mid-rise buildings to provide
adequate accuracy [6]. Thus 23 representative ground motions have been selected from NGA
database in accordance with the design response spectral of CSDE GB50011-2010 to take the random
nature of earthquakes into consideration as shown in Table 3.
1850 Progress in Industrial and Civil Engineering III

Table 3 Selection of ground motions

Distance PGA
Num. Time Station Magnitude
(km) (gal)
USA00005 1951.10.07 CITY HALL, FERNDALE, CA 6.0 53 110
USA00011 1952.07.21 TAFT LINCOLN SCHOOL, CA: TUNNEL 7.7 43 196
USA00067 1933.10.02 HOLLYWOOD STORAGE BLDG, LOS ANGELES 5.4 38 32
USA00068 1933.10.02 HOLLYWOOD STORAGE BLDG, LOS ANGELES 5.4 38 26
USA00095 1965.04.29 OLYMPIA, WASHINGTON HWY TEST LAB 6.5 61 -194
USA00113 1966.06.27 CITY RECREATION BLDG, SAN LUIS OBISPO 6.0 81 11
USA00118 1968.04.08 SOUTHERN CA EDISON, NUCLEAR POWER PLANT 6.7 134 40
USA00220 1972.02.09 4680 WILSHIRE BLVD, LOS ANGELES 6.6 39 -238
USA00398 1971.02.09 1800 CENTURY PARK EAST, LOS ANGELES 6.6 31 -82
USA00508 1971.02.09 6074 PARK DRIVE, WRIGHTWOOD 6.6 61 42
USA00523 1971.02.09 1880 CENTURY PARK EAST, LOS ANGELES 6.6 31 -114
USA00581 1971.02.09 HOSE STORAGE ROOM, HEMET FIRE STATION, HEMET, 6.6 151 38
USA00688 1971.02.09 6200 WILSHIRE BLVD, LOS ANGELES 6.6 29 124
USA00707 1971.02.09 1177 BEVERLY DRIVE, LOS ANGELES 6.6 39 -108
USA00832 1952.07.23 FIRE HOUSE, TEHACHAPI 5.0 30 -48
USA01605 1975.08.02 CDMG STATION 1, OROVILLE 5.2 11 -29
USA02101 1980.01.24 SAN RAMON FIRE STATION, SAN RAMON 5.9 17.2 45
USA02131 1980.01.27 SAN RAMON FIRE STATION, SAN RAMON 5.2 25 45
USA02347 1983.05.02 PARKFIELD, GOLD HILL 4W 6.5 54 93
USA02359 1983.05.02 PARKFIELD, STONE CORRAL 3E 6.5 46 100
USA02544 1978.08.13 SANTA BARBARA COURT HOUSE 5.5 7.9 -99
USA02545 1979.08.06 GILROY ARRAY 1, GAVILAN COL WATER TWR 5.9 15.7 -111
USA02581 1979.10.15 EL CENTRO ARRAY 8, CRUICKSHANK RD, EL CENTRO 6.9 27 258

Performance Levels
It is recognized that the selection of performance levels may vary in various studies and the predefined
damage states that affects the probability of structural performances significantly is very crucial. The
limit state, as the index of damage measure, is a scalar quantity used to classify the seismic response
of the building structures since the direct consequence of the defined damage states to the final
fragility curves place an utmost importance in defining these damage states accurately and
realistically.

Table 4 Performance levels

Damage state Performance Level Inter-storey Drift Limit
None Fully operational (FO) Drift<0.2%
Repairable Immediate operational (IO) Drift<0.5%
Irreparable Life safety (LS) Drift<1.5%
Severe Collapse prevention (CP) Drift<2.5%

Generally, the maximum inter-storey drift ratio was accepted as the damage measure and each
performance level has a prescribed limit value. These performance levels are directly related to the
extent damage sustained by the building during a damaging earthquake. Therefore, four different
performance levels: fully operational (FO), immediate operational (IO), life safety (LS), collapse
prevention (CP), corresponding to four damage states: none, repairable, irreparable and severe, are
specified in Table 4.

Incremental Dynamic Analysis

Incremental dynamic analysis (IDA) is a parametric analysis method and is very useful to estimate the
structural performance with the increasing of ground motion intensity. The method has been
discussed comprehensively by Vamnatsikos and Cornell [7], involving one or more points of damage
measure versus intensity measure under the effect of scaled ground motions as a result of several
 Applied Mechanics and Materials Vols. 638-640 1851

non-linear dynamic analyses. So the maximum interstorey drift ratio is assumed to be the damage
indicator and the peak ground acceleration is selected as the ground motion intensity measure. An
increment of 0.02g for the peak ground acceleration is selected to capture the four performance levels
with a reasonable sensitivity. The maximum interstorey drift ratio is recorded during the time history.
The relationship between the peak ground acceleration and the maximum interstorey drift is shown in
Fig. 3. It is observed that the distribution band of the maximum inter-storey drift ratio becomes
relatively wider as the increasing of peak ground acceleration.
0.014 0.014
Slope Angle=5 Slope Angle=10

0.012 0.012

Inter-storey Drift Ratio (%)

Inter-storey Drift Ratio (%)

0.01 0.01

0.008 0.008
IO=0.5% IO=0.5%
0.006 0.006

0.004 0.004
FO=0.2% FO=0.2%

0.002 0.002

0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
PGA (g) PGA (g)

(a) IDA curve of hill building on 5o slope (b) IDA curve of hill building on 10o slope
0.015 0.016
Slope Angle=15 Slope Angle=20
0.014
Inter-storey Drift Ratio (%)

Inter-storey Drift Ratio (%)

0.012
0.01
0.01

IO=0.5% 0.008
IO=0.5%
0.006
0.005
FO=0.2% 0.004
FO=0.2%
0.002

0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
PGA (g) PGA (g)

(c) IDA curve of hill building on 15o slope (d) IDA curve of hill building on 20o slope
0.018 0.02
Slope Angle=25 Slope Angle=30
0.016 0.018

0.014 0.016
Inter-storey Drift Ratio (%)

Inter-storey Drift Ratio (%)

0.014
0.012
0.012
0.01
0.01
0.008
IO=0.5% 0.008 IO=0.5%
0.006
0.006
0.004 0.004
FO=0.2% FO=0.2%
0.002 0.002

0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
PGA (g) PGA (g)

(e) IDA curve of hill building on 25o slope (f) IDA curve of hill building on 30o slope
Fig. 3. IDA curves of hill buildings with uneven ground column heights

Fragility Curves
Fragility curves represent the probability of structural damage with respect to the ground motion
indices. It is generally assumed that the fragility curves are expressed in the form of two parameter
lognormal distribution functions. The two parameters needed for the construction of lognormal
distribution are mean and standard deviation, values of which corresponding to typical peak ground
acceleration values. The fragility curves for those hill buildings are shown in Fig. 4 ~Fig. 7 in terms of
peak ground acceleration, where the exceedance probability increases with the increasing of peak
1852 Progress in Industrial and Civil Engineering III

ground acceleration. It is apparent that at four performance levels, the exceedance probability of the
hill building on 30o slope is much higher than that of building on other slopes. The exceedance
probability increases with the increasing of slope angles, especially at the life safety level and the
collapse prevention level.

1 1

0.9 0.9

0.8 0.8

Exceedance Probability
Exceedance Probability

0.7 0.7

0.6 0.6

0.5 0.5

0.4 0.4
5 5
0.3 10 0.3 10
15 15
0.2 0.2
20 20
0.1 25 0.1 25
30 30
0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
PGA (g) PGA (g)
Fig. 4. Fragility curves at fully operational level Fig.5. Fragility curves at immediate operational level
-3
x 10
0.25 7
5 5
10 10
15 6 15
0.2
20 20
Exceedance Probability
Exceedance Probability

25 5 25
30 30
0.15
4

3
0.1

2
0.05
1

0 0
0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3
PGA (g) PGA (g)

Fig.6. Fragility curves at life safety level Fig.7. Fragility curves at collapse prevention level

Conclusions
Based on the analytical results, the following conclusions can be drawn: The maximum interstorey
drift of the hill buildings with uneven ground column heights is prone at the ground storey due to
uneven column height. The exceedance probability at the same performance level increases with the
increasing of slope angle. Since the result of fragility analysis indicates the slope has considerable
influence on the seismic performance, therefore, a control method to determine the optimal layout
may need to be further developed.

Acknowledgements
This work was financially supported by the Chongqing Natural Science Foundation
(cstc2014jcyjA0682) and the Chongqing Natural Science Foundation (cstcjjA30006).

References
[1] Liping Wang and Chaolie Ning: Advanced Materials Research, (2014)
[2] Liping Wang and Yingmin LI: J. Xi'an Univ. of Arch. & Tech Vol.41 (2009), p. 822 (in Chinese)
[3] Liping Wang and Yao Zhao: Advanced Materials Research,(2012) ,p. 2279
[4] Liping Wang, Heping Zhong and Linqing Huang: Advanced Materials Research, (2013)
 Applied Mechanics and Materials Vols. 638-640 1853

[5] Pu Yang, Yingmin Li and Ming Lai: China civil engineering journal Vol. 33 (2000), p. 33 (in
Chinese)
[6] E.Borgonovo, I.Zentner, A.Pellegri, S.Tarantola and E. de Rocquigny: Reliability Engineering
and System Safety Vol. 109 (2013), p. 66
[7] Vamvatsikos D, Cornell AC. Earthq Eng Struct Dyn Vol. 31 (2002), p. 491
Progress in Industrial and Civil Engineering III
10.4028/www.scientific.net/AMM.638-640

Seismic Fragility Analysis of Hill Buildings with Uneven Ground Column Heights
10.4028/www.scientific.net/AMM.638-640.1848

DOI References
[6] E. Borgonovo, I. Zentner, A. Pellegri, S. Tarantola and E. de Rocquigny: Reliability Engineering and
System Safety Vol. 109 (2013), p.66.
http://dx.doi.org/10.1016/j.ress.2012.08.007