J. Basic. Appl. Sci. Res., 2(7)6566-6573, 2012
© 2012, TextRoad Publication
ISSN 2090-4304
Journal of Basic and Applied
Scientific Research
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Assessment of Collapse Safety of Stiffness Irregular SMRF Structures
According to IDA Approach
Mohammad Hossein Cheraghi Afarani1, Ahmad Nicknam2
1, 2
Department of Civil Engineering, Iran University of Science and Technology, Tehran, Iran
ABSTRACT
Stiffness irregularity is one of vertical irregularity branches according to ASCE/SEI 7-05. Stiffness-Soft
irregular story have lateral stiffness less than 70% of above story or 80% of average stiffness of the three above
stories. Stiffness-Extreme irregular story have lateral stiffness less than 60% of above story or 70% of average
stiffness of the three above stories.
In this study incremental dynamic analysis is applied to estimate response of stiffness irregular structures to
seismic loads. IDA presents response of structure from elastic to dynamic instability. According to Performance
Based Earthquake Engineering, collapse capacity of structure is established. Collapse capacity is minimum
ground motion intensity which causes dynamic instability of structural system and inability of structural
components to carry gravity loads. By comparison of expected collapse capacity of maximum considered
earthquake and stiffness irregular structures, collapse safety is evaluated. Lower ratio between MCE and
considered irregular structures indicate to lower collapse safety of structure.
For this purpose by focusing on eight stories steel moment resisting frame structure with stiffness irregularity, it
is understood which location of stiffness irregular story is important. Stiffness irregularity in lower stories has
more effects on response of structure than intermediate stories and leads to lower collapse safety.
KEY WORDS: Collapse Safety, Stiffness Irregularity, SMRF Structure, Incremental Dynamic Analysis,
Performance Based Earthquake Engineering.
1. INTRODUCTION
Evaluation of seismic response of structures is one of the popular subjects in recent years. Stiffness
irregularity in structures is one of challenges in earthquake and structural engineering. Al-Ali and Krawinkler
studied ten stories building with separate effects of mass, stiffness and strength irregularities [1]. They pointed
out that strength irregularity has more effects in performance of structure than stiffness irregularity. Also
combination of stiffness and strength irregularities is more effective than the others. Kien Le-Trung studied
twenty stories MRF building and found out that irregularity in bottom level of building would be more effective
than higher level of building. They said that if stiffness irregularity is located in one story, 70% limitation of
seismic provision (IBC2000, [2]) would be too conservative [3]. Chitanapakdee & Chopra found out that stories
with stiffness and strength irregularities needed much more drift demands than other stories [4]. Valmudsson &
Nau researched about mass, stiffness and strength irregularities and recognized that equivalent lateral force
procedure wouldn’t estimate response of irregular structures correctly [5].
In accordance with ASCE\SEI 7-05 [6], if lateral stiffness of story be less than 70% of lateral stiffness
of above story or less than 80 percent of the average lateral stiffness of the three stories above, the structure is
stiffness-soft story irregular. For stiffness-extreme story irregularity above percents are equal 60% and 70%
respectively.
In this study by applying IDA approaches, response of stiffness irregular structures is established and
collapse safety coefficient is evaluated. Lower collapse safety coefficients of stiffness irregular structures than
regular structure indicate to more effects of irregularity to collapse capacity of structures.
2. Incremental Dynamic Analysis (IDA)
Incremental Dynamic Analysis (IDA) is a powerful method for description of response of structures to
earthquake loads that extended by Vamvatsikos and Cornell [7, 8, 9]. IDA consists of series of dynamic nonlinear
analysis of structure under scaled levels of ground motion records. Records are scaled by Intensity Measures (IMs)
of ground motions to cover all range of structural response from elastic to finally dynamic instability.
IDA results generally are gathered in IDA curves. IDA curves display Engineering Demand Parameters (EDPs)
versus IMs. In this study, first mode spectral acceleration (Sa(T1)) is assumed as IM and maximum interstory
drift ratio (θmax) as EDP.
3. Collapse Capacity
One of the Performance Based Earthquake Engineering (PBEE) framework goals is description of
performance limit states on structural response. Immediate Occupancy (IO), Life Safety (LS) and Collapse
Corresponding Author: Mohammad Hossein Cheraghi Afarani, M.Sc. Student of Earthquake Engineering, Department of Civil
Cheraghi Afarani and Nicknam 2012
1.
2.
3.
Prevention (CP) are three structural performance limit states according to FEMA-273 [10] and FEMA-356 [11].
Related damages to structural components of these limit states are as below:
IO: minor local yielding at a few places without permanent distortion of members.
LS: formation of hinges, local buckling of some beams, without failure of shear connections and a few members
may experience partial failure.
CP: extensive distortion of beams and column panels. Many fractures at moment connections without shear
connection failure.
This study focus on collapse prevention limits state. Collapse in structures has two modes; vertical collapse
and sidesway collapse [12]. Direct loss of gravity load carrying capacity in several structural components is
vertical collapse and sidesway collapse comes from successive reduction of load carrying capacity of structural
components to final effects of second orders to overcome load resisting. P-Delta effect is one of second orders
to overcome load resisting. For assessment of collapse, we concern to sidesway collapse which small increment
in intensity measure cause to increase interstory drifts without bounds [13]. For IDA method, collapse may
occur in large interstory drift ratios that lead to definition of limit for collapse interstory drift ratio. Limit state of
CP occur where slope of IDA curve decrease to 20% of elastic slope or θmax is equal to 10%, whichever occur
first in IM terms [8, 14].
4. Ground Motion Records
A suit of ground motion records are needed for sufficient accuracy of IDA. Ground motion records must
be suitable with site conditions. For this propose, we select eighteen ground motion records from Pacific
Earthquake Engineering Research Center (PEER) database [15]. Records are Far-Field with distance more than
10 km from site and have Richter magnitudes of 5 to 8 on firm soil type D of USGS with Vs< 180 m/s. Ground
motion records are listed in Table 1.
Table1: Selected ground motion records
No
Event
year
Station
Component
1
Kobe
1995
Shin-Osaka
0
2
Kobe
1995
Shin-Osaka
90
3
Kobe
1995
Kakogowa
0
4
Kobe
1995
Kakogowa
90
5
Kobe
1995
Nishi-Akashi
0
6
Kobe
1995
Nishi-Akashi
90
7
Loma Prieta
1989
Larkspur Ferry Terminal
270
8
Loma Prieta
1989
Larkspur Ferry Terminal
360
9
Loma Prieta
1989
Apeel2 Redwood City
43
10
Loma Prieta
1989
Apeel2 Redwood City
133
11
Loma Prieta
1989
Treasure Island
0
12
Loma Prieta
1989
Treasure Island
90
13
Northridge
1994
Montebello-Bluff
206
14
Northridge
1994
Montebello-Bluff
296
15
Superstition Hills
1987
SLT
225
16
Superstition Hills
1987
SLT
315
17
Westmorland
1981
Salton Sea
225
18
Westmorland
1981
Salton Sea
315
5. Structural Modeling
For study the effect of stiffness irregularity in structures, an eight stories regular structure is considered.
Regular structure has 2 bays with 4 m width in y direction and 4 bays with 3 m width in x direction and similar
stories with 3 m height. Dead load of 2 ton/m is distributed on beams. Structure is symmetric to avoid torsional
effects and has steel moment resisting frames which are designed according to IBC 2006 [16] and ANSI/AISC
360-05 [17]. P-Delta Effect is considered. Figure 1 shows three dimensional modeling of eight story structure.
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J. Basic. Appl. Sci. Res., 2(7)6566-6573, 2012
Figure 1: Three dimensional model
If lateral stiffness of stories be equal to k, stiffness irregularity is applied in three levels of 30%, 40%
and 60% changes of K. Stiffness changes are considered in two levels height of structure; once in first story and
another in two stories of 4 and 5. Table 2 display considered models. For example in this table, 8ST-0.4k-1
model’s tag lead to stiffness irregular structure with 60% changes to lateral stiffness (stiffness is equal to 0.4 of
k) in the first story.
Table 2: Regular and irregular models
Number
1
2
3
4
5
6
7
Model Tag
8ST
8ST-0.7k-1
8ST-0.7k-4,5
8ST-0.6k-1
8ST-0.6k-4,5
8ST-0.4k-1
8ST-0.4k-4,5
Original regular and irregular structures are modeled in SeismoStruct-V5 software that is able to
perform nonlinear dynamic analysis [18]. Steel behavior of Elastic Perfectly Plastic (EPP) hysteresis are used
for material modeling of steel sections without experience of local and lateral buckling and connections failure
according to FEMA 440 [19]. Figure 2 shows considered steel behavior of this study.
Cheraghi Afarani and Nicknam 2012
Figure 2: Elastic Perfectly Plastic (EPP) hysteresis of steel modeling
6. ANALYSIS AND RESULTS
Incremental dynamic analysis is applied to regular and irregular models until response of structures
change from elastic to collapse and dynamic instability. Figures 3 to 9 show IDA curves in form of maximum
interstory drift ratios versus first mode spectral acceleration. Collapse points are specified on IDA curves as dot.
Figure 3: IDA Curve and collapse points 8ST-0.4k-4,5
Figure 4: IDA Curve and collapse points 8ST-0.7k-4,5
Figure 5: IDA Curve and collapse points 8ST-0.6k-4,5
Figure 6: IDA Curve and collapse points 8ST-0.4k-1
J. Basic. Appl. Sci. Res., 2(7)6566-6573, 2012
Figure 7: IDA Curve and collapse points 8ST-0.6k-1
Figure 8: IDA Curve and collapse points 8ST-0.7k-1
Figure 9: IDA Curve and collapse points 8St
From IDA curves and collapse points, fragility curves can be employed. Fragility curves are evaluated
through Cumulative Distribution Function (CDF). Figure 10 displays fragility curves of regular and stiffness
irregular models. Fragility curve is achieved from fitting lognormal distribution through collapse points.
Figure 10: Fragility curve of regular and stiffness irregular models
Cheraghi Afarani and Nicknam 2012
Median collapse capacity of SCT (SC50%) is intensity measure which probability of exceedance is equal to 0.5
and in IDA curves means half of ground motions cause structure to collapse [20]. Also 16% and 84% collapse
probability of exceedance (SC16% and SC84%) can be evaluated from fragility curves. Table 3 summarizes median,
16% and 84% exceedance collapse probability of models.
Table 3: Collapse capacity, 84%, 50% and 16% probability of exceedance
Tag
SC84%
SCT= SC50%
SC16%
8ST-0.4k-1
0.275
0.421
0.645
8ST-0.4k-4, 5
0.285
0.442
0.680
8ST-0.6k-1
0.313
0.475
0.725
8ST-0.6k-4,5
0.315
0.478
0.720
8ST-0.7k-1
0.340
0.480
0.675
8ST-0.7k-4, 5
0.330
0.492
0.721
8ST
0.320
0.483
0.722
Also the collapse capacity of structure under Maximum Considered Earthquake (MCE) can be approximated
from ASCE/SEI 7-0. For this purpose we need parameters of table 4 for establishing MCE collapse capacity in
site class E of ASCE/SEI 7-05 [6].
Table 4: MCE spectral parameters
SC
SC ≤0.250
SC =0.5
SC =0.75
SC =1
1.250≤ SC
Fa
2.500
1.700
1.200
0.900
0.900
SMS
0.625
0.850
0.900
0.900
1.125
Fv
3.500
3.200
2.800
2.400
2.400
SM1
0.875
1.600
2.100
2.400
3.000
T0
0.280
0.376
0.467
0.533
0.533
Ts
1.400
1.882
2.333
2.667
2.667
According to parameters of table 4, MCE collapse capacity (SMS) can be calculated. Table 5 shows MCE
collapse capacity (SMS) for regular and irregular structures according to first mode structural period (T1).
Table 5: MCE collapse capacity (SMS) for regular and irregular structures
SC ≤0.250
Tag
T1
SC =0.5
SC =0.75
SC =1
1.250≤ SC
SMS
8ST-0.4k-1
0.900
0.625
0.850
0.900
0.900
1.125
8ST-0.4k-4, 5
0.960
0.625
0.850
0.900
0.900
1.125
8ST-0.6k-1
0.870
0.625
0.850
0.900
0.900
1.125
8ST-0.6k-4,5
0.960
0.625
0.850
0.900
0.900
1.125
8ST-0.7k-1
0.860
0.625
0.850
0.900
0.900
1.125
8ST-0.7k-4, 5
0.880
0.625
0.850
0.900
0.900
1.125
8ST
0.840
0.625
0.850
0.900
0.900
1.125
The ratio between collapse capacity margins (SCx%) and MCE collapse capacity is Collapse Margin Ratio, CMR.
CMR is the primary factor for characterization of collapse safety of structures, [20]. CMR can be evaluated
through equation 1 as below:
%
=
%
(1)
For considered collapse margins of 84%, 50% and 16%, CMRs are calculated and summarized in Table 6 to 8
as below:
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J. Basic. Appl. Sci. Res., 2(7)6566-6573, 2012
Table 6: CMR ratios of 84% collapse margin
SC ≤0.250
SC =0.5
Tag
SC =0.75
SC =1
1.250≤ SC
CMR84%
8ST-0.4k-1
0.440
0.324
0.306
0.306
0.244
8ST-0.4k-4, 5
0.456
0.335
0.317
0.317
0.253
8ST-0.6k-1
0.501
0.368
0.348
0.348
0.278
8ST-0.6k-4,5
0.504
0.371
0.350
0.350
0.280
8ST-0.7k-1
0.544
0.400
0.378
0.378
0.302
8ST-0.7k-4, 5
0.528
0.388
0.367
0.367
0.293
8ST
0.512
0.376
0.356
0.356
0.284
Table 7: CMR ratios of 50% collapse margin
SC ≤0.250
SC =0.5
Tag
SC =0.75
SC =1
1.250≤ SC
CMR50%
8ST-0.4k-1
0.674
0.495
0.468
0.468
0.374
8ST-0.4k-4, 5
0.707
0.520
0.491
0.491
0.393
8ST-0.6k-1
0.760
0.559
0.528
0.528
0.422
8ST-0.6k-4,5
0.765
0.562
0.531
0.531
0.425
8ST-0.7k-1
0.768
0.565
0.533
0.533
0.427
8ST-0.7k-4, 5
0.787
0.579
0.547
0.547
0.437
8ST
0.773
0.568
0.537
0.537
0.429
Table 8: CMR ratios of 16% collapse margin
SC ≤0.250
SC =0.5
Tag
SC =0.75
SC =1
1.250≤ SC
CMR16%
8ST-0.4k-1
1.032
0.759
0.717
0.717
0.573
8ST-0.4k-4, 5
1.088
0.800
0.756
0.756
0.604
8ST-0.6k-1
1.160
0.853
0.806
0.806
0.644
8ST-0.6k-4,5
1.152
0.847
0.800
0.800
0.640
8ST-0.7k-1
1.080
0.794
0.750
0.750
0.600
8ST-0.7k-4, 5
1.154
0.848
0.801
0.801
0.641
8ST
1.155
0.849
0.802
0.802
0.642
The collapse capacity of 84%, 50% and 16% probability of exceedance from table 3 show that structures
with 60% changes in stiffness (8ST-0.4k-1 and 8ST-0.4k-4,5) have maximum effects in collapse performance
level. In this case of irregularity, collapse occurs in lower spectral acceleration than regular structure. Also it is
obviously form lower CMRs according to table 6 to 8. Lower CMRs lead to lower collapse safety of structure.
For irregular structures with 40% changes in stiffness (8ST-0.6k-1 and 8ST-0.6k-4,5), CMRs are lower
than regular structure in 84% and 50% probability of exceedance according to tables 6 and 7. But for 16%
probability of exceedance, 40% change in stiffness has similar effects with regular structure.
As be seen in tables 6 to 8, CMRs of 50% and 16% collapse margins for 30% changes in stiffness at first
story (8ST-0.7k-1) are lower than regular structure and for 84% collapse margin are more than regular structure.
It lead to low collapse safety in 16% and 50% margins. Also for 8ST-0.7k-4,5 structure, almost the CMRs are
more than regular structure and cause to more safety collapse.
CMRs of structures which stiffness irregularity is applied in first story are approximately lower than
CMRs of structures which stiffness irregularity is applied in two intermediate story of structure.
7. Conclusions
Estimation of exceedance probability of collapse and collapse safety of structures are two important
parameters for assessment of performance of stiffness irregular structures. In this case, IDA approach is used as
a baseline procedure to evaluate response of structures under seismic loads which able to reflect the structural
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Cheraghi Afarani and Nicknam 2012
response with high accuracy. In this article, 40%, 60% and 70% of story stiffness are applied to irregular stories.
Finally, the results can be noted as below:
1- Approximately by increasing of Stiffness irregularity, collapse safety decrease which expects as ACSE 7-05
classification.
2- Stiffness irregularity in lower stories of structures has more effects on collapse safety than stiffness irregularity
applied in two intermediate stories of structure. It leads to importance of location of stiffness irregularity in
structural height and needs more studies about effects of location of stiffness irregularity in structural height.
8. Acknowledgment
The authors would like to thank Mohammad Ali Cheraghi Afarani and Ehsan Noroozinejad Farsangi for
best help in preparing this article.
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