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Seismic Fragility Assessment of Reinforced Concrete High-Rise Buildings Using the Uncoupled Modal Response History Analysis (UMRHA)

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  • COMSATS University Islamabad, Wah Campus Wah Cantt, Pakistan

Abstract and Figures

In this study, a simplified approach for the analytical development of fragility curves of high-rise RC buildings is presented. It is based on an approximate modal decomposition procedure known as the Uncoupled Modal Response History Analysis (UMRHA). Using an example of a 55-story case study building, the fragility relationships are developed using the presented approach. Fifteen earthquake ground motions (categorized into 3 groups corresponding to combinations of small or large magnitude and source-to-site distances) are considered for this example. These ground motion histories are scaled for 3 intensity measures (peak ground acceleration, spectral acceleration at 0.2 s and spectral acceleration at 1 s) varying from 0.25 to 2 g. The presented approach resulted in a significant reduction of computational time compared to the detailed Nonlinear Response History Analysis (NLRHA) procedure, and can be applied to assess the seismic vulnerability of complex-natured, higher mode-dominating tall reinforced concrete buildings.
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Seismic Fragility Assessment of Reinforced Concrete High-rise Buildings
using the Uncoupled Modal Response History Analysis (UMRHA)
Muhammad Zain
National University of Sciences and Technology (NUST), Islamabad, Pakistan, zainaltaf@hotmail.com
Naveed Anwar
Asian Institute of Technology (AIT), Bangkok, Thailand, nanwar@ait.asia
Fawad A. Najam
Asian Institute of Technology (AIT), Bangkok, Thailand, fawad.ahmed.najam@ait.ac.th
Tahir Mehmood
COMSATS Institute of Information Technology (CIIT), Wah Cantt, Pakistan, tahirmeh@gmail.com
ABSTRACT:
In this study, a simplified approach for the analytical development of fragility curves of high-rise RC buildings is
presented. It is based on an approximate modal decomposition procedure known as the Uncoupled Modal
Response History Analysis (UMRHA). Using an example of a 55-story case study building, the fragility
relationships are developed using the presented approach. Fifteen earthquake ground motions (categorized into 3
groups corresponding to combinations of small or large magnitude and source-to-site distances) are considered for
this example. These ground motion histories are scaled for 3 intensity measures (peak ground acceleration, spectral
acceleration at 0.2 sec and spectral acceleration at 1 sec) varying from 0.25g to 2g. The presented approach resulted
in a significant reduction of computational time compared to the detailed Nonlinear Response History Analysis
(NLRHA) procedure, and can be applied to assess the seismic vulnerability of complex-natured, higher mode-
dominating tall reinforced concrete buildings.
Keywords: Seismic Risk Assessment, UMRHA, NLRHA, Fragility Relationships, High-rise RC Buildings
1. INTRODUCTION
Due to social and business needs, most of the population migrates towards the urban areas resulting in
an increased risk associated with structural collapse/failures. The need and complexities of high-rise
buildings are also rapidly increasing in densely populated urban areas. New approaches are emerging to
tackle the risks associated with design and construction of structures, such as Consequence-Based
Engineering (CBE). CBE is gaining popularity in structural engineering community which enables the
engineers to explicitly account for the uncertainty and risk aspects in their practice by employing the
probabilistic safety assessments. The ability of CBE to cover more than case-by-case scenarios to
mitigate future losses is one of the major attractions offered by this approach. Vulnerability (usually
described in terms of fragility relationships) demonstrates the probability that at certain level of intensity
of ground motion damage will occur. Therefore, fragility assessment is one of the integrated portions of
CBE and it represents the vulnerability information in the form conditional probability of exceedance
of particular damage states for particular given seismic intensity, as shown in equation 1. Fragility
relationships can be employed to perform both pre-earthquake planning, as well as for the post-
earthquake loss estimation.
    
(1)
The fragility relationships (also known as hazard-vulnerability relationships) also serve as one of the
best available means to select the most suitable and appropriate retrofitting strategies for structural
systems. Different approaches are employed by various researchers to derive these relationships which
can be classified into four major categories: (1) Empirical Fragility relationships: these are generated by
employing the statistical analysis of the data obtained from previous earthquakes. (2) Judgmental
Fragility Curves: such curves are primarily based upon the experts’ opinions. (3) Analytical Fragility
Curves: these curves can provide the most reliable results if sufficient data from past earthquakes is
available. However, there is still a margin of uncertainty involved due to limitations and uncertainties
in modelling the nonlinear response of RC structures. Since these curves are developed by simulating
the expected nonlinear behaviour of the structure itself, they demand a huge computational effort and
resources. (4) Hybrid Fragility Curves: these curves are made through the combination of the other two
or three types of fragility curves. The process of fragility derivation consists of thorough evaluation of
the uncertainties which are classified into two major categories, aleatory and epistemic. The former one
indicates the uncertainties associated intrinsically with the system i.e. the uncertainties involved in the
ground motions, while the later one characterizes the uncertainties that are mostly due to the deficiency
and lack of knowledge and data about the structural properties and behaviour (Gruenwald, 2008).
Various researches have proposed simplified methodologies for developing the analytical fragility
relationships, but most of the studies have focused on low-rise and mid-rise structures. Relatively less
work has been carried out to develop fragility curves for high-rise structures. Jun Ji et al. (2007)
presented a new methodology for developing the fragility curves for high-rise buildings by making a
calibrated and efficient 2D model of the original structure using genetic algorithms, considering the
PGA, Sa at 0.2 sec, and Sa at 1.0 sec as the seismic intensity indicators. High-rise buildings are very
complex structural systems, composed of many structural and non-structural components. The seismic
response of high-rise structures remains convoluted as the many vibration modes other than the
fundamental mode participate significantly in their seismic response. Among various numerical analysis
procedures for evaluating seismic performance of high-rise buildings, the Nonlinear Response History
Analysis (NLRHA) procedure has been widely considered and accepted as the most reliable and accurate
one. However, the procedure is computationally very expensive, and it does not provide much physical
insight into the complex inelastic responses of the structure. Nonlinear Static procedures (NSP) on the
other hand, accounts only for the response contribution of fundamental vibration mode and are not
considered suitable for evaluating higher-mode dominating structures. In this study, a simplified
procedure called the “Uncoupled Modal Response History Analysis (UMRHA)” (Chopra, 2007)
procedure is used as tool to develop the fragility curves for high-rise buildings. In UMRHA, the
nonlinear response contribution of individual vibration modes are computed and combined into the total
response as explained in next section.
2. BASIC CONCEPT OF THE UNCOUPLED MODAL RESPONSE HISTORY ANALYSIS
(UMRHA) PROCEDURE
The UMRHA procedure (Chopra, 2007) can be viewed as an extended version of the classical modal
analysis procedure in which the overall complex dynamic response of a linear Multi-Degree-of-Freedom
(MDOF) structure is considered as a sum of contributions from only few independent vibration modes.
The response behavior of each mode is essentially similar to that of a Single-Degree-of-Freedom
(SDOF) system governed by a few modal properties. Strictly speaking, this classical modal analysis
procedure is applicable to only linear elastic structures. When the responses exceed the elastic limits,
the governing equations of motion become nonlinear and consequently, the theoretical basis for modal
analysis becomes invalid. Despite this, the UMRHA procedure assumes that even for inelastic
responses, the complex dynamic response can be approximately expressed as a sum of individual modal
contributions (assuming them uncoupled). The number of modes included in the analysis are selected
based on cumulative modal mass participation ratio of more than 90%. The governing equation of
motion of SDOF subjected to a horizontal ground motion  can be written as:
 
(2)
Where   and
; and are the natural vibration frequency and the damping
ratio of the ith mode, respectively. Equation (2) is a standard governing equation of motion for inelastic
SDOF systems. To compute the response time history of  from this equation, one needs to know
the nonlinear function . A reversed cyclic pushover analysis is performed for each important
mode to identify this nonlinear function . The cyclic pushover analysis for the ith mode can
be carried out by applying a force vector with the ith modal inertia force pattern
  (where
is the mass matrix of the building and is the ith natural vibration mode of the building in its linear
range. The relationship between roof displacement, obtained from the cyclic pushover (denoted by
)
and is approximately given by
 

(3)
where
is the value of at the roof level. The relationship between the base shear  and 
under this modal inertia force distribution pattern is given by
 
(4)
By this way, the results from the cyclic pushover analysis are first presented in the form of cyclic base
shear ()roof displacement (
) relationship, and then transformed into the required 
relationship. At this stage, a suitable nonlinear hysteretic model can be selected, and its parameters can
be tuned to match with this  relationship. The response time history of as well as 
can then be calculated from the nonlinear governing equation (2). The response of each mode belongs
to the ith vibration mode and can be generally represented by . By summing the contributions from
all significant modes, the total response history is obtained as follows.
 

(5)
where is the number of significant vibration modes.
3. METHODOLOGY AND STRUCTURAL MODELING
A UMRHA-based methodology is intended to include higher mode effects in the process of developing
the fragility relationships. Computational effort is always considered as one of the major concerns in
seismic fragility analysis. The current methodology focuses towards the reduction of the computational
effort by proposing a theoretically close, yet simpler method (UMRHA) to replace full nonlinear
response history analysis (NLRHA). A 55 story core-wall high-rise building, located in a seismically
active area (Manila, Philippines) is selected for application of presented methodology. Full 3D nonlinear
finite element model (shown in figure 1) was created in PERFORM 3D (CSI, 2000) employing the
concepts of capacity-based design (all the primary structural members are not allowed to undergo shear
failure). The whole core wall and link beams are modeled as nonlinear, while the other components are
kept linear. Complete hysteretic behaviors (F-D Relationships) were assigned to all nonlinear structural
components as well as materials to explicitly account for stiffness degradation and hysteretic damping.
Table 1 and Figure 1 show the major characteristics of the selected building in terms of geometry and
material properties of key structural elements, respectively.
Table 1: Material properties of key structural elements
Member
Nominal Concrete
Strength, psi (MPa)
Lower Basement to 11th Level
10000 (69)
12th Level to 21st Level
8500 (59)
21st Level to Roof Deck Level
7000 (48)
Foundation to 40th Level
6000 (41)
40th Level to Roof Deck Level
5000 (34)
Figure 1. The 3D analytical model of a 55-
story case study building (CSI, 2000)
Figure 3 shows the proposed methodology in the form of a flow chart. The procedure requires to perform
both monotonic and reversed cyclic pushover analyses to identify strong/weak direction and damage
states of building, and to develop complete hysteretic behaviors for equivalent single-degree-of-freedom
(SDOF) systems, respectively. For case study building, the procedure is validated by comparing
Podium Plan Area = 92m x 54m
Tower Plan Area = 40m x 39m
Total Height = 163m
Number of Stories = 55
Typical Story Height = 2.9m
First Story Height = 4.7m
displacement histories obtained from UMRHA and NLRHA (presented in section 6). Selected ground
motions are applied to full analytical model as well as to all equivalent SDOF systems. The SDOF
displacement histories (obtained in terms of spectral displacement) are converted back to actual
displacements and are added linearly to obtain a complete displacement history.
For the case study structure, equivalent SDOF systems are created for the first four modes (shown in
figure 3) providing the modal mass participation ratio of more than 90%. A computer program
RUAUMOKO 2D (Carr, 2004), developed at University of Canterbury, is used for the solving SDOF
systems. It provides a convenient interface and allows user to assign and control a reasonably large
number of hysteretic behaviors to various nonlinear components.
Figure 2. First 4 mode shapes of the considered building in weaker direction
Figure 3. Proposed methodology for analytical fragility assessment of high-rise buildings.
4. UNCERTAINTY TREATMENT
Probabilistic nature of seismic fragilities is greatly influenced by the uncertainties (aleatory or epistemic)
and assumptions involved in the process. Wen et al. (2003) describes the aleatory uncertainty as the one
Mode 1, T = 4.67 sec Mode 2, T = 1.12 sec Mode 3, T = 0.51 sec Mode 4, T = 0.30 sec
Reference structure selection
Static pushover analysis for 1st
modes in both directions
Static pushover analysis for higher
modes in selected direction
Reverse cyclic pushover analysis
for determining base shear vs. roof
drift relationship
Definition of limit states Selection of damage measure
Uncertainty modeling
Construct equivalent SDOF
systems for all considered modes
FD relationship for equivalent SDOF system
Perform NLRHA for all
combinations of SDOF systems
Perform nonlinear response history analysis
(NLRHA) for structure using a suit of ground
motions and compare results with combined
response obtained from the UMRHA procedure
Validation of procedure, if desired
Intensity measure definition and
scaling
Post-processing of response data
Development of fragility curves
Calculation of direct sampling probabilities
Check for material uncertainty
Ground motion selection and scaling
Check weaker direction of structure
Combine modal results to
determine overall response
Uncoupled modal response history
analysis (UMRHA) framework
which can be explicitly recognized by a stochastic model, whereas those which exist in the model itself
and its parameters, are epistemic. In this case, aleatory uncertainties represent the uncertainty associated
with the ground motions (existing intrinsically in the earthquakes), while epistemic uncertainties
represent the existence of ambiguity in the structural capacity itself (mainly due to the lack of knowledge
and possible variation in construction process). The current study focuses towards the consideration of
both types of uncertainties involved in the seismic fragility assessment by considering 15 ground
motions and varying the materials’ strengths.
4.1 Material Uncertainty
The intrinsic variability of material strengths is also one of the major sources of uncertainty involved in
the seismic fragility assessment. In this study, the compressive and tensile strengths of concrete, as well
as, the yield strength of steel are considered as random variables. The studies by Ghobarah et al. (1998)
and Elnashai et al. (2004) employed statistical distributions to define the uncertainty involved in the
yield strength of steel. There seem to be a consent in terms of employing normal and log-normal
distributions to elaborate the variability in the yield strength of steel with coefficient of variation (COV)
ranging from 4% to 12%. Bournonville et al. (2004) evaluated the variability of properties of reinforcing
bars, produced by more than 34 mills in U.S. and Canada. The current study employs the research
outcome from Bournonville et al. (2004). The mean and COV for the yield point are 480 MPa and 7%
respectively, while the mean value and COV for the ultimate strengths are 728 MPa and 6% respectively.
The intrinsic randomness concrete strength can be captured using experimental data. Hueste et al. (2004)
studied the variable nature of concrete strength, including the experimental results from the testing of
higher strength concretes. The current study utilizes the results reported by Hueste et al. (2004).
4.2 Selection of ground motions accelerograms
Bazzuro and Cornell (1994) suggested that seven ground motions are sufficient for covering the aspect
of uncertainty from the earthquakes while the recent Tall Building Initiatives (TBI) Guidelines (2010)
also recommends the same number of ground motions. Some researchers also have demonstrated the
use of simulated ground motion histories e.g. Andrew et al. (2004) used simulated histories for fragility
derivation using single specific criteria (compatibility of ground motions with the site-specific response
spectrum). Shinozuka (2000b) also used the simulated ground motions by Hwang and Huo (1996) for
developing the analytical fragility curves for bridges. Jun Ji et al. (2009) presented new criteria for
ground motion selection based on earthquake magnitude, soil conditions and source-to-site distance.
Table 2. Selected ground motion records
Category
Earthquake
Magnitude
Distance to Rupture
Soil at Site
1
Chi-Chi, Taiwan
7.6
7.30
Stiff
Imperial Valley
6.5
2.50
Soft
Kobe, Japan
6.9
1.20
Soft
Loma Prieta, USA
6.9
5.10
Stiff
Northridge
6.7
17.5
Stiff
2
Aftershock of Friuli EQ, Italy
5.7
10.0
Soft
Alkion, Greece
6.1
25.0
Soft
Anza (Horse Cany)
4.9
20.0
Soft
Caolinga
5.0
12.6
Stiff
Dinar, Turkey
6.0
1.02
Soft
3
Chi-Chi, Taiwan
7.6
39.3
Soft
Kobe, Japan
6.9
89.3
Stiff
Kocaeli, Turkey
7.4
76.1
Stiff
Kocaeli, Turkey
7.4
78.9
Soft
Northridge
6.7
64.6
Soft
In this study, 15 ground motion records (organized in to 3 categories) are selected following the same
criteria as proposed by Jun Ji et al. (2009) i.e earthquake magnitude, site soil conditions and source-to-
site distance. The larger magnitude earthquake histories usually contain several peaks compared to
moderate and small earthquakes mostly causing the structure to undergo a significant extent of
nonlinearity. The source-to-site distance influences the filtration of frequency fractions during the
process of wave propagation, while soil conditions are mainly held responsible for the amplification or
dissipation of seismic waves. Based on these considerations, the selected ground motions are divided
among 3 categories each corresponding to an adequate variation in the magnitude, distance to source,
and soil conditions i.e Near-source and Large-magnitude, Near-source and Moderate-magnitude, and
Distant-source and Large-magnitude. The selected ground motions and their properties, with reference
to their categories, are enlisted in the table 2.
5. DEFINITION OF LIMIT STATES AND SEISMIC INTENSITY INDICATORS
The definition of limit states of the structure is a fundamental component in seismic fragility assessment.
For high-rise buildings, there is no universally acceptable and consistently applicable criterion to
develop a relationship between damage and various demand quantities. Several researchers have
proposed different performance limit states of buildings, usually classified in two major categories
(qualitative and quantitative). In qualitative terms, HAZUS (1999) provides four limit states of building
structures (Slight, Moderate, Major, and Collapse). Smyth et al. (2004) and Kircher et al. (1997) also
used four damage states; slight, moderate, extensive, and collapse. Whereas, the quantitative approach
describes the damage states in terms of mathematical representations of damage, depending upon some
designated and specific structural responses. Different researchers have employed different damage
indicators for representing damage at local and global levels. Shinozuka et al. (2000a) used ductility
demands as damage indicators for prescribed damage states. Guneyisi and Gulay (2008) used inter-story
drift (ISD) ratio to develop the fragility curves. Although many others have used damage indicators
related to energy and forces, but ISD is the most frequently used parameter and can correlate adequately
with both the non-structural and structural damage. This study also employs ISD ratios as the seismic
response (damage) indicator and defines the two limit states of case study building based on study
conducted by Jun Ji et al. (2009). The first one is “Damage Control”, and the second is “Collapse
Prevention”. Qualitative definitions of the considered limit states are provided in table 3.
Table 3. Definitions of considered limit states
Level
Limit State
Definition
Limit State 1
(LS 1)
Damage Control
The very first yield of longitudinal steel reinforcement, or the
formation of first plastic hinge.
Limit State 2
(LS 2)
Collapse
Prevention
Ultimate strength/capacity of main load resisting system.
A nonlinear static pushover analysis for first 4 modes in weaker direction was conducted for full 3D
nonlinear model. The definitions of the prescribed limit states are then applied to the results of the
pushover analysis to obtain the quantitative definitions of limit states in terms of ISD ratios. It should
be noted that limit states of case study building are defined for each mode separately to include the
higher modes effects on the selected damage criteria. Another essential step in the fragility analysis is a
proper selection of an intensity measure to relate structural performance. An adequate intensity measure
would correlate well between the structural response and the associated vulnerability (Wen et al., 2004).
In earlier studies, peak ground acceleration (PGA) was one of the frequently used seismic intensity
indicator. Other widely used measures involve Modified Mercalli Intensity (MMI), Arias Intensity (AI),
and Root Mean Square (RMS) Acceleration (Singhal and Kiremidjian, 1997). Some studies include
spectral acceleration (Sa) as intensity measure at fundamental period of structure (Kinali and
Ellingwood, 2007), Sa at 0.2 sec and Sa at 1.0 sec. In this study, PGA, Sa at 0.2 sec and Sa at 1.0 sec
are considered as seismic intensity measures considering the idea that PGA alone may not serve as an
accurate intensity indicator to correlate theoretically computed structural damage with observed
performance (Sewell, 1989).
6. RESULTS AND DISCUSSION
This section presents the results obtained from UMRHA and NLRHA with the view to develop fragility
curves for the case study tall building. However, first the effect of uncertainties in material strengths
will be presented to check the sensitivity of results.
6.1 Effect of Material Strength Uncertainties
It is preferable to concentrate on dominant factors that can play an influential role in the probabilistic
variation of the response; therefore, the results’ sensitivity prior to performing the complete UMRHA is
checked, and only that type of uncertainty is considered that can cause a significant variation in the
building response. The sensitivity of the results in response of the variation in material strengths requires
complete analytical simulations. Ibarra and Krawinkler (2005) described that the uncertainty in ductility
capacity and post-capping stiffness generate the principal additional contributions to the dispersion of
capacity, especially near the collapse. The former one is more important when P- effects are large. The
study further suggested that the record-to-record variability is also the major contributor to total
uncertainty.
Figure 4. Sensitivity of roof drift (%) to variation in material properties when the model was subjected to a
ground motion history
In the case of high-rise structures, it may not be suitable to conduct Monte Carlo simulation as it requires
a large number of analyses for reasonably accurate results, and a run time of each analysis is around 40
hours for 3D structural model. In this study, 5 pairs of material strengths (concrete and steel) are selected
and the results are presented here based on “worst-case-scenario”. Figures 4 and 5 show the sensitivity
of the results in response to the variation of material properties, when 3D analytical model was subjected
to the application of a ground motion history and the first modal pushover analysis respectively. The
steel strength is mean (µ) + 1 standard deviation (), while the concrete strength is mean (µ) - 1 standard
deviation (). This pair of strength is selected considering that high steel strength attracts more
earthquake forces, and the reduction in concrete quality decreases the shear capacity. The results show
that even with this much variation in material strengths, the response of building does not vary
significantly. Thus, the material uncertainty can be treated as an epistemic uncertainty.
Figure 5. Sensitivity of first modal monotonic
pushover curve from variation in material properties
Figure 6. Envelope of reversed cyclic pushover curve
for the first mode of the case study building
6.2 Fragility Derivation
For case study tall building, dynamic response histories were determined using both NLRHA and
UMRHA procedures to validate the methodology. Before UMRHA, a reversed cyclic pushover analysis
for all 4 modes was conducted to obtain global hysteretic behavior which was converted later to an
idealized force-deformation model to construct equivalent SDOF systems. As an example, figure 6
shows the envelope of cyclic pushover curve for the first mode in weaker direction. Table 4 shows some
-2
-1
0
1
2
0 5 10 15 20 25
Roof Drift (%)
History of Actual Structure - "μ" Values
History of Actual Structure, Steel Strength = μ + 1ϭ, Conc. Strength = μ -
Time (sec)
0
10
20
30
40
50
60
70
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Mode 1, Steel Strength = μ + 1ϭ, Conc. Strength = μ - 1ϭ
Mode 1 - "μ" Material Values
Roof Drift (%)
Base Shear (x103KN)
-80,000
-60,000
-40,000
-20,000
0
20,000
40,000
60,000
80,000
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
Base Shear (KN)
Roof Drift (%)Roof Drift (%)
of the important properties of equivalent SDOF systems determined from idealized force-deformation
model. Selected ground motions were then applied to each of the SDOF system (representing each
mode) and the individual response was linearly added to obtain overall displacement histories. Figure 7
shows the comparison of roof drift (%) history obtained from NLRHA of 3D analytical model and from
UMRHA. It can be seen that UMRHA response is reasonably matching with NLRHA. After getting a
reasonable degree of confidence on UMRHA validation, each ground motion history is scaled to eight
intensity levels (PGA, Sa at 0.2 sec and Sa at 1.0 sec) ranging 0.25g to 2.0g, with the increment of 0.25g.
In UMRHA, for each of equivalent SDOF system, 120 dynamic analyses are conducted making total
number of analyses as 480 for each seismic intensity measure. A total of 1440 analyses are conducted
to consider first 4 modes.
Table 4. Important characteristics of SDOF systems
Properties
Mode 1
Mode 2
Mode 3
Mode 4
1.50286
0.811329
0.604722
0.731589
 (mm)
490
395
310
200

(mm/sec2)
722.495
9073.197
49174.89
72711.33
Figure 7. Comparison between roof drift history of 3D analytical model from NLRHA and UMRHA
Once the results are obtained, the quantitative definitions of limit states are applied to assess the levels
of structure's performance subjected to ground motions with a particular intensity. When a computed
value approaches or surpasses the distinctly defined limit states (in any of the considered vibration
mode), that particular event is counted in the sample to compute the probabilities. This process of
probability calculation is performed for each limit state at each specific level of intensity (determined
by dividing the number of ground motions causing that specific limit state by the total number of
considered ground motions). Figure 8 shows the peak roof drift values for all three ground motion
categories against all considered levels of PGA. The random nature of peak dynamic response can be
seen for all 15 ground motions highlighting the importance of considering aleatory uncertainties in
seismic fragility analysis. Similar relationships were developed for other two intensity measures (Sa at
0.2 sec and Sa at 1.0 sec) and converted later to generalized fragility curves (as shown in figure 9). A
lognormal distribution is assumed to develop generalized fragility curves governed by equation 6 below.
 
(6)
Where  describes the probability of exceedance of a specific limit state of the building at a
particular intensity measure with an explicit intensity, whereas represents the standard normal
cumulative distribution function, and and are the controlling parameters, which represent the
slope of the curve and its median, respectively. Nonlinear curve-fitting techniques are utilized to make
an optimized estimation of the controlling parameters for each of the fragility curves. Figure 9 shows
the developed lognormal fragility relationships of the considered building at the intensity measures of
PGA, Sa at 0.2 sec, and Sa at 1.0 sec along with the direct sampling probabilities. The values of
controlling parameters are enlisted in table 5.
Table 5. Lognormal distribution parameters for fragility relationships
Limit State
PGA
Sa at 0.2 sec.
Sa at 1.0 sec.
LS 1
-0.3991
0.6390
0.3490
0.7990
-0.1624
0.6310
LS 2
0.6420
0.5799
1.3754
0.8015
0.9385
0.8450
-2
-1
0
1
2
0 5 10 15 20 25
Roof Drift (%)
History of Actual Structure History Obained from UMRHA
Time (sec)
Figure 8. Peak (and mean) roof drift values for all 3
ground motion categories
Figure 9. Developed fragility curves for the case
study building for defined intensity measures*.
*Solid lines represent the developed lognormal functions, and the dots show the directly calculated sampling probabilities.
7. CONCLUSIONS
In this study, a simplified approach based on the UMRHA procedure is proposed for the development
of analytical fragility curves of high-rise RC buildings. The methodology is executed on a 55 story case
study building to demonstrate the process. Following conclusions can be drawn from this study:
a) The damage limit states are generally defined only on the basis of first-mode pushover analysis
which are not able to account for the chances of secondary nonlinearity (e.g. secondary plastic hinge
developments) for the high-rise structures having significant response contribution from higher
vibration modes. The presented approach offers an advantage of defining the limit states, including
high-modes of vibrations and is expected to be more reliable compared to simplified methodologies
based on the damage limit states definitions for only first vibration mode.
b) Uncertainties arising from both seismic demand and structural capacity are evaluated and it is
concluded that the random nature of ground motion should be given due consideration while
developing generalized fragility functions.
c) Computational effort and cost is always an ever-growing challenge for analytical assessment of
seismic risk to existing and new buildings. The NLRHA procedure for one high-rise building
subjected to one ground motion record takes around 30 hours of computation time of a 3.4 GHz
processor and 4.0 GB RAM desktop computer. The processing of computed dynamic responses into
the required format takes another 4 to 5 hours. For the UMRHA procedure, it takes around one hour
0
0.25
0.5
0.75
1
1.25
1.5
00.25 0.5 0.75 11.25 1.5 1.75 2
Series1
Series2
Series3
Series4
Series5
mean
0
0.25
0.5
0.75
1
1.25
00.25 0.5 0.75 11.25 1.5 1.75 2
Series1
Series2
Series3
Series4
Series5
Series6
0
1.5
3
4.5
6
7.5
9
00.25 0.5 0.75 11.25 1.5 1.75 2
Series1
Series2
Series3
Series4
Series5
Series6
Chi
-Chi, Taiwan
Imperial
Valley, USA
Kobe,
Japan
Loma
Prieta, USA
Northridge,
USA
Mean
PGA (g)
Peak Roof Drift (%)
Aftershock
of Friuli EQ, Italy
Alkion,
Greece
Anza
(Horse Cany), USA
Caolinga
, USA
Dinar,
Turkey
Mean
Chi
-Chi, Taiwan
Kobe,
Japan
Kocaeli,
Turkey
Kocaeli,
Turkey
Northridge,
USA
Mean
PGA (g)
PGA (g)
Peak Roof Drift (%)
Peak Roof Drift (%)
Ground Motions Category 1
Ground Motions Category 2
Ground Motions Category 3
0
0.25
0.5
0.75
1
00.25 0.5 0.75 11.25 1.5 1.75 2
LS 1 - LogNorm
LS 2 - LogNorm
LS1 1 - Direct
LS 2 - Direct
0
0.25
0.5
0.75
1
00.25 0.5 0.75 11.25 1.5 1.75 2
LS 1 - LogNorm
LS 2 - LogNorm
LS 1 - Direct
LS 2 - Direct
0
0.25
0.5
0.75
1
00.25 0.5 0.75 11.25 1.5 1.75 2
LS 1 - LogNorm
LS 2 - LogNorm
LS 1 - Direct
LS 2 - Direct
PGA (g)
Sa at 0.2 sec (g)
Fragility
Fragility
Fragility
LS 1 - Direct
Sa at 1.0 sec (g)
to complete cyclic pushover analyses for three vibration modes, 10 minutes for the analysis of three
equivalent nonlinear SDOF systems, and 20 minutes for transforming and processing computed
responses into the final format. This low computational effort for the UMRHA (compared to the
NLRHA) may allow us to explore the nonlinear responses of tall buildings to a reasonably high
number of ground motions in a convenient and practical manner.
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... The dynamic response of these structures becomes highly intricate because of the significant contribution of higher modes (Rajbhandari, Anwar, & Najam, 2017). These complexities add to the challenges of accurately predicting the behaviour of these structures and the associated seismic demands (Zain, Anwar, Najam, & Mehmood, 2017). In such situations, the most desired analysis procedure is the nonlinear dynamic analysis. ...
... However, detailed nonlinear time history analysis (NLTHA) can be computationally very demanding and the computational cost increases significantly for complex structural systems (Lavaei & Lohrasbi, 2012). For instance, Zain et al. (2017) analysed a 55-story building using PERFORM 3 D (CSI, 2020) and reported a computational time of 30 hours to conduct the analysis using a desktop computer with a 3.4 GHz processor and 4.0 GB RAM. Accordingly, risk assessment of complex structures that integrates traditional NLTHA may be deemed unfeasible due to the associated computational cost. ...
... Soft computing methods, based on heuristic approaches that exploit the tolerance for imprecision and uncertainty, have been proposed to develop metamodels for various engineering problems. Artificial neural networks (ANNs), genetic algorithms, fuzzy logic, and decision tree analysis are among the popular methods in soft computing (Zain et al., 2017). Owing to their computational efficiency and ability to predict accurate relationship between data points (Alexandridis, 2013), ANNs have been widely used in solving structural engineering problems such as design optimization (e.g. ...
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