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

  • COMSATS University Islamabad, Wah Campus Wah Cantt, Pakistan

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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,
Naveed Anwar
Asian Institute of Technology (AIT), Bangkok, Thailand,
Fawad A. Najam
Asian Institute of Technology (AIT), Bangkok, Thailand,
Tahir Mehmood
COMSATS Institute of Information Technology (CIIT), Wah Cantt, Pakistan,
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
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.
    
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.
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:
 
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
 
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
 
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.
 
where is the number of significant vibration modes.
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
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.
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
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
Distance to Rupture
Soil at Site
Chi-Chi, Taiwan
Imperial Valley
Kobe, Japan
Loma Prieta, USA
Aftershock of Friuli EQ, Italy
Alkion, Greece
Anza (Horse Cany)
Dinar, Turkey
Chi-Chi, Taiwan
Kobe, Japan
Kocaeli, Turkey
Kocaeli, Turkey
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.
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
Limit State
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)
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).
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
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
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 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)
-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
Mode 1
Mode 2
Mode 3
Mode 4
 (mm)
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.
 
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
Sa at 0.2 sec.
Sa at 1.0 sec.
LS 1
LS 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.
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
00.25 0.5 0.75 11.25 1.5 1.75 2
00.25 0.5 0.75 11.25 1.5 1.75 2
00.25 0.5 0.75 11.25 1.5 1.75 2
-Chi, Taiwan
Valley, USA
Prieta, USA
PGA (g)
Peak Roof Drift (%)
of Friuli EQ, Italy
(Horse Cany), USA
-Chi, Taiwan
PGA (g)
PGA (g)
Peak Roof Drift (%)
Peak Roof Drift (%)
Ground Motions Category 1
Ground Motions Category 2
Ground Motions Category 3
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
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
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)
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. ...
Rapidly growing societal needs in urban areas are increasing the demand for tall buildings with complex structural systems. Many of these buildings are located in areas characterized by high seismicity. Quantifying the seismic resilience of these buildings requires comprehensive fragility assessment that integrates iterative nonlinear dynamic analysis (NDA). Under these circumstances, traditional finite element (FE) analysis may become impractical due to its high computational cost. Soft-computing methods can be applied in the domain of NDA to reduce the computational cost of seismic fragility analysis. This study presents a framework that employs nonlinear autoregressive neural networks with exogenous input (NARX) in fragility analysis of multi-story buildings. The framework uses structural health monitoring data to calibrate a nonlinear FE model. The model is employed to generate the training dataset for NARX neural networks with ground acceleration and displacement time histories as the input and output of the network, respectively. The trained NARX networks are then used to perform incremental dynamic analysis (IDA) for a suite of ground motions. Fragility analysis is next conducted based on the results of the IDA obtained from the trained NARX network. The framework is illustrated on a twelve-story reinforced concrete building located at Oklahoma State University, Stillwater campus.
... However, Zain et al. (2017) has presented that disparity in material characteristics does not bring any major inconsistency in terms of structural response and only variance in the time histories essentially be deliberated as the major factor in stimulating the structure's dynamic response, as variation in each time history may induce critical changes in the response of any building. ...
This study aims at determining the effect of masonry as an infill on the vulnerability of reinforced concrete frame buildings by using fragility assessment. Refined linear and nonlinear structural models were developed, from data collected through professional surveys, using the PERFORM-3D platform. Nonlinear-static and dynamic-analyses were carried out, for fifteen ground motions, to examine the plastic behavior of the models. Subsequently, the vulnerability was assessed using fragility relationships. The fragility parameters were determined by employing the Maximum Likelihood Method (MLM). The results indicated a decrease in the probability of exceedance for specific damage states of the structures with respect to seismic intensity for masonry infill frames. From fragility curves, it is concluded that although the use of masonry as an infill temporarily enhances the capacity of Reinforced Concrete (RC) Frame buildings as the probability of exceedance for masonry infilled RC frames is significantly reduced due to the increase in the overall stiffness of the structure.
Las medidas de intensidad sísmica vectorial han demostrado ser más eficientes en comparación con las medidas de intensidad sísmica tradicionales para predecir la respuesta de estructuras con comportamiento no lineal o aquellas dominadas por los modos superiores; sin embargo, pocos estudios han demostrado la habilidad de estas nuevas medidas para una estimación apropiada de la fragilidad sísmica de edificios. En el presente trabajo se analizaron ocho medidas de intensidad sísmica vectorial compuestas por dos parámetros. Para todos los casos se utilizó la seudoaceleración en el modo fundamental de vibración de la estructura, Sa(T1), como primera componente del vector y la Aceleración Máxima del Suelo (AMS), Velocidad Máxima del Suelo (VMS), duración efectiva (TD), potencial del movimiento sísmico (ID) y los parámetros de forma espectral RT1,T2, NpSa, Npv y NpSv, como segunda componente del vector. Para evaluar la eficiencia de las medidas de intensidad sísmica vectorial en el análisis de fragilidad sísmica, un edificio de concreto reforzado de 10 niveles es sometido a 30 registros sísmicos de banda angosta obtenidos en sueño blando de la Ciudad de México. Los resultados demuestran que la medida de intensidad sísmica vectorial que presenta una mejor relación con la probabilidad de falla es <Sa(T1), NpSa>, en comparación con las otras medidas, especialmente respecto a Sa(T1) que es ampliamente usada en los códigos de construcción vigentes. Por lo tanto, es deseable que en los futuros reglamentos de construcción se consideren medidas de intensidad sísmica más apropiadas.
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is properly cited. ick population density and its escalation propensity in seismically active regions of Pakistan has raised sincere concerns about the performance of building stock whose suboptimal performance and complete collapses led to a colossal number of casualties during the past earthquakes. .e current research is inspired by the Kashmir earthquake of 2005 which consumed more than 80,000 lives, out of which, approximately 19,000 were children due to wide spread collapse of school buildings. A new database for existing reinforced concrete (RC) school buildings in seismic zone 4 of Pakistan has been developed using the surveyed information and presented briefly. .e paper presents the statistics of the data collected through field surveys and professional interviews. It was found that the infrastructural authorities in the considered region developed some specific designs for school buildings, with varying architectural and structural configurations, which were eventually replicated throughout the area. In the current study, almost 2500 schools were surveyed for identifying versatile architectural and structural configurations, and subsequently, 19 different types had been identified, which were eventually used as representative stock for the schools in seismic zone 4 of Pakistan, Muzaffarabad district. .e results of the study yield the brief of the collected data from the field and a consolidated methodology for establishing the analytical fragility relationships for one of the 19 structural configurations of the school buildings. A sample building from the collected data has been selected by considering the maximum number of students, and afterwards, the vulnerability is assessed by employing incremental dynamic analysis (IDA) which constitutes the presented methodology. Finally, the fragility curves are developed and presented for the said building type. .e derived analytical fragility curves for the considered building type indicate its structural vulnerability and as a whole represent its satisfactory behavior. .e vulnerability assessment process and the fragility development are described in an easy manner so that the domestic practicing engineers can readily become able to extend the application towards other school buildings in the region. .e developed relationships can be employed for rational decision making so that essential disaster preparedness can be carried out by identifying any need for structural strengthening and interventions.
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There is an ever-increasing demand for assessment of earthquake effects on transportation structures, emphasised by the crippling consequences of recent earthquakes hitting developed countries reliant on road transportation. In this work, vulnerability functions for RC bridges are derived analytically using advanced material characterisation. high quality earthquake records and adaptive inelastic dynamic analysis techniques. Four limit states are employed, all based on deformational quantities, in line with recent development of deformation-based seismic assessment. The analytically-derived vulnerability functions are then compared to a data set comprising observational damage data from the Northridge (California 1994) and Hyogo-ken Nanbu (Kobe 1995) earthquakes. The good agreement gives some confidence in the derived formulation that is recommended for use in seismic risk assessment. Furthermore, by varying the dimensions of the prototype bridge used in the study. and the span lengths supported by piers, three more bridges are obtained with different overstrength ratios (ratio of design-to-available base shear). The process of derivation of vulnerability functions is repeated and the ensuing relationships compared. The results point towards the feasibility of deriving scaling factors that may be used to obtain the set of vulnerability functions for a bridge with the knowledge of a 'generic' function and the overstrength ratio. It is demonstrated that this simple procedure gives satisfactory results for the case considered md may be used in the future to facilitate the process of deriving analytical vulnerability functions for classes of bridges once a generic relationship is established.
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Conventional seismic hazard analysis methodology is generalized to estimate directly the annual seismic risk of exceeding a specified level of postelastic damage in real structures. The procedure makes use of empirical statistics of the nonlinear-response-based factor F-DM that is a measure of the damage potential of ground motions to multi-degree-of-freedom structures. Using a two-dimensional model of a jacket-type offshore platform and a large sample of observed time histories, it is confirmed that (as for single DOF systems): (1) There is no significant dependence of the average of F-DM on magnitude and distance; and (2) its coefficient of variation is relatively small. These two facts make the method practical. They were confirmed for both local (member-level) damage measures and global collapse, for variations in structure and foundation modeling, and for various durations. A companion paper presents applications to two large three-dimensional, multi-degree-of-freedom, nonlinear models of actual structures.
High-strength concrete (HSC) is widely used in prestressed concrete bridges. Current design provisions for prestressed concrete bridge structures, such as the AASHTO LRFD specifications, however, were developed based on mechanical properties of normal-strength concrete (NSC). As a first step toward evaluating the applicability of current AASHTO design provisions for HSC prestressed bridge members, statistical parameters for the mechanical properties of plant-produced HSC were determined. In addition, prediction equations relating mechanical properties with the compressive strength were evaluated. HSC samples were collected in the field from precasters in Texas and tested in the laboratory at different ages for compressive strength, modulus of rupture, splitting tensile strength, and modulus of elasticity. Statistical analyses were conducted to determine the probability distribution, bias factors (actual mean-to-specified design ratios), and coefficients of variation for each mechanical property. It was found that for each mechanical property, the mean values are not significantly different among the considered factors (precaster, age, specified strength class) or a combination of these factors, regardless of the specified design compressive strength. Overall, the 28-day bias factors (mean-to-nominal ratios) decrease with an increase in specified design compressive strength due to the relative uniformity of mixture proportions provided for the specified strength range. Nevertheless, the 28-day bias factors for compressive strength are greater than those used for the calibration of the AASHTO LRFD specifications. With few exceptions, the coefficients of variation were uniform for each mechanical property. In addition, the coefficients of variation for the compressive strength and splitting tensile strength of HSC in this study are lower than those for NSC used in the development of the AASHTO LRFD specifications.
This paper describes building damage functions that were developed for the FEMA/NIBS earthquake loss estimation methodology (Whitman et al., 1997). These functions estimate the probability of discrete states of structural and nonstructural building damage that are used as inputs to the estimation of building losses, including economic loss, casualties and loss of function (Kircher et al., 1997). These functions are of a new form and represent a significant step forward in the prediction of earthquake impacts. Unlike previous building damage models that are based on Modified Mercalli Intensity, the new functions use quantitative measures of ground shaking (and ground failure) and analyze model building types in a similar manner to the engineering analysis of a single structure.
This paper presents a statistical analysis of structural fragility curves. Both empirical and analytical fragility curves are considered. The empirical fragility curves are developed utilizing bridge damage data obtained from the 1995 Hyogo-ken Nanbu (Kobe) earthquake. The analytical fragility curves are constructed on the basis of the nonlinear dynamic analysis. Two-parameter lognormal distribution functions are used to represent the fragility curves with the parameters estimated by the maximum likelihood method. This paper also presents methods of testing the goodness of fit of the fragility curves and estimating the confidence intervals of the two parameters (median and log-standard deviation) of the distribution. An analytical interpretation of randomness and uncertainty associated with the median is provided.