Mohebi Et Al. - A New Damage Index For Steel MRFs Based On Incremental Dynamic Analysis

Journal of Constructional Steel Research 156 (2019) 137–154
Contents lists available at ScienceDirect
Journal of Constructional Steel Research
A new damage index for steel MRFs based on incremental

dynamic analysis
B. Mohebi a,⁎, A.H. Chegini. T b, A.R. Miri. T b
a
Department of Civil Engineering, Faculty of Engineering and Technology, Imam Khomeini International University, PO Box 34149-16818, Qazvin, Iran
b
M.Sc. Earthquake Engineering, Faculty of Civil Engineering, K. N. Toosi University of Technology, PO Box 15875-4416, Tehran, Iran
a r t i c l e i n f o a b s t r a c t
Article history: The present study introduced a damage index based on a relationship between two values: the maximum
Received 7 August 2018 interstory drift ratio and spectral acceleration at the first-mode period of the structure. The proposed damage
Received in revised form 17 January 2019 index was defined so as it equals 0 in the yield point of the structural system, while it equals 1 in the collapse
Accepted 6 February 2019
point. The incremental dynamic analysis was used to compute the constituent parameters of the damage
Available online 15 February 2019
index in both yield and collapse points. It was also used to observe the change trend of the index through varying
Keywords:
intensities of the earthquake ground motion records. A number of steel models, including single degree of free-
Damage index dom models, stick models and moment resisting frames served as different examples to exhibit the performance
Incremental dynamic analysis of the damage index as well as to compare it with a number of recognized damage indices. Non-linear models
Twisting pattern were developed in OpenSees. Stiffness and strength deterioration in these models were included using the
Sharp softening pattern concentrated plasticity model. The results indicated that unlike the comparative damage indices, the proposed
Steel moment resisting frame damage index was not influenced by successive hardening and instantaneous softening. In addition, once the
intensity of ground motion records increased, the index maintained its ascending trend. Besides, increasing the
period as well as the number of floors along with the presence of some levels of inconsistency in structures
caused a decline in the correlation between the proposed damage index and the comparative indices.
© 2019 Published by Elsevier Ltd.
1. Introduction Given various Engineering Demand Parameters (EDPs), there exist

several DIs proposed for concrete and steel structures to provide a
Typical scientific observations on devastating earthquakes indicate precise description of damage levels. These EDPs take in displacement,
structural damage imposed on certain buildings designed in accordance ductility, interstory drift ratio (IDR), element rotation, strain, strength
with current codes [1,2]. Accordingly, developing effective methods for and stiffness, dissipated energy, cyclic fatigue, etc. Taking into account
calculating the damage threshold of a building during its lifetime is essen- a combination of these EDPs, seismic DIs fall into two classes:
tial. Moreover, to judge the reparability of the elements, it is vital to assess
the level of imposed damage on a structural system. Performance-Based A) Local damage indices (LDIs): These are associated with an ele-
Seismic Design (PBSD) is among the efforts in this regard. This design ment or section and are subcategorized as non-cumulative and
concept adds consideration of displacement or damage to the code- cumulative. Non-cumulative indices regard displacement and
based design procedure. Recently, the methodology of PBSD has been ductility as the key parameters, which connect the damage
developed by some researchers [3–8]. These studies have braced the level merely to the maximum deformation [11–13], while
idea that various structural systems may produce an improved perfor- cumulative indices are subdivided as deformation-based
mance if the PBSD is adopted in preference to Strength-based methods. [14–16], hysteretic energy-based [17–19] and a combination of
The PBSD approach is grounded on attaining performance objectives deformation and energy absorption [20–24].
associated with certain damage levels [9,10]. Damage indices (DIs) are B) Global damage indices (GDIs): A GDI is developed to describe the
used to quantitatively describe the damage levels in a structure or ele- overall damage state of a structure by weighting the local dam-
ment as well as to make a significant connection between performance age of each element in a certain way [25,26] or using a specific
objectives and damage levels. response value of the entire structure [27–29]. It is shown that
the methods proposed for weighting the effects of LDIs cannot
explain the structural system behavior appropriately and may
⁎ Corresponding author. lead to ambiguous results. Therefore, the global response of
E-mail address: mohebi@eng.ikiu.ac.ir (B. Mohebi). the structural system is more trustworthy [28]. It is worth
https://doi.org/10.1016/j.jcsr.2019.02.005
0143-974X/© 2019 Published by Elsevier Ltd.
138 B. Mohebi et al. / Journal of Constructional Steel Research 156 (2019) 137–154
mentioning that the process of reaching the components of a GDI Table 1

equation has a significant effect on the accuracy of the results Applications of DIs in Structural Health Monitoring (SHM).
provided by that DI. Some researchers have proposed GDIs Researchers Research summary
based on the pushover analysis [29,30]. This method of analysis H.R. Ahmadi et al. [44] A new algorithm called EFTFD was proposed to detect
is built upon incorrect assumptions that do not take into account damage in bridge piers. The algorithm was defined
the duration and cyclic effects. Hence, it cannot be an appropriate using a square time-frequency distribution. In
method to study the damage under seismic and dynamic loads. addition, two new DIs were proposed based on
time-frequency matrices.
The non-linear dynamic analysis under earthquake effects can
M. Zhang and R. Schmidt A new approach was proposed to detect the damage
provide more accurate EDP values in DI equations. By extending [45] based on the auto correlation function. The relative
the results of the non-linear dynamic analysis at various intensity change of a normalized vector called Auto Correlation
levels, conducted by the Incremental Dynamic Analysis (IDA) Function at Maximum Point Value Vector (AMV) was
[31], the overall behavior of the structure can be studied compre- used as the DI to locate the damage before and after
damage.
hensively from the elastic state to the ultimate collapse [32]. As A. Eraky et al. [46] Dynamic computer simulation techniques were used
an inherent flaw, these DIs cannot detect the location and extent to develop non-destructive damage detection
of damage in the structure effectively, while calculating the DIs methodology for beam and plate structures. The
for the whole structure leads to more reliable results than those proposed procedure was built upon the comparison of
modal strain energy for different structure conditions
obtained through weighting. As a result, they may not provide
from which the DI can be calculated.
an accurate estimation of how the imposed damage affects an el- J.P. Amezquita-Sanchez A new methodology was presented for detecting,
ement and how it is distributed across the structure. To estimate and H. Adeli [47] locating, and quantifying the damage severity in a
the deterioration in a structure effectively, using a combination smart high-rise building structure. Fractality
of the structural global response and other damage assessment dimension (FD) was employed to detect features to be
used for damage detection. In addition, a new
tools, such as LDIs is proposed [33].
structural DI was proposed based on the estimated FD
values. Further, the damage was located using the
The current study presents a DI for earthquake-excited structures changes of the estimated FD values.
based on a specific combination of general response and local damage Z.X. Tan et al. [48] A technique was presented for predicting the
early-stage damage for both single and multiple
parameters of the structure. These parameters include the spectral ac-
damage scenarios in a steel beam. This technique uses
celeration at the first-mode period of the structure (Sa (T1)) and the the modal strain energy-based DI β to treat single
maximum interstory drift ratio (MIDR) in the structure, respectively. damage scenarios and Artificial Neural Network (ANN)
Expressly, the proposed DI equation includes the MIDR in the structure incorporating β to quantify damage severities under
during the non-linear dynamic analysis and Sa (T1) as the main param- multiple damage scenarios.
eters. In addition, the IDA analysis is applied to study the change trend
in the proposed DI at different Ground Motion (GM) intensities, and
to specify the parameters of the DI when the structure yields and col-
lapses. The DI is proposed to explore the damage in steel structures,
since the models used in this study consider stiffness and strength dete-
rioration only via backbone curves associated with steel structures. The Table 2
behavior of the proposed DI when examining damage in reinforced con- Applications of DIs in Performance-Based Seismic Design (PBSD).
crete structures can be examined in future studies. Researchers Research summary
A. Sharifi et al. The aspects of a robust and practical damage-based

2. Applications of the concept of DI [49] deterministic seismic design approach for reinforced concrete
buildings were discussed. The proposed performance-based
DIs are advantageous tools in various fields of structural and seismic design approach can be used in different structural systems
using fundamentals of reinforced concrete behavior and
engineering. Structural Health Monitoring (SHM) is among the central
practical analyses.
research fields, which has benefited from the concept of DI. Damage M.S. Alhaddad A Ductility-Based Damage Indices (DBDI) assessment
identification in Structural Health Monitoring (SHM) is the focus in et al. [50] procedure was presented to check the seismic adequacy of RC
many articles which have presented or reviewed certain new methods special moment resisting frames (SMRF) at the final design
for damage identification steps. These steps include (a) signal acquisi- stage. It can be used to improve the seismic performance of the
frames in the context of performance-based seismic design
tion, (b) signal processing [34–37], and (c) feature extraction and inter- (PBSD) process.
pretation [38–40]. Besides, many researchers have developed G. Nie et al. [51] A DI was defined based on maximum deformation and
methodologies for System Identification (SI) [41] in SHM, using modal cumulative deformation to explore different degrees of
parameters [42,43]. In addition to the foregoing steps, selecting a prac- damage of structural members caused by cyclic loading.
Furthermore, two structural damage models were defined
tical DI is essential for a numerical presentation of the damage level.
based on limit state criteria to quantify seismic damage while
An appropriate DI has an ability to reflect the local status of an element the structure was subjected to severe seismic loading. Finally,
or the overall state of a structure after a severe loading such as an earth- vulnerability analysis was conducted using structural damage
quake. Application of DIs in damage detection of structures has been model at different degrees of structural damage for seismic
studied by several researchers (Table 1). performance evaluation and risk assessment.
W. Huang et al. Based on the classical Park-Ang [20] damage model, a
According to Section 1, the significance of DIs is axiom in [52] consistent modification was proposed for structural members
Performance-Based Seismic Design (PBSD) of structures. Various dam- of different materials. Furthermore, the specific limit values of
age models and DIs have been defined to evaluate the seismic perfor- the damage model were calculated at various performance
mance of structural elements, thereby enhancing the methods of levels.
R. Pang et al. [53] A seismic fragility analysis method was extended based on
PBSD. Some research studies on DIs-PBSD relationships are presented
incremental dynamic analysis (IDA) to evaluate the seismic
in Table 2. performance of high concrete face rockfill dams (CFRDs).
Another benefit of the DI concept can be related to finding appropri- Permanent deformation and face slab DI using a modified
ate Intensity Measures to estimate the damage imposed on structures. generalized plasticity model for rockfills and a plastic-damage
Assessments of damaged structures after an earthquake show that model for face-slabs were included to be dam damage
measures (DMs) after defining a new face-slab DI.
there is interdependency between ground motion parameters and
B. Mohebi et al. / Journal of Constructional Steel Research 156 (2019) 137–154 139
structural response. Therefore, it is essential to explore the association during the design phase, or in case of post-earthquake reliability as
between seismic intensity parameters and damage criteria [54]. Accord- well as problems with repairability. To serve these purposes, GDIs
ingly, identification of an optimal intensity measure, which sufficiently should be both observable for practical purposes and a non-decreasing
correlates with an appropriate Engineering Demand Parameter (EDP), function of time unless the structure is repaired or strengthened [62].
is of high prominence. Many studies attempted to provide the best A GDI could be obtained from special combinations of local damage
seismic parameters to represent potential structural damage. Table 3 measures. Using a weighting scheme is the simplest technique for com-
indicates some research studies on the issue. bining LDIs. The weighting factor can reflect the replacement cost and/
or the relative importance of the member or substructure in maintain-
3. The proposed DI and the comparative DIs ing the integrity of the structure. For example, the lower story of a
building might be allocated more importance than the upper ones.
To assess the behavior of the proposed DI, a number of commonly The weighting factor for any story could also depend on the magnitude
used DIs were opted to be compared with the proposed DI. These DIs of the corresponding DI, so that severely damaged stories are weighted
(labeled comparative DIs) and their equations are defined in detail in more heavily [28].
Section 3.1. In addition, Section 3.2, which introduces the proposed DI, Park et al. [25] and Bracci et al. [26] have proposed weighted averag-
states the necessities of proposing this index initially, and then the ing methods to integrate the damage of single elements for each indi-
corresponding equation and its calculation procedure for a structural vidual story followed by measuring the global damage for the overall
system are discussed in detail. structure. The method proposed by Park et al. [25] is based on the
value of hysteretic energy dissipated by elements, whereas Bracci
3.1. Quantitatively compared DIs et al. [26] designated the gravity load as the averaging operative.
In the current paper, the Park et al. [25] method was employed as the
In the current study, three DIs were picked to be compared with the averaging method, to achieve the damage level in each story (Eq. (1))
proposed DI quantitatively. These included the DIs of Cosenza et al. [13], and the overall structure (Eq. (3)). In Eq. (2), E denotes the total energy
Kunnath et al. [21] and Ghobarah et al. [28]. The first two are LDIs and dissipated by the element and n is the number of elements of an individ-
that of Ghobarah et al. [28] is global. Generally, a DI is local when it re- ual story. In addition, in Eq. (4), E represents the total energy absorbed
fers to a single point, section, member, or structural part, whereas it is by the story and n is the number of stories. Finally, DIcomponent, DIstory
considered global when it defines the state of the entire structure [60]. and DIoverall are the damage level in a component, a story and the
Local damage at a cross-section of the structure can be measured whole structure respectively.
adequately by the degradation of bending stiffness and moment capac-
ity of that cross-section [19]. The analysis of LDIs depicts weak or vul- X
n
DI story ¼ λi;component :DIi;component ð1Þ
nerable elements that should be retrofitted [61]. However, it is i¼1
challenging to get a clear idea of a structural response to a given input
" #
ground motion from a long list of element DIs [28]. Stiffness and Ei
strength deterioration of the whole structure comprise the overall effect ðλi Þcomponent ¼ n ð2Þ
∑k¼1 Ei component
of local damage at various locations. A GDI can then be defined as a func-
tion of such continuously distributed local damage, characterizing the n n o
X
overall damage state as well as serviceability of the structure. GDIs are DI overall ¼ ðλi Þstory ðDIi Þstory ð3Þ
response quantities, determining the damage state of the structure i¼1
after an earthquake excitation and can be used for decision-making " #
Ei
ðλi Þstory ¼ n ð4Þ
∑k¼1 Ei story
Table 3
Applications of DIs in finding appropriate Intensity Measures.
The DI proposed by Cosenza et al. [13], is presented in Eq. (5):
Researchers Research summary
Y. Lieping et al. [55] The correlations between some existing intensity indices μ−1
DI ¼ ð5Þ
and major seismic responses of structures (as μ u;mon −1
representatives of the degree of structural damage) were
studied based on an analysis of elastoplastic SDOF and
MDOF systems, leading to several recommendations.
where μ is the maximum ductility during the loading history and μu,mon
M. Kumar et al. [56] The impact of frequency content and properties of the is the maximum allowable value of ductility equals to uu,mon/ uy while uu,
structure on global and interstory drift for steel frames mon is the ultimate displacement under monotonic loading and uy is the
designed in accordance with Eurocode 8 provisions was yield displacement. For flexure-resisting components μ, μu,mon, uu,mon
investigated using the incremental dynamic analysis.
and uy are replaced with μθ, μθ,mon, θu,mon and θy, respectively. μθ, is the ro-
V.V. Cao and H.R. 1040 far-fault motions were applied to obtain the
Ronagh [57] maximum interstory drifts and the Park-Ang [20] DIs in a tation ductility during the loading history and μθ,mon is the maximum al-
structure. Then, the correlation coefficient was calculated lowable value of rotation ductility under monotonic loading, while θu,
to attain the degrees of correlation between the damage to mon and θy are the ultimate and the yield rotation respectively.
the structure and the seismic parameters. Taking into account both deformation and hysteretic energy,
K. Kostinakis et al. The correlation between 19 ground motion IMs and the
[58] damage state of four medium-rise 3D RC buildings was
Kunnath et al. [21] proposed a damage model which is a modified
estimated and significant information relative to the form of the Park and Ang [20] DI. Eq. (6) presents the formulation of
efficiency of the commonly used damage indicators as well Kunnath et al. [21] DI:
as the influence of the seismic orientation with regard to
structural axes was provided. θm −θy E
A. Massumi and F. Regression coefficients were computed to acquire DI ¼ þβ h ð6Þ
θu −θy M y θu
Gholami [59] expedient ground motion parameters that well
characterize the potential damage caused by earthquakes.
Analysis was conducted on four RC frames under a set of where θm is the maximum experienced rotation of an element in a
85 ground motion records. The structural damage system subjected to an earthquake, θy is the yield rotation under mono-
considered was interstory drift, roof drift and the Park and tonic loading, θu is the ultimate rotation under monotonic loading, Eh is
Ang [20] index.
the hysteretic energy dissipated by the element, My is the yield moment
and β is a parameter calibrated in accordance with experiments to re-

flect the effect of repeated loading. In the current paper, β was set as
0.025, which is a common value for steel components [63].
The DI, which is proposed by Ghobarah et al. [28], is based on stiff-
ness reduction of the whole structure. The proposed DI is defined in
Eq. (7):
K final
DI ¼ 1− ð7Þ
K initial
where Kinitial is the initial slope of the base shear-top displacement curve
obtained from the pushover analysis of the structure before subjecting it
to the earthquake excitation while Kfinal is the initial slope of the same
curve after subjecting the structure to the earthquake GM. The value
of this DI is between 0 and 1. A value of 0 indicates the structure is linear
and a value of 1 indicates the structure is in danger of collapse. However, Fig. 2. The sharp softening pattern in an IDA curve.
collapse may occur at lower damage values or within the domain of lin-
ear behavior of the structure because of the construction faults. Fig. 1 illustrates the twisting pattern. The pattern is, in fact, the repeti-
tion of the hardening and softening phenomenon in a succession. The
3.2. The proposed DI cause of the hardening phenomenon is that an increase in the scale fac-
tor of the record makes the small cycles of the structural response so
In this section, the proposed DI is introduced. To calculate the pro- strong at the beginning of the time history that the structure is dam-
posed DI, from among the known EDPs, Sa (T1) and the MIDR in the aged, and as a result, the structural characteristics are changed in the
structure under any excitement applied to the structure are required. next strong cycles [31]. The occurrence of severe hardening phenome-
The IDA analysis has been introduced as an appropriate tool for studying non can prevent or even reverse the accumulation of damage instanta-
the structural behavior, and to calculate the EDPs, which considers the neously, therefore, can locally cause the curve to move toward a smaller
parameters from the elastic state to the yield point, non-linear domain, damage measure. Expressly, under a greater intensity of seismic load-
and finally the dynamic instability [32]. The IDA analysis involves ing, a damaged structural system displays the same or smaller response
performing a number of non-linear time history analyses where the compared to the preceding step. The phenomenon can be observed in
intensity of GM record, selected for the study of damage, is gradually in- segment BD in Fig. 1. In the following, assuming the possibility of creat-
creased until it reaches the global collapse capacity point of the struc- ing a collapse mechanism in the structural model, the final softening
ture. The IDA analysis has been proved as a time-consuming method segment is seen in the curve, indicating the onset of the dynamic insta-
due to high volumes of calculations. However, nowadays, the analysis bility (segment EF in Fig. 1).
can be performed at very high speeds owing to computing advances. The equations related to DIs provided by Kunnath et al. [21] and
The curve derived from the IDA analysis demonstrates an intensity Cosenza et al. [13] indicate that in Fig. 1, in the hardening segment
parameter of the GM (such as the spectral acceleration response, Sa (BD) associated with a decrease in the maximum rotation and displace-
(T1)) against a damage parameter (such as the MIDR) called Intensity ment, the level of these DIs also decreases. It is also expected that
Measure (IM) and Damage Measure (DM), respectively [31]. Therefore, Ghobarah et al. [28] DI, which is related to the initial stiffness of the
the curve obtained from the IDA analysis can have a positive effect on structure after the earthquake, will decrease in the hardening and rever-
the calculation process of the proposed DI. Figs. 1 and 2 represent exam- sal domain of the IDA curve slope. This phenomenon causes these indi-
ples of IDA analysis curves for a Single Degree Of Freedom (SDOF) struc- ces to exhibit fewer values toward the end of the hardening segment
ture. In the following section, the detailed account of the design and compared to the beginning of this segment. However, as the hardening
modeling of this structure and other structures is provided. As men- segment ends and with a slight increase in Sa (T1), the structure reaches
tioned earlier, in the current study, the spectral acceleration response point E and enters the final softening step, or specifically, the dynamic
of the structure, Sa (T1), was selected as IM and the MIDR as DM. instability.
The IDA curves reveal different behaviors of the structures under The sharp softening is another common pattern in IDA curves that
various GM records. Successive hardening and softening (the twisting can occur for a variety of reasons, including the presence of construction
pattern) and instantaneous dynamic instability of the structure (the faults in the structure, formation of a weak story, and the P-Delta effect.
sharp softening pattern) in structures are among these behaviors. The sharp softening pattern is seen in Fig. 2. As the figure depicts, the
structure suffers from severe softening after passing point G, and a
sharp increase in the MIDR occurs with a small increase in the value of
Sa (T1). If the above indices are used for seismic assessment of this pat-
tern, they are expected to provide small values at lower GM intensities,
be accompanied by a sudden increase at higher intensities of GM, and
reach their maximum value. Consequently, when these DIs are used in
the seismic assessment of structures and the current status of the struc-
ture under the intended GM record is followed by the presence of a
twisting pattern or sharp softening, they can provide a misleading
view of the status and capacity of the structure, especially when ap-
proaching the dynamic instability. In the current paper, an index is pre-
sented to resolve this problem. The index is completely independent of
the weighted averaging method and consists of a local damage param-
eter and a global response parameter of the structure, so as to be able to
interpret the status of the structure correctly with an appropriate com-
bination of these two parameters in a non-linear dynamic analysis
Fig. 1. The twisting pattern in an IDA curve. under each GM record. The MIDR as one of the most commonly used
Fig. 3. Sequence of steps for calculating the proposed DI.
seismic assessment factors for structures was selected as the local dam- explanations provided, the steps for calculating the proposed DI under
age parameter, based on which important DIs have been presented any GM record are as follows:
[64,65]. Sa (T1) was used to be applied in the proposed DI among overall
response parameters of the structural system. The reason for using Sa 1. Determining the spectral acceleration at the first-mode period of the
(T1) is that, among the global response parameters of the structure, it structure (Sa (T1)).
has reported the highest correlation with Park and Ang [20] DI (one of
2. Performing the non-linear dynamic analysis under the GM record
the most widely used and reliable LDIs) [54,57]. The proposed DI is rep-
and obtaining the MIDR (d) in the structure.
resented by Eq. (10):
3. Performing the IDA analysis through selecting parameters Sa (T1)
and d, as IM and DM, respectively.
d−dy
DI1 ¼ ð8Þ 4. Specifying the yield (Say,dy) and collapse (Sau,du) points on the IDA
du −dy
curve and calculating the DI through Eq. (10).
Sa−Say
DI2 ¼ ð9Þ
Sau −Say Fig. 3 clarifies the sequence of these steps. The equation for the
proposed DI is defined such that if the structure does not reach the
8 9
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < If DI1 and DI2 ≤0 then : DI ¼ 0 = yield point (A), the damage level equals 0. Besides, if the structure
DI ¼ DI1 DI 2 ð10Þ crosses the collapse point (E), the damage level equals 1, indicating
: ;
If DI1 and DI2 ≥1 then : DI ¼ 1 the dynamic instability of the structure. Therefore, the proposed DI ex-
amines the condition of the structure in the segments between the
where Sa and d are the spectral acceleration at the first-mode period of yield point and the collapse point. Given that, no significant change
the structure (Sa (T1)) and MIDR for each accelerogram, respectively. occurs in the Sa (T1) level by passing the collapse point, and this is
In fact, these parameters are obtained by conducting the non-linear the MIDR that has a sudden increase, DI2 does not exceed 1. As can
dynamical analysis under the selected accelerogram. Say and dy denote be seen, the geometric mean of Eqs. (8) and (9) was used to attain
the spectral acceleration at the first-mode period and the MIDR in the the final equation of the DI. Of the advantages of geometric mean
structure at the yielding state, respectively. In other words, parameters compared to other averaging methods, is that its result is less influ-
Say and dy correspond to the first point where the IDA curve has no enced by too much or too little data [66]. This feature can help pre-
longer the linear state indicated by point A in Fig. 1. Sau and du also vent high number of fluctuations in the values of the DI during
represent the spectral acceleration and MIDR in the structure at the col- severe hardening or softening.
lapse state of the structure (point E in Fig. 1), respectively. After this In order to show that the problems associated with the indices intro-
point, the IDA curve flattens out in a plateau. According to the duced in Section 3.1 are removed by the proposed DI, the value of this DI
Fig. 4. Comparison of the comparative DIs with the proposed one for the IDA curve of Fig. 1 Fig. 5. Comparison of the comparative DIs with the proposed one for the IDA curve of Fig. 2
(the twisting pattern). (the sharp softening pattern).
Table 4
Qualitative Comparison of DIs.
Proposed Ghobarah Kunnath Cosenza et al. Amezquita-Sanchez Rajeev & Wijesundara Yu. Chen & Yi. Chen
et al. et al. & Adeli
DI typea global global Local Local Local and global Local Local
c
Loading or Analysis type IDA Push Over Monotonic Monotonic loading FD Analysis Continuous wavelet Monotonic loading
Analysis Analysis loading / THb / TH Analysis transform and IDA and Cyclic loading
Analysis
Considering the stiffness and strength YES NO NO NO NO YES YES
deterioration
Influenced by successive hardening/ NO YES YES YES NO NO YES
the sharp softening patterns
more accurate estimation near the YES NO NO NO NO NO NO
collapse state
Considering the design and YES NO NO NO NO YES NO
construction faults
A combination of global response and YES NO NO NO YES YES YES
local damage parameters
Influenced by Weighted averaging NO NO YES YES NO NO YES
methods
Considering the frequency content YES NO NO NO YES YES NO
effect
a
Defined in Section 3.1.
b
Time History.
c
Fractality Dimension.
was computed for each step of the curves of Figs. 1 and 2 and then com- At this step, a qualitative comparison of the proposed DI and other
pared with the comparative DIs in Figs. 4 and 5, respectively. DIs could improve demonstrating the advantages to the proposed DI.
As indicated in Figs. 4 and 5, when hardening or softening occur All the comparative DIs, along with DIs of Rajeev and Wijesundara
(segment BD in Fig. 1 and segment GH in Fig. 2, respectively), the pro- [17], Yu. Chen and Yi. Chen [24], and Amezquita-Sanchez and Adeli
posed DI values are not influenced and can provide the necessary infor- [47] were compared with the proposed one. The effort was made to con-
mation on reaching the collapse point of the structure in the segments sider numerous comparison fields and to choose different types of DIs.
near this point. However, the comparative DIs are influenced by the The effect of stiffness and strength deterioration, hardening and sharp
IDA curve fluctuations. For example, point C in Fig. 1, whose correspond- softening phenomenon, proximity to the collapse state, and construc-
ing point in Fig. 4 is specified as C′, is located in the hardening segment. tion faults were discussed. Furthermore, the efficiency of each DI with
Fig. 4 depicts that all the DIs of Kunnath et al. [21], Ghobarah et al. [28], regard to considering both local and global damages, the frequency con-
and Cosenza et al. [13] provide a lower damage value for that level of Sa tent, and the use of weighted averaging methods were investigated.
(T1) compared to the proposed DI, and also the damage level at this Table 4 presents the assessment of the mentioned DIs and compares
point is less than that in the hardening starting point (B). The IDA the characteristics thereof.
curve indicates that a slight increase in Sa (T1) to reach point E exposes
the structure to collapse. Thus, the three comparative DIs may not be 4. Structures and non-linear modeling
able to warn about reaching this point. With regard to uncertainties in
modeling and the assumed parameters, in reality, the value of Sa (T1) For the purpose of the present paper, the analyzed structures are di-
could be slightly greater than that obtained in the analysis. As shown vided into three types: A) Single Degree Of Freedom (SDOF) models,
in Fig. 2, this slight increase leads to a high increase in the damage pa- B) stick models [67,68] and C) moment resisting frames, the results of
rameters of the structure or even to global instability of the structure. which are discussed in the following section. All the structures were
While none of the introduced DIs, which do not consider a little ahead made of steel and designed in accordance with the procedures in
and a little behind the current point, could correctly estimate the dam- ASCE/SEI 7–10 [69]. Furthermore, the structures were assumed being
age level, i.e. far less than it may actually occur by taking the precise con- located in an area having a Risk-Targeted Maximum Considered
dition of the structure into account. (MCER) and with site class D. For the design of all the structures, the im-
Therefore, the use of DIs that are influenced by the fluctuations in the portance factor, I = 1, and the seismic design parameters, SDS = 1.25 g
IDA curve may provide limited understanding of the structure's condi- and SD1 = 0.6 g were included.
tions when exposed to different GM records. As fluctuations of the IDA A) SDOF models were assumed with periods of 0.5 s, 1.0 s, and 2.0 s
curve have much less effect on the results of the proposed DI in each and a height of 3000 mm. Also, the ductility ratio, miu = δ / δy, for these
step of this analysis, one can examine the approaching of the global in- structures was assumed 2, in that these structures represent steel ordi-
stability of the structure more accurately using the proposed DI. nary moment resisting frames. The schematic view of the SDOF models
is presented in Fig. 6. Williamson [70] used a number of SDOF models
under various GM records to study the role of the P-Delta effect in re-
sponse of non-elastic systems and concluded that the P-Delta effect is
very important in the structural response and has a significant effect
on the behavior of the analyzed structure. Therefore, for all type
(A) models, the P-Delta effect was included.
B) This group includes structures assigned to the stick model, which
are used as a simplified model of Multi Degree Of Freedom (MDOF)
structures and moment resisting frames. Two-dimensional stick
models, which were 2-, 6-, and 10-story, were applied for damage as-
sessment. The period of each of them equals 0.2 N, where N is consid-
Fig. 6. The schematic view of the SDOF models. ered as the number of stories [67,68]. The height of the stories in all of
the models was assumed 3600 mm. The typical floor plan of the struc-
tures is shown in Fig. 7. Gravity loads were applied to all floors as the
dead load of 5.5 kN/m2 and the live load of 2 kN/m2. Given the increase
in the number of stories in comparison to type (A) models and, conse-
quently, the increased impact of the P-Delta effect, for this type of
models, the P-Delta effect was also taken into account. Thus, according
to the ASCE/SEI 7–10 [69], the gravity load of DL + 0.25LL was applied
to all the structures where DL is the dead load and LL is the live load.
2- and 6-story models were assumed steel intermediate moment
resisting frames and the ductility ratio of the members for these models
was considered to be 4. Therefore, according to ASCE/SEI 7–10 [69], the
value of the response modification factor (R) for them is 4.5. Besides, a
steel special moment resisting frame was selected for the 10-story
model, in which the ductility ratio of the members is 6 and the response
modification factor (R) is 8. Fig. 8 illustrates the 2-, 6- and 10-story stick
models. As mentioned earlier, the P-Delta effect was included in all of
the models, but it was excluded from the analysis of this structure by
Fig. 7. Typical floor plan of the structures used for creating the stick models.
creating a new model in order to further examine the P-Delta effect
on the results of the DIs in the 10-story stick model. Then the results
were compared to those of the condition in which the P-Delta effect
had been included.
C) A number of steel moment resisting frames with different geo-
metric and strength characteristics designed by Kitayama and
Constantinou [71] were analyzed. The results are discussed in the next
section. At first, two 3- and 6-story steel moment resisting frames
were selected as the benchmark models, both of which were steel spe-
cial moment resisting frames. W-type sections were used for their
cross-sections. In both models, the height of the first story was assumed
4420 mm and the height of the remaining stories was considered
4304 mm. The frames had three spans, each with a length of 8230 mm.
Fig. 9a and b represent the elevations of 3- and 6-story frames, re-
spectively. In modeling of these structures, in order to consider the P-
Delta effect, leaning columns were used, which is a common method
for taking into account the P-Delta effect in two-dimensional frames.
The loads applied to the structure, as well as the details of the design of
these models, are presented in Kitayama and Constantinou [71]. Then,
two changes were made in the benchmark frames, each aimed at model-
ing the possible faults in the design and construction of structures. This
was done to investigate the damage estimation capability by the pro-
posed DI in structures, which are not fully complying with the codes.
One of these changes is an increase in the height of the first story of the
studied frames, which increases the potential for weak story formation
in the structure. This change was applied to both the 3- and 6-story
frames. The elevation of these frames is shown in Fig. 10a and b.
The second change is the redesign of the frames so that the strong
Fig. 8. The 2-, 6- and 10-story stick models. column-weak beam principle is not observed in the structure, the
Fig. 9. Illustration of the benchmark moment resisting frames: (a) the 3-story model (b) the 6-story model [71].
Fig. 10. Illustration of the moment resisting frames with a changed column height: (a) the 3-story model, (b) the 6-story model.
columns have less resistance than the beams, and it is more likely Fig. 13 illustrates the backbone curve of this model. In this figure, the
that plastic hinges are created in columns. The redesign was per- strength parameters include: My = effective yield moment, Mc = cap-
formed in both structures according to ASCE/SEI 7–10 [69], and no ping moment strength, Mr = residual moment, and the deformation pa-
changes were made to the geometric characteristics, loading way, rameters include: θy = yield rotation, θp = pre-capping plastic rotation
and design method. for monotonic loading (difference between yield rotation and rotation
The cross-sections of the redesigned structures are shown in Fig. 11a at maximum moment), θpc = post-capping plastic rotation (difference
and b. Finally, the modified models were compared with the benchmark between rotation at maximum moment and rotation at the point of
models and the results were interpreted. Each of the models in groups complete loss of strength), θu = ultimate rotation capacity. Also, Ke =
A, B and C were labeled according to their characteristics (Table 5). My / θy is the elastic stiffness.
The distributed plasticity and the concentrated plasticity are two In order to find the value of deformation parameters, the equations
common methods for modeling the plasticity of the elements in non- presented by Lignos and Krawinkler [76] were used, in which geometric
linear analyses. Here the concentrated plasticity method was applied. characteristics and material type of the member are determinant. Lignos
In this method, modeled by OpenSees software [72], in each of the and Krawinkler [76] obtained these equations by analyzing the database
beams and columns, an Elastic Beam Column element is used in the cen- results found in Newell and Uang's studies [79].
ter, two Zero Length Elements are used at the two ends, and the plastic To enhance the reliability of the IDA analysis, the structural models
behavior occurs merely in the Zero Length Elements. The stiffness of were analyzed under records with various specifications. For type
Elastic Beam Column Element and Zero Length Elements are considered (A) and (B) structures, 22 pairs of GM records were used according to
serial [73]. Fig. 12 displays the schematic representation of the concen- FEMA P695 [80]. All records were scaled to different intensity levels
trated plasticity model in a single element. until collapse occurred in the structure. For simplicity, details of all
In order to define the plastic behavior (here, the moment-rotation), in GM records are not provided. The IDA analyses were performed for
Zero Length Elements, the cyclic degradation of stiffness and strength type (C) structures using eight records from among the listed records
was included. Many researchers have used deteriorating material models (Table 6). The Hunt and Fill [81] algorithm was used to reach the col-
to study the structural damage [74–78]. In the current study, the modi- lapse point of structures in each IDA analysis. To do this, MATLAB [82]
fied deterioration model of Ibarra and Krawinkler [75] was used. and OpenSees [72] were used simultaneously.
Fig. 11. Illustration of the redesigned moment resisting frames: (a) the 3-story model, (b) the 6-story model.
Table 5
Labels of the models.
Model type Label Explanation
A (SDOF SDOF-0.5 Single degree of freedom model with a 0.5 s period

models) SDOF-1.0 Single degree of freedom model with a 1.0 s period
SDOF-2.0 Single degree of freedom model with a 2.0 s period
B (Stick models) ST-2 2-story stick model
ST-6 6-story stick model
ST-10 10-story stick model
ST-10-NP 10-story stick model without considering the
P-delta effect
C (Moment MRF-3 Benchmark 3-story 3-span moment resisting
resisting frame
frames) MRF-3-RD Re-designed 3-story 3-span moment resisting
frame
MRF-3-CH 3-story 3-span moment resisting frame with a
changed column height in the first floor
MRF-6 Benchmark 6-story 3-span moment resisting
frame
MRF-6-RD Re-designed 6-story 3-span moment resisting
frame
MRF-6-CH 6-story 3-span moment resisting frame with a
changed column height in the first floor
Fig. 13. Modified Ibarra and Krawinkler deterioration model [75].
5. Results and discussion

accurate comparison between this index and the existing indices, and to
This section presents and discusses the results obtained from the as- have more comparison points in the correlation graphs. In other words,
sessment of the proposed DI and the three comparative DIs for the actually, it is necessary to obtain the damage level under the effect of a
models mentioned in Section 4. Initially, the results of the SDOF models certain GM record with a specific Sa (T1) in order to obtain the damage
(type (A)) and the effect of period changes in these models are ex- level in a structure after an earthquake. However, for a better assess-
plored. Next, the results of the stick models (type (B)) are presented ment, it was necessary to obtain the damage level for each GM record
and the effect of changing the number of stories for these models is in various spectral accelerations. Using this technique, the number of
discussed. Besides, the results of the stick model analyzed without the comparison points for the evaluation of the proposed DI was multiplied.
P-Delta effect are discussed and the role of the P-Delta effect is studied.
Finally, the results of the moment resisting frames (type (C)) are pre- 5.1. SDOF models
sented and the changes made in the results are explained in details as
comparing the benchmark frames with other ones. Caused by space lim- As mentioned in the previous section, the SDOF models with periods
itations, the curves for all of the models are not fully represented. In- of 0.5 s, 1.0 s, and 2.0 s were analyzed by the IDA analysis using 22 GM
stead, some are shown as instances. The complete results are record pairs introduced by FEMA P695 [80]. As an example, Fig. 14
presented in the tables. To provide a more complete explanation of the shows the IDA curves of the SDOF-0.5 model under the records stated
method used in this study, it is worth noting that during the assessment by FEMA P695 [80], indicating the MIDR in the structure versus the Sa
procedure, it was assumed that at each step of the IDA analysis, the goal (T1). All of the IDA curves presented in this study are based on these
was to obtain the value of damage in the same step so as to make a more two parameters. In each curve, a flattening occurs after passing a certain
value of Sa (T1), indicating the dynamic instability phenomenon in the
structure.
Performance of the indices was studied by the IDA analysis under all
of the applied GM records while the occurrence of events such as hard-
ening or sharp softening, and their effect on the damage level declared
by each index. Fig. 15 depicts the results of the proposed DI and the
comparative DIs under the GM record No.1 of FEMA P695 [80] by the ap-
plication of the IDA analysis for the SDOF-0.5 model.
As presented in Fig. 15, when the SDOF-0.5 model was subjected to
this GM record, the results of the proposed DI and the comparative DIs
indicated relatively close values. The same trend occurred for the
other GM records, which are further discussed by examining the
Table 6
Characteristics of the GM records used for type (C).
No. Date Record name Station name Magnitude
1 1990 Manjil, Iran Abbar 7.4

2 1999 Duzce, Turkey Bolu 7.1
3 1994 Northridge Canyon Country - WLC 6.7
4 1971 San Fernando LA-Hollywood Stor 6.6
5 1999 Kocaeli, Turkey Arcelik 7.5
6 1999 Kocaeli, Turkey Duzce 7.5
7 1999 Chi-Chi, Taiwan CHY101 7.6
8 1999 Hector Mine Hector 7.1
Fig. 12. The schematic view of the concentrated plasticity model in a single element.
Fig. 14. IDA curves for the SDOF-0.5 model under 22 GM record pairs mentioned in FEMA
P695 [52]. Fig. 17. Correlation between the proposed DI and that of Cosenza et al. [13] for the SDOF-
0.5 model.
proposed DI. This can lead to a more precise judgment on the forward
and backward steps by the proposed DI. In other words, the proposed
DI can warn about the approaching of the collapse point through its
gradual increase.
Figs. 16-18 represent the correlation between the results of the
proposed DI and those of the comparative DIs for the SDOF-0.5 model.
To represent the numerical value of correlation, the coefficient of deter-
mination (R2) was used. Besides, a linear relationship with the y = m.x
pattern was obtained for correlation with each index by performing a
linear regression analysis and using the least square method, between
the proposed DI and the comparative DIs. In this equation, x is the
value of the proposed DI, y is the value of the compared index and m
is the line slope, which is another parameter used to evaluate the corre-
lation of the results.
Fig. 15. Comparison of the comparative DIs with the proposed one for the SDOF-0.5 model If parameters R2 and m become closer to unity, the greater correla-
under the GM record No.1 of FEMA P695 [80]. tion between the results of the proposed DI and the comparative DIs is
achieved. Figs. 16-18, which demonstrate the results obtained from
correlation parameters. The important point in Fig. 15 is that the hard- the IDA analysis under 22 pairs of GM records, suggest that the pro-
ening phenomenon in the structures led to a decline in the DIs of posed DI exhibits a high correlation with the comparative DIs for the
Kunnath et al. [21], Cosenza et al. [13] and Ghobarah et al. [28]. It was SDOF-0.5 model. Studying the correlation between the results of the
also observed that the final softening, which leads to the collapse of proposed DI and those of the comparative DIs showed that the results
the structure, is shown with a steep increase in slope, while, for the pro- of the SDOF-1.0 and SDOF-2.0 models were in accordance with the re-
posed DI, the descending branch is not observed in the hardening seg- sults of the SDOF-0.5 model. In Table 7, the results of parameters R2
ments, and the corresponding graph reaches its maximum value from and m are provided for all three models, namely SDOF-0.5, SDOF-1.0,
0 with a gradual increase. Besides, the final softening segment was not and SDOF-2.0.
associated with a sudden increase in the DI. Therefore, according to The results indicate that in type (A) models, as the period of the
the observations, the successive hardening and softening can be con- structure increased, the value of both parameters R2 and m for all the
cluded to have a very lower effect on the damage estimation by the existing indices decreased. For example, R2 value between the proposed
Fig. 16. Correlation between the proposed DI and that of Kunnath et al. [21] for the SDOF- Fig. 18. Correlation between the proposed DI and that of Ghobarah et al. [28] for the SDOF-
0.5 model. 0.5 model.
Table 7
Parameters R2 and m between the proposed DI and the comparative DIs for type
(A) models.
R2 m
Kunnath Cosenza Ghobarah Kunnath Cosenza Ghobarah
SDOF-0.5 0.979 0.973 0.878 0.903 0.872 0.677

SDOF-1.0 0.936 0.928 0.861 0.893 0.862 0.664
SDOF-2.0 0.867 0.862 0.739 0.760 0.753 0.620
DI and Cosenza et al. [13] DI is 0.973, 0.968 and 0.862 for the SDOF-0.5,
SDOF-1.0 and SDOF-2.0 models, respectively, indicating that a 4.6% de-
crease occurred for the SDOF-1.0 model and a reduction of about
11.4% occurred for the SDOF-2.0 model compared to the SDOF-0.5
model. In addition, to examine Cosenza et al. [13] DI, parameter m for
the SDOF-0.5 model equals 0.872, while this parameter for the SDOF- Fig. 20. Comparison of the comparative DIs with the proposed one for the ST-10 model
1.0 model is 0.862, and for the SDOF-2.0 model, it is 0.753, representing under the GM record No.16 of FEMA P695 [80].
1.1% and 13.6% reductions in correlation compared to that of the SDOF-
0.5 model. Accordingly, it can be inferred that as the period in type type (A) models, the ascending trend in the proposed DI occurs from
(A) models increased, the value of results obtained from the proposed 0 to 1, while in those ranges of Sa (T1), where hardening and softening
DI diverged from the results of the comparative DIs. The reason can be occur on the IDA curves, fluctuations are perceived on the curves of
the greater influence of P-Delta effect on higher periods, which in- Kunnath et al. [21], Cosenza et al. [13], and Ghobarah et al. [28] DIs.
creases the likelihood of dynamic instability in structures. The weaker This confirms that as hardening and sharp softening phenomena occur
consideration of the P-Delta effect by the comparative DIs can be the in stick models, the proposed DI reports a lower level of affectability. It
outcome of this phenomenon. means that a more accurate judgment can be made on the capacity of
the structural system in these segments, since previous and subsequent
5.2. Stick models steps are also included for exploring the structure status, along with ex-
amining the DIs in this segment. The same phenomenon was observed
This section deals with examining the results of the stick models. At for the other GM records in the other models of type (B).
first, models with 2, 6, and 10 stories that have the periods of 0.4 s, 1.2 s Figs. 21-23 indicate the correlation between the results of the pro-
and 2.0 s, respectively, were analyzed by the same 22 pairs of GM re- posed DI and those of the three comparative DIs for the ST-10 model.
cords mentioned in the previous sections. In addition, the results of In these figures, which were produced using the IDA analysis results
the DIs in these models were studied. Next, to study the role of the P- from the above-mentioned 22 pairs of the GM records, coefficients R2
Delta effect, the results of the DIs in a 10-story model, in which the P- and m between the proposed DI and the comparative DIs are listed in
Delta effect was excluded (ST-10-NP), were examined and compared addition to the graphs for each index. Moreover, coefficients R2 and m
with the results obtained from the ST-10 model. are presented in Table 4 for the quantitative representation of the corre-
lation between the proposed DI and the comparative DIs for the ST-2,
5.2.1. The effect of number of stories ST-6, and ST-10 models as well as to examine the effect of increased
Among the models studied in this section, the IDA curves of the ST- number of stories on the correlation between the results of the DIs.
10 model are shown in Fig. 19. As seen, when each graph flattens, the According to Figs. 21-23, a lower correlation is seen between the re-
collapse of the structural system occurs in the IDA analysis under the sults of the proposed DI and the comparative DIs for the ST-10 model.
GM record associated with that curve. Low correlation, especially in parameter m (the slope of regression
Fig. 20 is presented to examine and compare the change trend in the line), which represents the ratio of the results of the compared index
proposed DI and the comparative ones by increasing the spectral accel- to those of the proposed DI, is clearly visible for all the indices. According
eration and also to examine the sudden hardening and softening phe- to Table 8, the correlation established for the ST-2, ST-6 and ST-10
nomena in the stick models. models between the proposed DI and the DIs of Cosenza et al. [13],
The DIs are presented as a function of Sa (T1) under the GM record Kunnath et al. [21], and Ghobarah et al. [28], reveals that an increase
No.16 of FEMA P695 [80] for the ST-10 model. As with the results of in the number of floors in the stick models, leads to a decrease in both
Fig. 19. IDA curves for the ST-10 model under 22 GM record pairs mentioned in FEMA Fig. 21. Correlation between the proposed DI and that of Kunnath et al. [21] for the ST-10
P695 [80]. model.
promising result. However, the proposed DI attains the correct damage

level in the collapse state. Consequently, in structures with a higher pe-
riod, this modeling approach leads to a lower correlation between the
value of the proposed DI and the comparative DIs. In addition, an in-
crease in the number of stories leads to an increase in the level of hori-
zontal displacement of the floors due to an earthquake and the role of
the P-Delta effect becomes more significant. Therefore, it can lead to a
reduction in the correlation parameters. In other words, it caused a di-
vergence among the results of the proposed DI and those of the compar-
ative DIs. In the following, the influence of disregarding the P-Delta
effect in one of the models on the results of the DIs is examined.
5.2.2. Evaluation of the P-Delta effect

When an earthquake occurs, the P-Delta effect can be mentioned as
one of the factors affecting the behavior and response of structures. An
Fig. 22. Correlation between the proposed DI and that of Cosenza et al. [13] for the ST-10
increase in the number of stories i.e. the height of the structure enlarges
model. the effect of this phenomenon on the behavior of structures. In
Section 5.1, it was also observed that as the period in a structural system
increases, the P-Delta effect becomes more vital. Consequently, explor-
ing the P-Delta effect in a model with the high number of stories (or a
high period) can further reveal the importance of this effect. In this sec-
tion, the results of the correlation between the proposed DI and the
comparative DIs for the ST-10-NP model were obtained and compared
with the results of the ST-10 model.
Fig. 24 embraces the IDA analysis curves for the ST-10-NP model
under the above-mentioned 22 pairs of the GM records. Comparison
of Figs. 19 and 24 reveals that if the P-Delta effect is excluded, the struc-
tural system under the same GM records reaches the dynamic instabil-
ity in a higher value of MIDR and Sa (T1) on average. This comparison
displays the higher importance of the P-Delta effect in the behavior of
structures exposed to various earthquakes.
The change trend of the proposed DI and the comparative DIs against
the change in Sa (T1) value for the ST-10-NP model is shown under the
GM record No.7 of FEMA P695 [80] in Fig. 25. A comparable trend also
occurred for the other GM records. The correlation between the pro-
Fig. 23. Correlation between the proposed DI and that of Ghobarah et al. [28] for the ST-10 posed DI and the comparative DIs derived from 22 pairs of GM records
model.
are shown in Figs. 26-28. The values of parameters R2 and m in these fig-
ures show a high correlation between the results of the proposed DI and
R2 and m values for all the comparative DIs. For instance, the value of R2 those of the comparative DIs. In order to explore the influence of exclud-
between the proposed DI and Kunnath et al. [21] DI is 0.945 for the ST-2 ing the P-Delta effect, the results of the ST-10 and ST-10-NP models are
model, while it decreases to 0.850 and 0.715 for the ST-6 and ST-10 presented in Table 9.
models respectively. These values decreased about 10.0% and 24.3%, re- By comparing the values of R2 and m between these two models, it is
spectively, compared to the corresponding values in the ST-2 model. For observed that as the P-Delta effect was excluded, the correlation of the
Kunnath et al. [21] DI, the values of parameter m in the ST-2, ST-6 and proposed DI with the comparative DIs increased, particularly, this in-
ST-10 models were obtained as 0.654, 0.569, and 0.508, which they decrease in m was significant. The P-Delta effect with the creation of the
crease about 13.0% and 22.3% for the ST-6 and ST-10 models respec- secondary moment in nodes and a big change in their rotation rate, es-
tively. This trend may occur as an increase in the number of stories pecially in columns, causes a complete change in the formation of the
improves the effectiveness of the weighted averaging method devel- plastic hinges in the model.
oped by Park et al. [25], and reduces the correlation parameters. In ac-
cordance with the method used to design the stick structures,
overstrength was regarded as a constant factor. Due to this assumption,
collapse occurs suddenly in one of the stories, which leads to the overall
instability of the structure. Nevertheless, in real structures, increasing
the number of stories causes overstrength to be of higher value [83].
For this reason, the stick models with a high period are exposed to sud-
den collapse and the weighted averaging method fails to reach a
Table 8
Parameters R2 and m between the proposed DI and the comparative DIs for the ST-2, ST-6
and ST-10 models.
R2 m
ST-2 0.945 0.930 0.872 0.654 0.597 0.809

ST-6 0.850 0.843 0.738 0.569 0.528 0.655
Fig. 24. IDA curves for the ST-10-NP model under 22 GM record pairs mentioned in FEMA
ST-10 0.715 0.705 0.567 0.508 0.484 0.347
P695 [80].
Fig. 25. Comparison of the comparative DIs with the proposed one for the ST-10-NP model Fig. 28. Correlation between the proposed DI and that of Ghobarah et al. [28] for the ST-10-
under the GM record No.7 of FEMA P695 [80]. NP model.
Table 9
Parameters R2 and m between the proposed DI and the comparative DIs for the ST-10 and
ST-10-NP models.
R2 m
ST-10 0.715 0.705 0.567 0.508 0.484 0.347

ST-10-NP 0.817 0.831 0.721 0.799 0.753 0.733
DIs. Then, the models of Table 5, which were included for comparison
with the 3- and 6-story benchmark models (MRF-3-RD, MRF-3-CH,
MRF-6-RD, MRF-6-CH), were subjected to the same eight records, by
the application of the IDA analysis. Similar to the process applied for
the benchmark models, the damage level was calculated for each
index in these models as well. As mentioned in Section 4, the changes
Fig. 26. Correlation between the proposed DI and that of Kunnath et al. [21] for the ST-10-
made in the benchmark models were implemented in order to investi-
NP model. gate the performance of the proposed DI and the other DIs, if there are
faults related to the design and construction of the structure. In fact,
the 3- and 6-story frames were examined as case studies in Sections
5.3. Moment resisting frames 5.3.1 and 5.3.2, respectively.
In this section, the results of the steel moment resisting frames,

which are referred to as type (C) models in Table 5, were examined. 5.3.1. 3-Story frames
First, the IDA analyses were performed for the MRF-3 and MRF-6 This section includes the results of the MRF-3, MRF-3-RD, and MRF-
models, as the benchmark structures, through the eight records intro- 3-CH models to evaluate the behavior of all the DIs. The IDA curves of
duced in Table 6. The level of damage was calculated for these frames the MRF-3 model, which is the benchmark model in this section, are
in each step of the IDA analysis by the proposed DI and the comparative shown in Fig. 29 for the GM records of Table 6.
Fig. 27. Correlation between the proposed DI and that of Cosenza et al. [13] for the ST-10-
NP model. Fig. 29. IDA curves for the MRF-3 model under the eight GM records of Table 6.
Fig. 30. Comparison of the comparative DIs with the proposed one for the MRF-3 model Fig. 32. Correlation between the proposed DI and that of Cosenza et al. [13] for the MRF-3
under the GM record No.1 of Table 6. model.
In Fig. 30, the DIs of Section 3.1 along with the proposed DI are
shown as a function of Sa (T1) for the MRF-3 model, using the IDA anal-
ysis under the GM record No.1 of Table 6. In this curve, the fluctuation
due to hardening can be observed in the curve of Kunnath et al. [21], Co-
senza et al. [13] and Ghobarah et al. [28] DIs, while, a gradual increase in
the curve of the proposed DI occurs similar to previous examples, and
the value of this index rises from 0 to 1. This behavior occurred in the
other models of this section and for the other GM records as well.
Figs. 31-33 display the correlation between the proposed DI and the
comparative DIs for the MRF-3 model under all the eight records of
Table 6. According to these figures, parameters R2 and m show satisfac-
tory values for the MRF-3 model. In fact, these figures indicate that the
results of the proposed DI and the comparative DIs are very close and
highly correlated for this 3-story model, which was designed with com-
plete compliance with the code criteria.
In the following, the MRF-3-RD and MRF-3-CH models were sub-
Fig. 33. Correlation between the proposed DI and that of Ghobarah et al. [28] for the MRF-3
jected to the same eight records mentioned above, by the IDA analysis,
model.
and the values of the proposed DI and the comparative DIs were calcu-
lated. In order to compare the results from these models and the bench-
mark model (MRF-3), coefficients R2 and m were calculated between from the results of these two indices, and the ratio of values of these
the proposed DI and the comparative DIs for these two models. The re- two indices to the proposed DI decreased. A reduction in parameter m
sults are mentioned in conjunction with the results of the MRF-3 model may not have the same interpretation for all the indices. As seen, the
in Table 10. value of m between the proposed DI and Ghobarah et al. [28] DI de-
Comparison of the results of the MRF-3 and MRF-3-RD models in creases from 1.22 to 0.93. This means that, totally, the ratio of Ghobarah
Table 10 indicates that by changing the design and excluding the strong et al. [28] DI values to the proposed DI values decreases. Nevertheless,
column-weak beam principle in the design of the 3-story model, values despite the reduction in m, the results converge as this parameter is ap-
R2 and m decreases from the proposed DI to the Kunnath et al. [21] and proaching 1.
Cosenza et al. [13] DIs.This might mean that by making this change in The results of the MRF-3 and MRF-3-CH models in Table 10 indicate
the quality of design, the response rate of the proposed DI diverged that, as the height of the first column increases, or, expressly, as the po-
tential for formation of weak story enhances, the value of R2 between
the proposed DI and Kunnath et al. [21], Cosenza et al. [13], and
Ghobarah et al. [28] DIs increases up to 3%, 3.2%, and 2.4%, respectively.
Since the difference of R2 between the models of MRF-3 and MRF-3-CH
is small, the results of this parameter can be assumed approximately
equal, while parameter m decreases by 17.7%, 16.4%, and 8.2%, respec-
tively. As is shown, the decrease in m is more significant than the in-
crease in R2. The sharp decline in m shows that, in general, with an
increase in the height of the columns of the first story in the 3-story
Table 10
Parameters R2 and m between the proposed DI and the comparative DIs for the 3-story
models.
R2 m
MRF-3 0.864 0.846 0.830 0.754 0.706 1.226

MRF-3-RD 0.818 0.792 0.811 0.412 0.387 0.935
Fig. 31. Correlation between the proposed DI and that of Kunnath et al. [21] for the MRF-3
MRF-3-CH 0.898 0.873 0.856 0.627 0.591 1.126
model.
Fig. 34. IDA curves for the MRF-6 model under the eight GM records of Table 6. Fig. 36. Correlation between the proposed DI and that of Kunnath et al. [21] for the MRF-6
model.
model, the ratio of the results of the comparative DIs to those of the pro-
posed DI dropped. This decreased ratio in Kunnath et al. [21] and Co-
senza et al. [13] DIs means that the results are also diverging, since the
value of m distances from unity. However, like the previous example
of this section, here, Ghobarah et al. [28] DI exhibited a different behav-
ior, and the decrease in the value of m between this index and the pro-
posed DI led to the convergence of the results of these two indices, as
the values of this parameter for the MRF-3 and MRF-3-CH models are
1.22 and 1.12 respectively. This indicates that the results of the pro-
posed DI and Ghobarah et al. [28] DI are more correlated in the MRF-
3-CH model. Therefore, according to the information provided in this
section, it can be inferred that parameters R2 and m were usually re-
duced by making changes, which were pertinent to the design and con-
struction weaknesses. However, this phenomenon does not seem to be
true for Ghobarah et al. [28] DI, as the correlation increased with
changes in the structural model. Given that the results of Ghobarah
et al. [28] DI, which is a GDI, and the proposed DI are obtained using
the pushover analysis and the IDA analysis, respectively, the results of Fig. 37. Correlation between the proposed DI and that of Cosenza et al. [13] for the MRF-6
the proposed DI can be more reliable. The comparative DIs underesti- model.
mate the actual damage level in the structure with the high potential
of weak story formation (the MRF-3-CH model). For further discussion,
the results of the 6-story models are examined in Section 5.3.2. Fig. 35 demonstrates the proposed DI and the comparative DIs ver-
sus Sa (T1) for the MRF-6 model under the GM record No.7 of Table 6.
Figs. 36-38 illustrate the correlation of the results for the MRF-6
5.3.2. 6-Story Frames model. In these figures, the results of the proposed DI are compared
In this section, the IDA analyses were performed for the 6-story with those of the comparative DIs for each step of the IDA analyses,
models (MRF-6, MRF-6-RD, MRF-6-CH) by the eight GM records of and the values of parameters R2 and m are listed alongside each figure.
Table 6. As with the 3-story models, the proposed DI and the compara- Furthermore, The MRF-6-RD and MRF-6-CH models, whose details
tive DIs were calculated for these models in each step of the analysis. were presented in Section 4, were subjected to the above-mentioned
Fig. 34 displays the IDA curves for the MRF-3 model (the benchmark eight GM records by the IDA analysis and the values of all the indices
model), under all the eight records. were calculated for them in order to compare the correlation of results
Fig. 35. Comparison of the comparative DIs with the proposed one for the MRF-6 model Fig. 38. Correlation between the proposed DI and that of Ghobarah et al. [28] for the MRF-6
under the GM record No.7 of Table 2. model.
Table 11 and a sudden increase of the DI with a small increase in Sa (T1) value.
Parameters R2 and m between the proposed DI and the comparative DIs for the 6-story The occurrence of these phenomena in calculating the damage level
models.
can provide a misleading view of the status and capacity of the struc-
R2 m ture, especially when approaching the dynamic instability. Another im-
Kunnath Cosenza Ghobarah Kunnath Cosenza Ghobarah portant point is that the use of the weighted averaging method to
expand LDIs to the whole structure can also lead to erroneous results.
MRF-6 0.811 0.774 0.861 0.666 0.630 1.049
MRF-6-RD 0.769 0.747 0.707 0.651 0.615 0.773 For the same reasons, the proposed DI was defined such that it can re-
MRF-6-CH 0.746 0.730 0.847 0.555 0.543 0.816 move the effects of the above-mentioned issues on the values of DIs.
Further, it can warn about approaching the structure collapse via exam-
ining previous and subsequent steps creating an ascending trend in
of each model with those of the MRF-3 model. As mentioned in the pre- their values. For this reason, a more accurate judgment can be made
vious sections, this is done in order to evaluate the effects of design and about the capacity of the structural system in these segments. In order
construction faults. Table 11 shows the parameters R2 and m between to examine the level of correlation between the results of the proposed
the proposed DI and the comparative DIs for the MRF-6, MRF-6-RD, DI and the comparative DIs, two parameters of R2 and m were used. It
and MRF-6-CH models. was seen that as the period and/or the number of floors increased, the
The trends of results are similar to those of Section 5.3.1, but in con- results of the proposed DI diverged from those of the comparative DIs.
tradiction to them, no exception in the values of R2 and m is seen for any In the higher structures, the role of overstrength and the P-Delta effect
of the models and the indices studied. Expressly, by making both the is more significant. Therefore, as explained earlier, the precision of the
intended changes in the benchmark model, the correlation between comparative DIs decreased. In particular, this phenomenon is more pal-
the proposed DI and all the comparative DIs dropped. pable at the collapse state. In addition, as the P-Delta effect was ex-
According to the results of Section s 5.3.1 and 5.3.2, it can be concluded, the correlation between the proposed DI and the comparative
cluded that in general, if a kind of inconsistency exists in the structural DIs increased. Moreover, the comparative DIs cannot offer an accurate
system indicating the weaknesses of design or construction, the correla- description of a sudden damage mechanism, a phenomenon related to
tion between the results of the proposed DI and the comparative DIs the instability of the whole structure.
mentioned in Section 3.1, may decrease. This suggests that the results Accordingly, due to the conceivable faults in the structure, the
of the proposed DI and the comparative DIs are diverging. By examining weighted averaging method could not result in the correct estimation
the frames with various inconsistency types, it is observed that there of the damage state. In other words, if the structural system complies
were various scenarios for different situations of the plastic hinges and with the current codes, a relatively high correlation between the value
eventually a sudden damage mechanism in the structure. In fact, differ- of the proposed DI and that of the comparative DIs would be observed.
ent collapse mechanisms and arrangement of plastic hinges can cause However, changing the design approach and excluding the strong
structural instability or only local damage to a structure under seismic column-weak beam principle in the design, which occasionally leads
loads. The changed models were designed so that the probability for to the formation of plastic joints with various arrangements in different
the occurrence of an instability in these models is more than the bench- points of the structure and global instability, decreases the correlation
mark models due to a weakness in columns. When dynamic instability parameters between the proposed DI and the comparative DIs. The pro-
occurred, that was more likely to be found in the changed models, the posed DI has the ability to overcome these shortcomings and address
correlation between the results of the proposed DI and those of the more accurate damage levels. It can be used to achieve a better estima-
comparative DIs, compared to the benchmark models, decreased. tion of the structure performance after earthquakes.
Cosenza et al. [13] and Kunnath et al. [21] DIs that calculate the damage
in the whole structure by weighting the damage created in the ele-
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Mohebi Et Al. - A New Damage Index For Steel MRFs Based On Incremental Dynamic Analysis

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Journal of Constructional Steel Research 156 (2019) 137–154

Contents lists available at ScienceDirect

Journal of Constructional Steel Research

A new damage index for steel MRFs based on incremental

1. Introduction Given various Engineering Demand Parameters (EDPs), there exist

mentioning that the process of reaching the components of a GDI Table 1

crete structures can be examined in future studies. Researchers Research summary

A. Shariﬁ et al. The aspects of a robust and practical damage-based

and β is a parameter calibrated in accordance with experiments to re-

Fig. 3. Sequence of steps for calculating the proposed DI.

Model type Label Explanation

A (SDOF SDOF-0.5 Single degree of freedom model with a 0.5 s period

Fig. 13. Modiﬁed Ibarra and Krawinkler deterioration model [75].

5. Results and discussion

No. Date Record name Station name Magnitude

1 1990 Manjil, Iran Abbar 7.4

Kunnath Cosenza Ghobarah Kunnath Cosenza Ghobarah

SDOF-0.5 0.979 0.973 0.878 0.903 0.872 0.677

promising result. However, the proposed DI attains the correct damage

5.2.2. Evaluation of the P-Delta effect

Kunnath Cosenza Ghobarah Kunnath Cosenza Ghobarah

ST-2 0.945 0.930 0.872 0.654 0.597 0.809

Kunnath Cosenza Ghobarah Kunnath Cosenza Ghobarah

ST-10 0.715 0.705 0.567 0.508 0.484 0.347

In this section, the results of the steel moment resisting frames,

Kunnath Cosenza Ghobarah Kunnath Cosenza Ghobarah

MRF-3 0.864 0.846 0.830 0.754 0.706 1.226

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