1. science
2. mathematics
3. statistics

Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
A numerical procedure for calibrating damage indices
Chara Ch. Mitropoulou, Nikos D. Lagaros, Manolis Papadrakakis
Institute of Structural Analysis & Antiseismic Research, National Technical University of Athens, 9, Iroon Polytechniou Str.,
Zografou Campus, 157 80 Athens, Greece.
email: {chmitrop, nlagaros, mpapadra}@central.ntua.gr
ABSTRACT: In this study a numerical calibration procedure is proposed for some of the most widely accepted damage indices
(DIs) used for quantifying the extent of damage in reinforced concrete structures. In particular, without loss of generality of the
applicability of the proposed procedure, the Park and Ang local damage index, its modified variant proposed by Kunnath,
Reinhorn and Lobo; the Chung, Meyer and Shinozuka local damage index; along with the maximum and final softening damage
indices proposed by Di Pasquale and Çakmak, are calibrated on the basis of the width of crack openings. Τhe proposed
procedure is performed in two steps that both are based on the incremental dynamic analysis. In the first one, the estimation of
the crack width is performed by means of detailed modelling with hexahedral finite elements for concrete and rod elements for
steel reinforcement. Due to the extensive computing demands the databank of values for the damage indices under investigation
is defined based on coarse models with beam-column elements, in the second step of the proposed calibration procedure.
Finally, the statistical characteristics of DIs are computed by means of horizontal statistics in conjunction with the maximum
likelihood function method and an optimization algorithm; the statistical characteristics are used for assessing the efficiency of
each DI.
KEY WORDS: Damage indices; numerical calibration; incremental dynamic analysis; detailed finite element modelling;
performance based design.
1
INTRODUCTION
Performance-Based Design (PBD) in earthquake engineering
is the state-of-the-art framework for future developments in
structural design. PBD in engineering practice requires the
definition of clearly determined levels of damage achieved for
different seismic intensity levels. In order to implement this
design framework, appropriate models for assessing the
structural damage, within the context of a random seismic
environment, are required. The idea of describing the state of
damage of the structure by one number on a defined scale in
the form of a damage index (DI) is attractive because of its
simplicity. So far, a number of researchers have studied
various DIs for reinforced concrete or steel structures [1-4].
Damage indices can be broadly divided into two classes [5]:
(a) strength-based DIs; and (b) response-based DIs. Strengthbased DIs do not require finite element (FE) analysis [6,7];
however, they must be calibrated against observed damages
using a large experimental database. Although the seismic
performance of the structures is commonly related to their
capacity to undergo inelastic deformations, experimental
studies have shown that ductility as well as alternative
measures of the structural performance in case of seismic
loading, based on the theory of low-cycle fatigue, do not seem
to provide a satisfactory index of the seismic damage [8].
These test results are consistent with the notion that failure of
brittle systems is caused by excessive deformation, while the
failure of ideal ductile systems is initiated by repeated
inelastic deformations. The damage indices, used for
structural systems that are neither ideal brittle nor ideal
ductile, need to account for the damage effect of both
excessive and repeated inelastic deformations [9]. Thus, there
is a need for more general and reliable indices able to
characterize the performance of the structures. The responsebased DIs can be divided into three groups according to the
quantity that the index accounts for [2]: (a) maximum
deformation indices [10-14]; (b) cumulative damage indices
[8,15]; and (c) combination of maximum deformation and
cumulative damage indices. In this work some of the most
widely accepted damage indices accounting for both
maximum deformation and cumulative damage are
considered. In particular, the Park and Ang [9] local damage
index, its modified variant proposed by Kunnath, Reinhorn
and Lobo [16], the Chung, Meyer and Shinozuka [17,18] local
damage index, along with the maximum and final softening
damage indices proposed by DiPasquale and Çakmak [19,20]
are studied in this work.
In this study a numerical calibration procedure is proposed
and it is applied to the above mentioned damage indices that
are used to quantify the extent of damage in reinforced
concrete structures. Furthermore, the width of the crack
openings is used as the basis for implementing the calibration
procedure. The estimation of the crack width is performed
using detailed numerical modelling with hexahedral finite
elements for the concrete and rod elements for the steel
reinforcement; while due to the computing demands the
databank of values for the damage indices under investigation
is defined based on coarse models with beam-column
elements. Both steps of the proposed procedure are based on
the incremental dynamic analysis (IDA) [21]. Next, the
statistical characteristics of the DIs are computed by means of
horizontal statistics in conjunction with the maximum
likelihood function method and an optimization algorithm.
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
The proposed numerical calibration procedure is not limited
neither to the DIs considered, nor to the quantity used as a
basis (crack width), or the material (reinforced concrete). For
the purposes of the current study one 3D framed structure is
considered.
2
CALIBRATION PROCEDURE
For the needs of a calibration procedure a significant number
of experimental tests are required that will provide snapshots
of the structural behavior corresponding to different damage
levels for a number of structural systems. In order to replace
the experimental tests, which are expensive and cannot be
performed for any structural system, the proposed calibration
procedure is based on numerical experiments. The idea behind
the proposed numerical calibration procedure is to correlate
levels of visible damages with engineering demand
parameters (EDPs) values (i.e. the damage indices
considered). The levels of the visible damages considered in
this study are based on the crack width openings observed in
experimental tests found in the literature [22].
For this purpose, the proposed numerical calibration
procedure of the damage indices is based on the incremental
dynamic analysis [21] procedure where Nincr nonlinear
dynamic analyses are performed, each for a specific
earthquake hazard level of increased intensity. For
demonstration purposes the five damage indices described
above, belonging to the category of maximum deformation
and cumulative damage, are considered; while in order to
implement the proposed numerical calibration procedure the
level of the structural damage is associated with the crack
width.
The main objective of the IDA-based study is to define a
correlation of the intensity level and the maximum seismic
response of the structural system and it is represented with a
curve. The intensity level and the seismic response are
described through an intensity measure (IM) and an
engineering demand parameter, respectively; while IDA is
implemented through the following five steps: (i) Define a
reliable nonlinear finite element model for performing the
required nonlinear dynamic analyses; (ii) select a suit of
natural records; (iii) select the proper intensity measure and
engineering demand parameter; (iv) employ a suitable
algorithm for selecting the record scaling factor in order to
obtain the IDA curve with the least required nonlinear
dynamic analyses and (v) employ a post processing technique
for exploiting the multiple records results.
Figure 1 depicts the calibration procedure, where the
seismic demand used for calibrating the damage indices DIi
(i=1,2,...,m) is defined with reference to the characterization
of the damage state associated with respect to the crack
openings (0.05 cm, 0.1cm, 0.3cm and 0.4cm), without loss of
generality of the proposed procedure. The crack openings are
monitored at the beam-column joints (at the end of the
structural elements). Furthermore, it should be noted that there
are cases where the width of diagonal cracks is much more of
a concern than the width of flexural cracks, for this reason
both diagonal and flexural cracks are monitored. Although
this approach (correlation of DIs with crack openings) can be
considered as a simplistic one, it is employed in this study
since the relation between the maximum crack openings and
284
the level of damage can be found in the literature [22].
However, the distribution of the cracks, or more accurately,
the distribution of the plastic strains within the structure can
also be used.
According to a recent work by Mitropoulou et al. [23] there
are cases where more than 80 natural records are required in
order to obtain a good estimation of the probabilistic
characteristics of a response quantity. For the purpose of this
study 100 natural records have been used, which are randomly
selected from the four lists of records. These records have
been selected from the PEER [24] strong-motion database
with the following features: (i) Events occurred in specific
area (longitude -124o to -115o, latitude 32o to 41o). (ii)
Moment magnitude (M) is equal to or greater than 5. (iii)
Epicentral distance (R) is smaller than 150 km. These records,
according to step (iv) of IDA, are scaled into multiple hazard
levels and the 16%, 50% and 84% fractile curves for each
damage index are developed. In order to compare the
observed structural damage (quantified in this study through
the crack openings) with the values of the damage indices, 3D
detailed modelling with solid finite elements is used.
Performing IDA over a pool of 100 natural records with
detailed finite element models requires extensive
computational effort which is substantially reduced with the
procedure presented in Figure 1:
(i) For each seismic intensity level of IDA the mean
response spectrum of the 100 natural records is obtained.
(ii) One artificial accelerogram is generated for each
intensity level to be consistent with the corresponding
mean response spectrum. This artificial accelerogram is
implemented in the framework of IDA for defining the
dynamic capacity curve (correlation pairs “SA(T1,5%)crack width”). For the test example considered in this
study, a constant incremental step size of 0.05 g is
applied up to failure, resulting into 100 to 120
increments, in order to obtain the database of the
“SA(T1,5%)-crack width” pairs uniformly distributed
over the range of crack openings. Following this
procedure the damage levels are correlated with the
hazard levels.
(iii) The correlation of the crack width with the values of the
DIs for the 100 natural records is achieved with the IDA
studies, performed for each DI implementing beamcolumn FE simulation of the structure.
(iv) The statistical characteristics of the DIs are computed by
means of horizontal statistics in conjunction with the
maximum likelihood function method and an
optimization algorithm. The statistical information
necessary to understand and quantify the behavior of
structural systems can be presented in different formats
depending upon the objective. For instance, if the issue
is loss assessment, where it is important to evaluate the
distribution, mean value and/or dispersion of a DI (or
EDP) for a given IM, the “horizontal” statistics format is
applied as the most appropriate one. However, if the
issue is conceptual design or fragility analysis, where the
designer’s main objective is to find the global strength
required to limit the value of a DI (or EDP) to a certain
quality, then “vertical” statistics are the most suitable.
“Vertical” statistics are also used to quantify the ground
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
motion intensity at which a system approaches a certain
limit state. The terms “horizontal” and “vertical”
presuppose that DIs (or EDPs) are plotted on the
horizontal axis and IMs on the vertical axis. Since this
study is concerned with the correlation of damage levels
with response quantities, the most appropriate format is
the “horizontal” statistics for the hazard levels in order to
calculate the statistical parameters of the response
quantities i.e. the DIs via the maximum likelihood
function method and an optimization algorithm.
The likelihood function in our implementation is expressed
as follows:
Ν rec
L( μDI ,σ DI )   f S A , j ( DI i , μDI ,σ DI )
(1)
i 1
where the DIs are assumed to follow the lognormal
distribution, fSAj is the lognormal density function of the DI
corresponding to the jth hazard level characterized with the
first mode spectral acceleration value SA,j and Nrec is the total
number of records considered. The two parameters μDI and σDI
of Eq. (1) are computed as the values that maximize ln(L) by
implementing a metaheuristic optimization algorithm [25].
be equal to 1.5kN/m2 and for the load combination the dead
and live loads are multiplied by 1.0 and 0.3, respectively.
For the two story 3D RC frame test example, the floor area
is 5×5 m2 and the height of each story is 3.5m. The
dimensions of the beams are 30×50cm2, with longitudinal
reinforcement LR: 3Ø18 top and bottom, and transverse
reinforcement TR: (2)Ø8/10cm. The dimensions of the
columns are 50×50cm2 with longitudinal reinforcement LR:
12Ø22 and transverse reinforcement TR: (2)Ø8/20cm.
Rayleigh damping is used for the first and the second mode
(T1=0.167 sec, T2=0.167 sec and ξ1=ξ2=5.0%). The numerical
modelling is performed with solid finite elements of size
10×10×10 cm3 for the concrete resulting into 12,400
hexahedral finite elements with 19,056 nodes and 57,168 dof,
while 13,280 embedded strut elements with 13,120 nodes and
78,720 dof are used for modelling the steel reinforcement. The
detailed FE mesh is shown in Figure 2.
SA(T1,5%)
beam-column finite
element discretization
(a)
0.4
0.3
0.1
0.05
SA,0.1
SA,0.05
Crack width
(cm)
. . .
SA,0.4
SA,0.3
SA,0.4
SA,0.3
SA,0.1
SA,0.05
DI1
SA(T1,5%)
SA(T1,5%)
5
}
3D solid finite element
discretization
SA,0.4
SA,0.3
SA,0.1
SA,0.05
Figure 2. 3D test example - FE mesh for the concrete.
(b)
DIm
Figure 1. IDA-based calibration procedure, (a) dynamic
capacity spectrum defined with detailed FE modelling and (b)
IDA studies performed for each DI with beam-column FE
modelling.
3
NUMERICAL TESTS
For the purpose of this study one test example is considered in
order to implement the proposed procedure. The test example
is simulated with 3D solid finite elements (hexahedral finite
elements) with embedded reinforcement for implementing
step (ii) of the calibration procedure, while the following
material properties are considered for the concrete: modulus
of elasticity Ec=30.0 GPa and characteristic compressive
strength at 28 days fcκ = 25.0 MPa (C25/30), while the
material properties of the reinforcing steel are: nominal yield
stress fyκ = 500 MPa (S500) and modulus of elasticity equal to
200 GPa, while steel hardening was taken into account equal
to 1.0%. The slab thickness is equal to 18cm, the self-weight
of the beams and the slab is taken equal to 24 kN/m3, while a
distributed dead load of 2kN/m2 due to floor finishing and
partition walls is also considered. The live load is assumed to
According to step (ii) of the proposed calibration procedure,
pairs of “SA(T1,5%)-crack width” are obtained using the IDA
curves defined with the detailed finite element models:
(2.96(g)-0.05cm,
6.34(g)-0.1cm,
10.35(g)-0.3cm
and
14.20(g)-0.4cm) for the 3D test example.
Following step (iii) of the calibration procedure, the IDA
procedure is performed over the set of 100 records
implementing the beam-column numerical simulation for the
structural elements. Combining the pairs “SA(T1,5%)-crack
width” obtained as described previously in step (ii) of the
procedure with the “SA(T1,5%)-DI” IDA curves defined with
the beam-column numerical models in step (iii), the crack
openings are correlated with the corresponding damage
indices.
Nonlinear static or dynamic analysis needs a detailed
simulation of the structure in the regions where inelastic
deformations are expected to develop. In order to consider the
inelastic behavior either the plastic-hinge or the fiber
approach can be adopted. In our implementation, the inelastic
cyclic response, for the beam-column simulation, is modelled
with distributed plasticity (fiber approach) which has certain
advantages compared to the concentrated plasticity models
(plastic hinge approach) [26]. According to the fiber
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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
approach, each structural element is discretized into a number
of integration sections restrained to the beam kinematics, and
each section is divided into a number of fibers with specific
material properties. Every fiber in the section can be assigned
to different material properties, e.g. concrete, or reinforcing
bar material properties, while the sections are located at the
Gaussian integration points of the elements. The main
advantage of the fiber approach is that every fiber has a
simple uniaxial material model allowing an easy and efficient
implementation of the inelastic behavior. All dynamic
analyses, implementing the beam-column numerical models,
are performed using the OpenSEES [27] simulation platform.
A bilinear material model with pure kinematic hardening is
adopted for the steel, while geometric nonlinearity is also
taken into account. In order to simulate the concrete the
modified Kent-Park model [28] is applied, where the
monotonic envelope of the concrete in compression follows
the model of Kent and Park [28] as extended by Scott et al.
[29] and it is implemented in Concrete02 model of OpenSEES
[27]. This model was chosen because it allows for an accurate
prediction of the demand for flexure-dominated RC members
despite its relatively simple formulation. The transient
behavior of the reinforcing bars was simulated with the
Menegotto-Pinto model [30] and it is implemented in Steel02
model of OpenSEES [27].
Park & Ang
18
18
16
16
14
14
12
12
SA(T1,5%) (g)
SA(T1,5%) (g)
Maximum interstorey drift
10
8
6
10
8
6
4
4
single records
MSDA 16%,84% fractiles
MSDA median
2
0
0
0.5
1
1.5
2
2.5
3
3.5
single records
MSDA 16%,84% fractiles
MSDA median
2
0
4
0
0.2
0.4
0.6
0.8
max (%)
(a)
16
16
14
14
12
12
SA(T1,5%) (g)
SA(T1,5%) (g)
1.4
1.6
1.8
10
8
10
8
6
4
4
single records
MSDA 16%,84% fractiles
MSDA median
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
single records
MSDA 16%,84% fractiles
MSDA median
2
0
2
0
0.2
0.4
0.6
0.8
DIKRL
1
1.2
1.4
1.6
1.8
2
DICMS
(c)
(d)
Max softening
16
16
14
14
12
12
SA(T1,5%) (g)
18
10
8
6
ACKNOWLEDGMENTS
This work has been supported by the European Research
Council Advanced Grant “MASTER–Mastering the
computational challenges in numerical modeling and optimum
design of CNT reinforced composites” (ERC-2011ADG_20110209).
10
8
6
4
4
single records
MSDA 16%,84% fractiles
MSDA median
2
0
0.2
0.4
0.6
0.8
1
DIms
1.2
CONCLUDING REMARKS
In this study a numerical calibration procedure is proposed.
The proposed procedure is used to calibrate damage indices
for different damage levels is performed by means of multiple
nonlinear dynamic analyses performed in different earthquake
hazard levels of increasing intensity. The first mode spectral
acceleration representing the seismic demand was used for
calibrating the damage indices implemented in two levels. In
the first level, pairs associating the first mode spectral
acceleration with crack openings where defined by means of a
single IDA curve implementing detailed finite element models
with solid elements and a single artificial accelerogram
generated on the basis of the mean response spectrum of 100
natural records In the second level, 100 natural records are
scaled in multiple hazard levels and the 16%, 50% and 84%
fractile curves for each damage index are obtained. The 100
seismic demand-damage index IDA curves are obtained
implementing crude finite element models with beam-column
elements. Combining the results of the two levels, the
calibrated values of the damage indices are defined based on
the 50% fractile with reference to the seismic demand. In
addition to the calibrated values of the five damage indices
defined on the basis of the 50% fractile IDA curve, the
coefficient of variation for each damage index and for each
damage state is defined.
For this purpose five damage indices, belonging to the
category of maximum deformation and cumulative damage,
are considered and they are calibrated numerically with
reference to different levels of structural damage. The
calibrated values can be used in the framework of
performance-based design in order to assess the structural
behavior under seismic actions. Furthermore, an additional
goal of this study is achieved by identifying the damage index
less influenced to the seismic action. This damage index is
considered to be the most reliable one for assessing the
structural behavior, among those considered in this study.
Based on the numerical investigation performed for the five
damage indices employed in this study the Park & Ang
depicts the lowest coefficient of variation for all limit states
considered, while similar performance depict Kunnath,
Reinhorn and Lobo along with the Chung, Meyer and
Shinozuka local damage indices.
Final softening
18
1.4
1.6
1.8
single records
MSDA 16%,84% fractiles
MSDA median
2
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
DIfs
(e)
(f)
Figure 3. 3D test example-IDA curves for the six damage
indices
The 16, 50 and 84 percentile “SA(T1,5%)-DI” IDA curves
of the 3D test example for all damage indices are provided in
Figures 3(a) to 3(f). DIs have a similar trend in pairs; in
particular the pairs DIPA-DIKRL and DIMS-DIFL have a similar
trend, while DICMS show a different trend. Based on the pairs
“SA(T1,5%)-crack width”.
286
2
Chung, Meyer & Shinozuka
18
6
SA(T1,5%) (g)
1.2
(b)
Kunnath, Reinhorn & Lobo
0
1
DIPA
18
0
4
2
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