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Dynamic analysis of single-degree-of-freedom systems (DYANAS):
a graphical user interface for OpenSees*
Georgios Baltzopoulos1, Roberto Baraschino1,2, Iunio Iervolino1, Dimitrios Vamvatsikos2
1 University of Naples Federico II, Department of Structures for Engineering and Architecture, Via Claudio 21, Naples, Italy.
2 National Technical University of Athens, Department of Civil Engineering, Heroon Polytechneiou 9, Athens, Greece.
* Preprint version of the article published in Engineering Structures (2018), 177: 395-408.
Abstract. Non-linear dynamic response of SDOF systems enjoys widespread application in earthquake engineering,
sometimes as a testing ground for cumbersome analytical procedures, but often as a direct proxy of first-mode-dominated
structures, within the family of simplified, pushover-based methods for seismic structural assessment and/or design. This
article presents DYANAS, a MATHWORKS MATLAB®-based graphical user interface that uses the OpenSees finite
element platform to perform nonlinear dynamic analysis of single-degree-of-freedom (SDOF) oscillators. The scope of this
open-source, freely distributed software is to serve as a tool for earthquake engineering research. The main advantages offered
by the DYANAS interface are ease in the definition of the required analysis parameters and corresponding seismic input,
efficient execution of the analyses themselves and availability of a suite of convenient, in-built post-processing tools for the
management and organization of the structural responses. The types of dynamic analysis frameworks supported are
incremental, multiple-stripe and cloud. Simultaneous consideration of pairs of uncoupled dynamic systems gives the
possibility for intensity measures to refer to bidirectional ground motion. In the paper, an outline of the types of dynamic
analysis frameworks typically used in performance-based earthquake engineering is provided, followed by a detailed
description of the software and its capabilities, that include an array of post-processing tools. In order to properly place this
software tool within its natural performance-based earthquake engineering habitat, some example applications are provided
at the end of the paper.
Keywords: performance-based earthquake engineering, pushover analysis, incremental dynamic analysis, multiple-stripe
analysis, cloud method.
1. Introduction
Earthquake engineering sees widespread usage of simple structural systems, such as single-degree-of-
freedom (SDOF) yielding oscillators, as a means of providing insights into the response of more complex
engineered systems, such as buildings or bridges. Although this type of simplification was initially
motivated by the lack of computational resources for performing dynamic analysis of larger numerical
models, in recent years the trend remains - only for different reasons. Advances in performance-based
earthquake engineering (PBEE, e.g., Cornell and Krawinkler, 2000) and seismic structural reliability
(e.g., Pinto et al., 2004) have brought forth the need for an adequate probabilistic representation of
structures’ post-elastic behavior during earthquake excitation. Due to the inherently stochastic nature of
this phenomenon, mainly (but not solely) attributable to the so-called record-to-record variability of
seismic response (Shome and Cornell, 1999), such a probabilistic description by analytical means may
require a prodigious number of dynamic analyses, putting a strain even on modern computational power.
For this reason, methods that employ SDOF-level approximations of structural systems have been duly
preserved in the arsenal of PBEE and have continued to undergo refinement and development.
Looking back, the dynamic response of yielding SDOF oscillators to base acceleration has been in the
limelight of earthquake engineering research since its early steps (Veletsos et al., 1965) and remained
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there throughout its evolution, as simplified methods for seismic assessment and design of buildings
appeared that were reliant on pushover analysis to define an equivalent, substitute SDOF structure (Vidic
et al., 1994; Fajfar and Gašperšič, 1996; Chopra and Goel, 1999; Miranda, 2001; FEMA, 2005; Priestley
et al., 2007; Guerrini et al., 2017). In more recent years, with the PBEE paradigm gaining an ever-steadier
foothold in the earthquake engineering community, new SDOF-based procedures began to surface that
additionally sought to map the probabilistic distribution of seismic structural response (e.g., Vamvatsikos
and Cornell, 2006; Fragiadakis and Vamvatsikos, 2010; Baltzopoulos et al., 2017). Other PBEE-related
contexts that employed yielding SDOF oscillators as structural proxies, include the direct use of inelastic
displacement as a measure of shaking intensity (Tothong and Cornell, 2006) and the critical scrutiny of
prevalent ground motion record selection and modification practices for conducting response history
analysis of structures (Iervolino and Cornell, 2005; Kwong et al., 2015).
Some of the issues that permeate all of the aforementioned cases are: the shape of the backbone curve of
the SDOF system, the hysteretic law it follows during cyclic loading and the potential presence of
strength and/or stiffness cyclic degradation in the hysteresis. The force-displacement relationship
describing response to monotonic static loading, i.e., the backbone curve, is typically an idealization of
the pushover curve obtained analytically for the corresponding structure. This idealization can be a
simple bilinear shape, or it can also include a descending, negative stiffness branch and even a residual
strength plateau. The appearance of negative stiffness in a structure’s pushover may be due to P-∆ effects
or material strength degradation (often both) and the importance of its representation on the equivalent
SDOF’s backbone has been strongly advocated by past research (Ibarra and Krawinkler, 2005; Adam
and Jäger, 2012).
With regard to the hysteretic behavior that ought to characterize a substitute SDOF system in order to
best represent global response of the corresponding structure, there was a shift in modern practice, from
the simple, kinematic hardening rules that saw extensive use in the past, towards phenomenological,
evolutionary hysteretic rules (e.g., Ibarra et al., 2005) that are more representative of real structural
behavior. Various proposals exist in the literature in this regard, typically based on experimental data
that refer to specific structural typologies: for instance, the studies of Takeda et al. (1970) for reinforced
concrete, Graziotti et al. (2016) for unreinforced masonry and Lignos and Krawinkler (2011) for steel
structures, are mentioned. Typically, the same model used for a structure’s numerical simulation at the
element level (especially in cases of concentrated plasticity modelling) is also assigned to the equivalent
SDOF oscillator for that structure. However, it should be noted that, in the case of parametric
phenomenological modelling of cyclic degradation effects, experimental model calibration at the element
level may not be adequately representative of the entire structure’s global behavior and a dedicated study
may be needed (e.g., Kampenhuber and Adam, 2014).
The present article introduces a new earthquake-engineering-oriented software: DYANAS, a
MATHWORKS MATLAB®-based graphical user interface (GUI) that expedites the definition of
inelastic SDOF systems in the OpenSees (Open System for Earthquake Engineering Simulation;
McKenna, 2011) finite element platform, streamlines the execution of dynamic analyses under suites of
base-acceleration time histories and allows for methodical processing of the final results. The analysis
method and post-processing aspects of the software are focused on facilitating dynamic analysis schemes
that draw on the PBEE paradigm. The graphical interface components of the software run in MATLAB
environment. Interaction with OpenSees is performed by manipulating Tcl/Tk scripts (Tool Command
Language a general scripting language that has been extended with OpenSees-specific commands;
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Ousterhout and Jones, 2009) via MATLAB functions, in a process that will be described in the following
sections.
OpenSees is an open-source, object-oriented finite element platform developed for PBEE. Since its
inception, OpenSees has seen extensive use for numerical analysis in structural, geotechnical and seismic
reliability problems (e.g., Scott and Haukaas, 2006). In the past, various GUIs have been built around
OpenSees, intended to facilitate the definition of numerical models and subsequent running of analyses.
Such examples are the Build-X software (Psyrras and Sextos, 2018) and the GID interface (Papanikolaou
et al., 2017) that are both intended for aiding the definition of multi-DOF building models, as well as the
web platform of Dolšek et al. (2015) and the II-DAP interactive interface (Elkady and Lignos, 2018) that
operate on the SDOF level. DYANAS is a new addition to this list and is, primarily, a PBEE tool that
incorporates a multitude of functions oriented towards efficient seismic analysis of simple structures
(e.g., Vamvatsikos and Cornell, 2004).
The software presented in this article, DYANAS, is freely available for research purposes as a software
package (http://wpage.unina.it/georgios.baltzopoulos/software/software_page.html) and is also
distributed as open-source code, obtainable at https://github.com/georgebaltz/SDOF-OSEES. DYANAS
allows users to easily define yielding SDOF oscillators with classical viscous damping, piece-wise linear
monotonic backbones and a choice of six evolutionary or hardening hysteretic rules that are already
implemented in OpenSees (to follow). Subsequently, the GUI gives users the opportunity to perform
non-linear dynamic analyses to large sets of accelerometric input, supporting incremental dynamic
analysis (IDA; Vamvatsikos and Cornell, 2002), multiple-stripe analysis (MSA; e.g., Jalayer and Cornell,
2009) and cloud analysis (e.g., Jalayer et al., 2015). For each analysis session, users may define a single,
or two uncoupled, SDOF oscillators. In the latter case, these can be run simultaneously, using
bidirectional ground motion records. Finally, the software has the capability to post-process results,
graphically represent them for the users and allows exporting figures and data files for further
elaboration. The remainder of this paper is organized in the following fashion: first, an outline of the
dynamic analysis methods prevalent in PBEE and supported by the GUI is given. Subsequently, a
description of the software’s function, structure and capabilities is provided. Then, a section dedicated
to a few illustrative applications that showcase specific aspects of the software’s utility as a PBEE tool
are to be found, followed by some closing remarks and discussion.
2. Dynamic analysis methods for PBEE
The primary use of dynamic structural analysis in PBEE is as a means of quantifying the ability of a
structure to meet certain performance objectives, in a probabilistic framework. An understanding of this
can be obtained through Eq. (1), that looks at part of the PEER framing equation (Cornell, 2004) to
provide the annual rate of a structure’s failure to meet some performance objective,
f
λ
, which is a
measure of seismic risk.
f f im
im
λ P EDP edp IM im
 
 
 
(1)
Eq. (1) is based on two fundamental variables, the so-called engineering demand parameter (EDP) and
the seismic intensity measure (IM) that will be both assumed scalar, for simplicity. In that case,
im
λ
represents the annual rate of exceeding a specific value
im
at the site of the structure and is therefore a
measure of seismic hazard (McGuire, 1995). Finally, the conditional probability that structural response
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given a certain IM level, exceeds the performance objective threshold,
edp
, f
P EDP edp IM im
 
 
 
is typically termed a fragility function (sometimes simply written as the probability of failure given IM,
P f IM im
 
 
). EDP is a generic term used for any measure of structural response of engineering
significance, that can be used to check or determine if the structure has failed to meet a given performance
objective due to, for example, excessive stress or deformation. Examples of often-used EDPs for building
structures are inter-storey drift ratio or roof-level displacement. In this context, failure (sometimes also
termed “exceedance of a limit state”) can be expressed as the exceedance of a predefined threshold EDP-
value. On the other hand, an IM can be some direct instrumental measure of ground motion amplitude,
such as peak ground acceleration (PGA), or a quantity of elastic response, such as spectral pseudo-
acceleration at various periods,
Sa T
.
While the concept of structure-specific fragility functions sees extensive use in PBEE and seismic risk
analysis, in order to determine such a functional relationship between EDP and IM as
f
P EDP edp IM im
 
 
 
by analytical means, engineers may need to resort to a number of nonlinear
dynamic analyses of a numerical model of the structure, according to some methodology that maps the
distribution of EDP for various IM levels. The methods used most frequently to obtain this EDP-IM
mapping for seismic response are incremental dynamic analysis, multiple-stripe analysis and cloud
analysis. The principal objective of this software is to streamline and facilitate the execution of these
types analyses using suites of ground acceleration records, when the structural model adopted is an SDOF
simplification of the original structure (or an actual SDOF system, naturally). For this reason, the three
aforementioned dynamic analysis methodologies (plus a variant of IDA) are briefly outlined below,
whilst other issues mentioned in this first part of the section, such as seismic fragility and hazard will
receive little more attention, by virtue of being beyond the scope of the software presented.
2.1 Incremental dynamic analysis
IDA consists of running a series of dynamic analyses for a non-linear structure, using a suite of
accelerograms that are progressively scaled upwards in amplitude in order to represent a broad range of
IM levels. Structural response, in terms of some EDP, is being continuously registered at each IM level;
thus, for each acceleration record, a continuous EDP-IM relationship is obtained, termed an IDA curve
(Figure 1a). A common way of summarizing IDA results for the entire suite of records, is to calculate
the 16,50 and 84% fractile IDA curves of EDP given IM or vice-versa. As can be seen in the figure,
while individual-record IDA curves can be non-monotonic, the fractile curves generally are. In fact, for
structures susceptible to dynamic instability, when a record is scaled up to levels that can induce such
behavior, the IDA curve tends to become horizontal, indicating almost-infinite increase in EDP for small
IM increments; thus IDA flat-lines are taken to indicate structural collapse (for further details the
interested reader is referred to Vamvatsikos and Cornell, 2002). It is mentioned in passing that IDA has
already found its way into guidelines for practicing engineers that espouse PBEE principles (e.g., FEMA-
P695, 2009b).
2.2 Multiple-stripe analysis
MSA, like IDA, has the objective of calculating EDP-responses of a structure at various, increasing IM
levels. The difference from IDA is that MSA does not necessarily resort to scaling the same suite of
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accelerograms to reach these IM levels, but may instead use different sets of (scaled or unscaled) records
at each IM level (e.g., Jalayer and Cornell, 2009). Ideally, these record sets should be selected to reflect
site-specific seismic hazard at each IM level, thus rendering the calculated seismic structural demand
hazard-consistent (e.g., Lin et al., 2013). A schematic example of MSA results is given in Figure 1b,
where it can be seen that EDP response samples, sometimes termed “EDP-stripes”, have been obtained
for six IM levels (note that in the context of IDA and due to the continuous nature of IDA curves, apart
from EDP-given-IM, one may also calculate “IM-stripes” for a given EDP value).
Figure 1 Illustrating the dynamic analysis methods most frequently used in PBEE: thirty IDA curves of a yielding SDOF
oscillator and corresponding 16,50,84% EDP-given-IM fractile curves (a); EDP responses of a non-linear structure at six
IM levels obtained via MSA (b); logarithmic-scale scatter plot of EDP-IM responses obtained via cloud analysis and
corresponding ordinary least squares regression line (c).
2.3 Cloud analysis
Cloud analysis gets its name from the fact that it uses sets of unscaled acceleration records, meaning that,
typically, only a single record will correspond to each IM level (besides it being not-very-likely to find
many records with, say, the exact same PGA or
Sa T
value, the very nature of this method calls for an
even spread of intensity values within the range of interest). This results in a “cloud” of points in the
EDP-IM plane, such as the scatter-plot shown in Figure 1c. With the cloud method, a probabilistic
relation between EDP and IM can be obtained by means of linear regression and its assumptions of
homoscedastic, Gaussian residuals (e.g., Jalayer et al., 2015). Regression of cloud analysis EDP-IM data
can find application in some simplified seismic reliability methods; the interested reader is referred to
Cornell et al. (2002) for further details.
2.4 Back-to-back incremental dynamic analysis
Besides traditional IDA, an extended version thereof has also been suggested in the past (Luco et al.,
2004). This extension of IDA was intended to provide a probabilistic description of seismic response for
structures that have already been damaged by a mainshock earthquake (MS) and are susceptible to the
effect of aftershocks (AS) before any repairs can take place. In this type of analysis, henceforth referred
to as back-to-back IDA, the analyzed structure is first subjected to an initial set of accelerograms, let us
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call them the MS record set, each scaled to produce the same predefined EDP level. At the end of each
single-run MS analysis, a damaged incarnation of the structure has been produced; this is allowed to
revert to at-rest conditions and is subsequently subjected to a second set of records, let us call these the
AS set, that are scaled upwards in the traditional IDA manner. Figure 2 provides a schematic
representation of this procedure. The end result, as in traditional IDA, is a set of continuous EDP-IM
curves, the difference being that these curves no longer describe the seismic behavior of the original
structure, but that of the structure that has been subjected to a specific damaged state. Also note, that the
MS-AS characterization of the two record sets employed in this discussion is purely conventional and
was used for designating the order with which the two shocks are considered in the analysis, without
implying that they must be part of the same seismic sequence.
Figure 2 Schematic representation of back-to-back IDA: a “mainshock” record acting on a structure in pristine condition
is scaled until a limit state threshold edpf is reached, then IDA is performed for a set of “aftershock” records, acting on the
structure damaged by the mainshock record; the procedure is repeated for a predefined number of “mainshock”
accelerograms, each followed by a set of “aftershock” accelerograms.
3. Software structure and graphical user interface
DYANAS is structured around two main pillars: the various MATLAB-coded pre- and post-processing
tools that are incorporated into the GUI and the suite of parametric scripts that run directly on the
OpenSees platform. In order to run an analysis defined by the user via the MATLAB GUI, OpenSees
must be provided with Tcl/Tk scripts that contain definition of the numerical model, loading, and the
analysis’ operational parameters, including what response information to save. These Tcl/Tk scripts are
divided into two categories: parametric and definition files. Parametric scripts contain all the information
necessary for model definition and execution of analysis in OpenSees, as functions of externally-defined
placeholder variables or as hard-coded analysis parameters; these files remain unchanged during any
analysis session of the software (unless the user decides to make a direct, outside intervention of course).
On the other hand, the definition files are constantly over-written by the GUI’s pre-processing tools,
according to user input and system definition. Once OpenSees concludes a packet of user-requested
analyses, output files are created, that are subsequently parsed by the GUI’s post-processing MATLAB
routines for further elaboration of the raw results. This procedure is schematically depicted in the
flowchart of Figure 3.
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The pre-processing, (system and analysis definition) functionalities of the GUI can be further split into
two main groups: definition of the SDOF system characteristics, such as dynamic properties, backbone
curve and the rules governing hysteretic behavior, and definition of the type of analysis, including
definition of controlling parameters and handling of dynamic input. The post-processing tools, on the
other hand, enable organizing, plotting and extracting specific subsets of the results, in various common
formats, useful within the PBEE framework. The remainder of this section describes the salient features
of the functionalities outlined above.
Figure 3 Operational flowchart of the DYANAS GUI’s interaction with OpenSees
3.1 Definition of oscillator characteristics, engineering demand parameter and seismic intensity
measure
The first step in any new analysis session is the definition of the system to be analyzed, that may either
be an SDOF oscillator or two uncoupled simple oscillators. In the latter case, two separate SDOF systems
are defined and analyzed simultaneously, without any interaction occurring between them. This can be
useful in the case of regular, symmetrical structures, for which equivalent SDOF systems may be defined
via pushover in both principal directions (Senevinatra and Krawinkler, 1997).
In all cases, the SDOF system definition requires the assignment of dynamic characteristics, such as
period of natural vibration
T
and viscous damping ratio
ν
(modelled as mass-proportional), backbone
curve and hysteretic rule. The backbone curve corresponds to a piece-wise linear idealization of the force-
displacement response of the system to monotonic loading and is defined by the yield strength and
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displacement,
y
F
and
y
δ
respectively, and up to four parameters for a quadrilinear case, shown in Figure
4.
Figure 4 Definition of non-dimensional parameters for the characterization of an SDOF system’s piece-wise linear
backbone curve (a); backbone curve with descending branch, residual strength plateau and peak-oriented hysteretic loops
(b); bilinear, kinematic hardening system (c); trilinear backbone exhibiting peak-oriented hysteretic behavior with
moderate cylcic strength degradation.
These are the hardening slope
h
(positive ratio of post-yield stiffness to elastic stiffness), the capping-
point ductility
c
(point where loss of strength with increasing deformation begins), the post-capping
slope
c
(negative slope corresponding the ratio of the negative post-capping stiffness divided by the
initial elastic stiffness), the height of the residual strength plateau
p
r
(ratio of residual strength divided by
yield strength) and the fracture ductility
f
(point corresponding to sudden, complete loss of strength),
with ductility being response displacement normalized by yield displacement,
y
 
and strength
ratio
R
as force normalized by yield strength
y
R F F
. A sample of the GUI’s dialogue windows
during the definition of backbone and other properties is provided in Figure 5.
As can be seen in the figure, the software permits the definition of asymmetric backbone curves, with
c
,
f
and
y
F
being allowed to differ in the two directions, while maintaining the same elastic stiffness.
Regarding the choice of hysteretic constitutive law, several options that have been implemented in
OpenSees in the past are available (Ibarra and Krawinkler, 2005; Ibarra et al., 2005; Lignos and
Krawinkler, 2011; Altoonash and Deierlein, 2004; Lowes et al., 2004; Perzyna, 1971). These are divided
into two broad categories of peak-oriented and hardening hysteretic rules and some of them allow for
evolutionary behavior, including cyclic strength and stiffness degradation. It should be noted that
association of a backbone curve with a descending branch is meaningful only for the peak-oriented
hysteretic behavior group; in such cases that include softening branches, in-cycle strength degradation
can also emerge during dynamic response (whenever cycles exceed the
c
limit see also FEMA-
 
1 1
h c
 
c
μ
f
μ
p
r
1
1
h
c
y
F
RF
y
μ
R
μ
R
μ
R
μ
(
)
a
( )b
( )c( )d
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P440A, 2009a). By means of the “quasi-static cyclic” analysis type, the software offers the user a means
to visualize the effect of each hysteretic rule on displacement-controlled cyclic loading.
Figure 5 Main GUI window and subsidiary dialogue windows during definition of a SDOF oscillator’s dynamic
characteristics, backbone curve (asymmetrical in this case) and hysteretic model.
The second preliminary step, prior to proceeding to the analysis, is to select the EDP and IM that will be
employed. For such simple structures as these SDOF oscillators, the choice of EDP is limited to the peak
transient displacement in either horizontal direction,
X
δ
and
Y
δ
, the residual displacements in both
directions and the demand over capacity ratio,
D C
. The
D C
ratio expresses, in non-dimensional form,
how close the structure comes to exhausting a threshold EDP. It also provides a means of defining an
EDP that takes response in both directions into account, according to the definition in Eq. (2), which is
valid for all converging runs, where
f
X
δ
and
f
Y
δ
are the “failure” displacements in the corresponding
directions (i.e., the EDP thresholds).
f f
X Y
X Y
D C max
δ δ ,δ δ
(2)
Although the actual choice of IM can be deferred until the analysis definition phase (the default option
being
Sa T
at the X-direction oscillator’s vibration period), in the case of bi-directional ground motion
(i.e., definition of two SDOF systems) the user should also determine how that IM is to be calculated:
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for example,
Sa T
can be taken as the maximum between the values of the two components,
max
Sa T
, or as the geometric mean of the two,
gm
Sa T
.
3.2 Running IDA and back-to-back IDA
Due to the particular shape of individual IDA curves, when attempting to adequately represent them by
scaling all records at a fixed, constant step in terms of IM, the number of dynamic analyses that may be
needed can prove overly time-consuming, even in the case of SDOF systems. DYANAS seeks to
optimize required analysis time, by incorporating a MATLAB-programmed implementation of the
“hunt-and-fill” algorithm (Vamvatsikos and Cornell, 2004;
http://users.ntua.gr/divamva/software/bundle_runIDA.zip), whereby the number of runs for each record
is determined on the fly during analysis. According to this algorithm, after a few test-runs at characteristic
IM levels, the software begins to determine the scale factor of the next run based on the history of
previous results, resulting on a set of EDP-IM points on the curve – black dots in Figure 6 – that may be
closer or farther apart between curves, in an attempt to use approximately the same number of runs for
all curves, regardless of flat-line height. Then, the complete IDA curve (intermediate points colored
segments in Figure 6) is obtained by means of piece-wise interpolation between these calculation points
on the IM-EDP plane (http://users.ntua.gr/divamva/software/bundle_postIDA.zip). Both spline and
linear interpolation are possible, with the former being the default option in DYANAS, as it results in
smoother IDA curves.
Figure 6 Conversion of IDA curves from five records between IMs: from
1 25 s
Sa T . (a) to PGA (b). The highlighted
calculation point on one of the five records used for this example (event, year and recording station reported in the legend)
shows the re-mapping of that point between the two panels, the PGA-to-spectral-ordinate ratio for that record being
2 911 63
. .
. Herein the hunt-and-fill algorithm was stopped prior to evening out the total number of runs per curve, when
each flatline was adequately captured.
CA Santa Susana ETEC FF
Coalinga(1983)
Northridge(1994)
Eastern Honshu(2011)
Darfield(2010)
Niigata(2004)
Parkfield Fault Zone 6
FKS015
Darfield High School
NIG014
1 63
a
S . g2 91PGA . g
(a) (b)
S
a
T
(
=
1
.
2
5
s
)
[
g
]
PGA [g]
11
An additional advantage, provided by this computational strategy, is the possibility for nigh-
instantaneous change of IM, once a set of IDA curves has been obtained. This IM-change operation can
prove useful in a variety of situations, such as when a set of distinct structures must all have their seismic
vulnerability functions expressed in a common IM (Kazantzi and Vamvatsikos, 2015) or when the same
requirement occurs for the two principal orientations of a single structure (Kohrangi et al., 2017) or
simply for studying the relative merits of choosing one IM over another (Luco and Cornell, 2007). This
is also demonstrated in Figure 6, where the conversion of a set of five IDA curves from spectral pseudo-
acceleration,
Sa T
, at period
1 25 s
T .
to PGA is shown. Given that each explicit calculation point
from the trace-IDA implementation, shown in the figure as a black dot upon the spline-interpolated curve,
corresponds to a specific scale factor applied to that record, each EDP value at any of these points can
be reassigned to the value that the new IM assumes on the record’s scaled spectrum. After this re-
mapping of the calculation points to the new IM (in this example PGA) is complete, spline interpolation
based on the new IM-EDP points provides the converted IDA curves.
The same computational strategy is also followed in the case of back-to-back IDA, with the only
difference being in the management/definition of seismic input (to follow). At the conclusion of either
of these types of analysis (IDA or back-to-back IDA) a set of post-processing options are available to the
user: change of IM or EDP, extraction of IM or EDP stripes that can be saved into MATLAB-variable
or text file formats and saving the analysis results for later use. Change of EDP offers a choice between
X
,
Y
, residual displacements and
D C
ratio. Switching EDP is trivial, as all necessary EDP responses
are recorded and saved during every single run. Conversion to different IMs, on the other hand, follows
the re-interpolation procedure outlined earlier. For single-component ground motion, change of IM
entails switching to a spectral ordinate at a different period
T
and/or different viscous damping ratio
ν
,
with defaults being the X-direction SDOF’s period of natural vibration and five percent, respectively. In
the case of bidirectional motion, the definition of the IM with respect to the two horizontal acceleration
components can also be changed among the already-mentioned alternatives being:
Sa T
of either single
component,
max
Sa T
or
gm
Sa T
.
Finally, the post-processing tools of the GUI can provide the extraction of text or MATLAB files
containing IDA results organized as either vectors of EDP responses given IM (“EDP stripes”) or IM
causing exceedance of a specific EDP value (“IM stripes”). Such output formats may come handy when
attempting to fit a parametric probabilistic model to a fragility function (the interested reader is referred
to Baker, 2015, for the nuances of fitting such a model in the IM- and EDP-based case). For IM stripe
extraction (example provided in Figure 7), the software finds the intersection of each spline-interpolated
IDA curve with the vertical line passing through the user-defined EDP threshold; the lowest IM value is
returned in the case of non-monotonic IDA curves that intersect the line at more than one point. For EDP
stripes, the user defines a vector of desired IM values and the software returns the intersections with the
corresponding horizontal lines; for IDA curves that have already flat-lined below a given IM level, the
information that the structure has collapsed is returned.
12
Figure 7 Extraction of an IM stripe from a set of IDA curves. The EDP threshold is defined in the homonymous dialogue
window and the intersections with the curves are displayed in the main GUI lower-right window.
3.3 Running multiple-stripe and cloud analysis
To conduct MSA, the user must define all IM levels for which EDP responses are needed beforehand,
via the dedicated dialogue window of the GUI (Figure 8), and subsequently assign a set of records to
each IM stripe. Each stripe must be assigned a specific IM value and all records in the stripe will be then
scaled to match that value (exception to that rule is cloud analysis, see below). Internally, analysis is run
using the same MATLAB and OpenSees (Tcl/Tk) routines as for IDA, with the difference that record
scaling is completely pre-determined at the start of the analysis and a different seismic input file has to
be read at each single run. Note that, in the case of MSA, a change of IM requires re-running the analysis
from scratch (unlike IDA see above) while EDP still can be changed at will after completion of the
analysis.
In the present GUI application and for reasons of convenience, cloud analysis is also managed via the
MSA window, as one may imagine it as a single-record-per-stripe MSA. The user simply has to define
a single IM stripe with a nominal intensity value, which is disregarded, and check the “do not scale
records” box; in that case the software will realize that could analysis is in order and will display the
results accordingly, as shown in Figure 8. For both MSA and cloud analysis, EDP stripes can be exported
in text- or MATLAB-file format, reducing to IM-EDP pairs in the latter case.
13
Figure 8 Main GUI window snapshots displaying the results of MSA (upper left panel) and cloud analysis (upper right
panel); MSA dialogue window, showing the setting-up of an analysis at ten levels of seismic intensity (lower panel). All
records in each stripe will be scaled to match that stripe’s predefined IM level, unless the “Do not scale” checkbox is
ticked, in which case cloud analysis runs.
3.4 Seismic input
Seismic input can be defined by selecting sets of ASCII files, each containing a recorded acceleration
time-history. The software can parse two standard accelerograms formats, namely the Pacific Earthquake
Engineering Research NGA database format (Ancheta et al., 2014) and the Engineering Strong Motion
database format (Luzi et al., 2016). In either case, the necessary information for running the analyses are
extracted/computed from the selected files (units, duration, sampling rate, elastic response spectrum) and
passed on to the corresponding MATLAB routines, while the actual accelerometric data are written into
temporary files suitable for being read by OpenSees. As an alternative to these standard formats, the user
may use simple “.txt” ASCII files that contain a single column of acceleration values and will be inquired
by the software to provide units and sampling rate.
The software, at this stage, is distributed equipped with two folders that already contain two sets of
accelerograms: one is the suite of thirty single-component records used for calibrating the SPO2IDA tool
(Vamvatsikos and Cornell, 2006) and the other is the twenty-two bidirectional ground motion “far-field”
14
set of FEMA-P695 (FEMA, 2009b). Both record sets consist of accelerograms recorded on firm soil,
predominantly during California events of magnitude six or greater.
For IDA or cloud analysis, a single set of records must be selected by the user, while, in the case of MSA,
different sets may be assigned to each predefined IM level (stripe), all containing the same number of
records. Cloud analysis in particular, is defined via the MSA control panel, by suspending record scaling,
as discussed above. Back-to-back IDA on the other hand, requires two sets of records: the first set,
referred to as the MS set, is used to perform a preliminary IDA analysis, that serves to determine the
scale factors necessary to induce the predefined EDP value, associated with some structural limit state,
across the entire MS record set. During any type of analysis, each record is followed by an appendix of
zero acceleration entries for a duration of five times the elastic period of the system, intended to provide
time for the residual velocity at the end of the excitation to be damped down to negligible values. This is
performed internally in OpenSees, without tampering with the records, so that the next shock will find
the system at rest. This procedure is implemented to achieve increased precision in the calculation of
residual displacements, which is especially important in the case of back-to-back IDA.
Figure 9 Selection of accelerograms for running back-to-back IDA; the MS record set list has been already filled and the
record-selection dialogue window is open for designating the AS set.
The second set of accelerograms, referred to as the AS set, is used to run an IDA that always follows one
of the MS records (the user is actually free to use the same set to represent both MS and AS if one so
desires, or two sets of different size). In other words, each scaled AS record acts on a structure that has
already experienced the predefined transient maximum EDP value and has had some time to come to
quasi-rest conditions. In Figure 9 the analysis-parameter definition panel for back-to-back IDA is shown,
15
with the AS acceleration-file selection-window open. The user is additionally given the option to define
the number of AS IDAs that are going to be run per MS record, ranging from one to the number of
records in the AS set. When the number of IDAs requested per MS is less than the number of AS
accelerograms defined, a random extraction is performed from among the AS records (independent
extractions are performed for each MS record). A final option available to the user for back-to-back IDA,
is that one may request that all MS-AS pairs be created so that no AS record is ever used twice, provided
that an adequate number of records has been provided.
In the case of bidirectional ground motion, record assignment proceeds as in the single-component-of-
motion case, with the difference that all record sets must be even in number, so that they may be divided
in two halves and re-assigned to each of the two directions. Pairing of the records follows the ASCII
dictionary alphabetical order of the filenames, by assigning any two consecutive records first in the X
and then the Y direction. This is intended to take advantage of the fact that, in most strong ground motion
databases, accelerograms recorded by two streams of the same station will be typically saved under
filenames differing by only a few characters.
4. Illustrative applications
In this section, some applications are provided as examples, in order to illustrate DYANAS’ functionality
in more practical terms. The first example uses a pair of relatively regular and symmetrical buildings, for
which equivalent SDOF structures are considered in each principal direction, to provide applications of
IDA and MSA. The second example deals with cloud analysis, while the third application presents the
use of back-to-back IDA. In order to consolidate the usefulness of the GUI as a PBEE tool, some of these
examples involve the calculation of fragility functions for SDOF structures, based on IDA results.
However, it should be clear that the primary scope of the software is to use non-linear dynamic analysis
to provide EDP-IM relationships, one natural use of which is the assessment of seismic structural
fragility.
4.1 Two equivalent SDOF systems representing both principal directions of a building
For this example application, four static pushovers were used, corresponding to the two principal
directions of two code-conforming, reinforced concrete (RC), moment-resisting frame buildings. These
two buildings, a three-storey frame with masonry infills and a six-storey bare frame, were designed
according to the current Italian seismic code and subsequently numerically modelled (Camata et al.,
2017). The scope of these numerical models, which were also used to obtain the pushover curves, was
the analytical assessment of seismic collapse risk implicit in the Italian code’s provisions, via non-linear
dynamic analysis (Iervolino et al., 2017). Considering both buildings in either direction, pushover
analysis was performed using a force distribution according to the modal load of each direction’s
predominant elastic mode shape. Thus, four equivalent SDOF systems with tri- or quadri-linear
backbones were derived from these static pushover curves (for further details, see Suzuki et al., 2018)
and a Clough-type, peak-oriented model without cyclic deterioration (Altoonash and Deirlein, 2004),
was assumed to govern hysteretic response.
The two equivalent SDOF oscillators corresponding to the three-storey infilled frame had vibration
periods of 0.28 and 0.31 s while those of the six-storey bare frame were 1.11 and 1.25 s. As can be seen
in Figure 10, the bare RC frame is characterized by similar maximum attainable base shear and
displacement capacity in both directions. On the other hand, the infilled frame exhibits higher maximum
16
resistance in one direction with respect to the other. This is due to the increased presence of openings in
the masonry infills along the direction parallel to the building’s facade, resulting in lower infill-
contribution to lateral strength in that direction. Furthermore, it can be seen that this building is
characterized by different ultimate displacement capacity per principal direction.
Figure 10 Piecewise-linear idealizations of the RC frames’ static pushovers: three-storey frame with masonry infills,
direction X (a) and direction Y (b), six-storey bare frame direction X (c) and direction Y (d). To the right of the pushovers,
the GUI main window is shown in post-analysis of the six-storey frame.
4.1.1 Incremental dynamic analysis for estimation of collapse fragility
In this exercise, each pair of equivalent SDOF systems, corresponding to a single RC building, was
subjected to IDA using a suite of thirty, two-component acceleration records. These accelerograms were
recorded on firm soil during events of magnitude ranging from 6.1 to 7.6, at distances from the fault
plane 44 km or less and are considered to not exhibit impulsive characteristics due to directivity; during
IDA, the same scale factor was applied to both horizontal components with each pair thus always
maintaining their as-recorded relative amplitudes. In order to be able to express the seismic collapse
fragility of these structures in terms of a common IM, the geometric mean of the two components’
spectral acceleration at one second period,
1.0 s
gm
Sa T , was arbitrarily selected and all IDA curves
were converted to that IM. For each building three collapse fragility functions were calculated, two
curves considering each principal direction X, Y separately and one corresponding to the entire building,
i.e., considering both directions at once. In the first two cases, the EDP was set to
X
δ
or
Y
δ
and IM stripes
were extracted at
f
X
δ
and
f
Y
δ
, respectively. In the third case, the
D C
ratio according to Eq. (2) was
taken as EDP and the IM stripe for
1
D C
was obtained. It follows from Eq. (2) that, if one were to
name failure in the two directions
X
f
,
Y
f
and since failure of the entire structure can be seen as the union
of the two events, 1
X Y
P f IM im P f f IM im P D C IM im
   
   
. In all cases,
lognormal seismic fragility functions are estimated according to Eq. (3), where
f
i
im
represents the i-th
record’s (lowest) scaled IM value causing exceedance of the threshold (i.e., the i-th element of the IM
stripe at that threshold, according to the nomenclature used earlier),
and
are the mean and standard
F
X
[
k
N
]
F
X
[kN]
δ
X
[m]
δ
X
[m] δ
Υ
[m]
δ
Υ
[m]
F
Y
[kN]
F
Y
[kN]
17
deviation, respectively, of the logs of
f
im
,
is the standard Gaussian function and
n
is the total
number of records.
 
 
1
2
1
ln
1ln
1ln
1
nf
i
i
nf
i
i
im
η
P f IM im β
η im
n
β im η
n
 
  
 
 
 
 
 
 
 
(3)
The resulting fragility curves are shown in Figure 11. At this point, it may be useful to recall that the
procedure outlined above (as well as those in the following illustrative applications) is inextricably linked
to the requisites behind conducting a pushover-based SDOF simplification of a real structure. The main
requisites are the predominance of the first mode going into the inelastic response range and that the
structure be characterized by sufficient symmetry and regularity for response in the two principal
directions to remain largely uncoupled and unaffected by torsion (e.g., Seneviratna and Krawinkler,
1997). For this reason, extending evaluations that are based on an equivalent SDOF systems’ seismic
response, to the actual corresponding multi-DOF structure, may require additional effort to account for
the influence of higher modes (see for example Vamvatsikos and Cornell, 2005; Kappos et al., 2013;
Baltzopoulos et al., 2017). This being an illustrative example intended to demonstrate the capabilities of
this software, with no pretense of using the result for seismic risk assessment of any real structure,
considerations need not and will not depart from the SDOF level.
With these premises in mind, it can be seen from the figure that, in the case of the infilled RC frame
(Figure 11a), separate evaluation of collapse fragility along the principal directions results in quite
different probability distributions of collapse intensity, with median collapse intensities in the X and Y
directions being
exp 1.97 g
X
η and
exp 1.19 g
Y
η, respectively. In this case of one direction
nominally exhibiting greater seismic vulnerability than the other by a wide margin (when the two are
viewed as separate structural systems during analysis) it happens that the entire system’s collapse
fragility almost coincides with the most vulnerable one, with
exp 1.17 g
η.
On the other hand, in the case of the bare RC frame (Figure 11b), collapse fragilities derived separately
for each direction are more uniform, with
exp 2.31 g
X
η and
exp 2.12 g
Y
η. However, in this
case, the simplified analysis executed for this example indicates that the total building vulnerability is
greater than the worst of the two nominal vulnerabilities obtained by the separate analyses, with a median
collapse intensity of
exp 1.86 g
η.
18
Figure 11 Collapse fragilities estimated separately for each principal direction and for the entire structure. Fragility curves
of the three-storey infilled RC frame (a) and those of the six-storey bare RC frame (b).
4.1.2 Multiple-stripe analysis using different sets of records per level of seismic intensity
In this example, a six-storey bare RC frame building is used, designed for a site in L’Aquila, Italy
(Camata et al., 2017). Details on the equivalent SDOFs in the two principal directions of the structure
can be found in Suzuki et al. (2018), while the backbone curve in the X direction is shown in the upper
left panel of Figure 8. In order to calculate site-specific seismic risk for that building, Iervolino et al.
(2017) selected hazard-consistent sets of records at ten different levels of seismic intensity, when said
intensity was expressed in terms of
max
1.0 s
Sa T . Thus, twenty pairs of bidirectional acceleration
time-histories were obtained, for each IM level, and were assigned to an MSA via the GUI (Figure 8). In
this case, definition of IM-type, i.e., maximum-among-horizontal-components spectral acceleration
max
Sa
, has to be defined prior to running the analysis, in order to ascertain correct scaling of the records.
EDP for the analysis was defined as a
D C
ratio by setting threshold displacements in both directions,
corresponding to a roof drift of the actual building of three percent. The resulting EDP vs. IM plot from
the analysis can be seen in Figure 12, where the dashed line at
1
D C
separates the converging runs
between those causing failure to the right and those who do not, to the left. Note that in the two highest
IM levels,
max
1.0 s 1.837 g
Sa T   and
3.52 g
, two and ten cases, respectively, reached collapse and
are thus not present on the plot; collapsed cases per stripe are reported separately by the GUI.
( =1 s) [g]Sa T
gm
0 1 2 3 4 5
P
f
0
0.2
0.4
0.6
0.8
1
X direction
Y direction
Entire building
( =1 s) [g]Sa T
gm
0 1 2 3 4 5
P
f
0
0.2
0.4
0.6
0.8
1
X direction
Y direction
Entire building
 
a
 
b
19
Figure 12 MSA results shown in a semi-logarithmic EDP-IM plot obtainable via the “export figures” option of the
software.
4.2 Running cloud analysis via the MSA panel of the GUI
In this brief example application, the trilinear-backbone SDOF oscillator shown in Figure 13 is used,
having vibration period
0.60 s
T
and characterized by a moderately-pinched peak-oriented hysteresis
(Lowes et al., 2004). As mentioned previously, in order to run cloud analysis the user has to define a
single nominal IM stripe in the MSA panel of the GUI, assign all records to that stripe and select the “do
not scale records” option.
Figure 13 Trilinear backbone curve of the SDOF oscillator used for cloud analysis and cyclic quasi-static response of the
peak-oriented pinched hysteretic model (a); cyclic response of the same to a specific accelerogram (b). Both plots can be
obtained via the GUI’s “export figures” option, following a “cyclic quasi-static” or “single-run dynamic” analysis,
respectively.
For this analysis, a set of forty-two single-component records is selected from within the Engineering
Strong Motion database (http://esm.mi.ingv.it/), with the criteria of covering a wide range of as-recorded
intensity in terms of
0.60 s
Sa T , never using both components of the same station for a single event,
not using more than three stations from the same event and always exceeding yield acceleration of the
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
10−1
100
Sagm ( =1.00 s , =5.0 % ) [g]T v
D C
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
[m]
[m]
-1000
-500
0
500
1000
F [kN]
−0.2 −0.1 0 0.1 0.2 0.3
Backbone curve
Quasi-static
cyclic response
Dynamic
cyclic response
(a) (b)
20
oscillator in Figure 13. It is evident that this type of analysis requires a notably lower number of runs
with respect to the preceding IDA and MSA examples. The results of the analysis, i.e., an EDP-IM scatter
plot forming the homonymous “cloud”, are shown in Figure 14.
Figure 14 Results of cloud analysis, shown in a logarithmic-scale EDP-IM scatter plot, obtainable via the “export figures”
option of the software.
4.3 Back-to-back incremental dynamic analysis for estimating state-dependent seismic fragility
State-dependent seismic fragility extends the traditional concept of fragility, expressed in Eq. (3) as the
probability of failure for a given IM level,
P f IM im
 
 
, to the probability of failure given both the
occurrence of a specific intensity and the structure already being in a generic limit state
i
S
,
i
P f S IM im
 
 
. Such state-dependent fragility functions can find application in seismic reliability
analysis that considers damage accumulation phenomena (e.g., Iervolino et al., 2016) and may be
analytically estimated by means of back-to-back IDA (see for example Raghunandan et al., 2015).
For this illustrative example, a simple SDOF structure is assumed, with vibration period
0.50 s
T
and
a horizontal post-yield segment stretching up to a capping ductility
6.0
c
μ
, after which lateral strength
drops gradually; hysteresis for this system follows the peak-oriented model of Lignos and Krawinkler
(2011) and exhibits mild cyclic strength degradation, as shown in Figure 15. Furthermore, it is assumed
that a generic limit state, corresponding to this structure having sustained some seismic damage and
indicated as
1
S
, is reached when maximum transient seismic ductility demand,
μ
, exceeds 9.6. The
example consists of using IDA to estimate the collapse fragility of this SDOF structure at its intact state,
indicated as
0
S
, and back-to-back IDA for estimating the same fragility under the condition that the
structure is in the
1
S
state due to one or more prior shocks.
10
-1
δ [m]
0.4
0.6
1.0
1.4
2.0
Sa ( =0.60 s , =5.0 % ) [g]T v
21
Figure 15 Backbone curve of the SDOF oscillator used for back-to-back IDA and cyclic quasi-static response of the peak-
oriented modified Ibarra-Medina-Krawinkler hysteretic model exhibiting mild strength degradation (a); cyclic response of
the same to a specific accelerogram (b). Both plots can be obtained via the GUI’s “export figures” option, following a
“cyclic quasi-static” or “single-run dynamic” analysis, respectively.
As mentioned in passing earlier, back-to-back IDA is preceded by an IDA performed with a suite of
accelerograms designated as the MS set. In this case, the MS set consisted of ten records and the
preliminary analysis provided the scale factors required to reach the
9.6
μ
threshold that was assumed,
by convention, to signify exceedance of
1
S
(see Figure 16a). During back-to-back IDA, each MS record
is followed by three AS records that are randomly sampled from within a pool of twenty accelerograms,
resulting in thirty curves. For this analysis, the thirty single-component records provided with the
software are used. It should be noted that the number of records and back-to-back IDA runs used in this
example is simply dictated by the need to maintain Figure 16 easily legible by avoiding overcrowding of
IDA curves. In fact, these numbers should be chosen on the basis of the confidence that the user desires
to be able to place on the estimated fragility parameters; the interested reader is referred to Iervolino
(2017) for more details.
By taking the IM stripes corresponding to the so-called flat-line heights of the IDA curves that, as
discussed previously, signify the onset of dynamic instability, collapse fragility curves can be obtained
according to Eq. (3) for both
0
S
and
1
S
initial states. These fragility curves are shown in Figure 16, in
terms of both
0.50 s
Sa T and
PGA
, with the latter being derived from the former almost-
instantaneously, via re-interpolation of the IDA curves. In either case, one observes the characteristic
shift of the
1
S
fragilities to the left of the
0
S
curves, quantifying the decrease in capacity (or increase in
vulnerability) of the structure, that has befallen a damaged state.
F [kN]
δ [m] δ [m]
22
Figure 16 Preliminary IDA used to determine intact state collapse fragility and intensities inducing the S1 limit state
threshold (a); back-to-back IDA used to determine collapse fragility given that the structure has previously reached the
limit state S1 (b); comparison of collapse fragility curves of the intact and damaged structure, expressed in terms of
0 50 s
Sa T . (c) and in terms of PGA (d), after implementation of the IM conversion tool.
5. Concluding remarks
This article was dedicated to the presentation of DYANAS, an open-source, earthquake engineering
software. The software is a MATHWORKS MATLAB®-based graphical user interface that interacts
with the OpenSees finite element framework and offers engineers an easy way to define and run dynamic
analysis of yielding single-degree-of-freedom systems subjected to large suites of earthquake-induced
ground motion records. Therefore, the main purpose of this software is to determine the relationship
between seismic intensity and engineering demand parameters using any one of several dynamic analysis
methods that are frequently used in performance-based earthquake engineering. Incremental dynamic
analysis is supported in its traditional form and also in a “back-to-back” format that could allow state-
dependent seismic fragility to be estimated. Multiple-stripe and cloud-type analyses are also supported,
in cases where the user does not desire to use scaled records to estimate seismic response at different
levels of shacking intensity. Much of the software’s utility stems from the post-processing routines, with
which it is equipped. These routines, which are also programmed in MATLAB and can be recalled via
the user interface, permit the re-interpolation of incremental dynamic analysis curves, thus providing
rapid conversion of the curves into alternative intensity measures, such as peak ground acceleration or
spectral ordinates at various periods. Furthermore, at the end of analysis, the user can obtain vectors of
intensity measure given engineering demand parameter, or vice versa, thus facilitating further
probabilistic seismic demand elaborations for these simple structural systems. Overall, the GUI was
max
y
δ
μδ
max
y
δ
μδ
(
=
0
.
5
s
)
[
g
]
S
a
T
( =0.5 s) [g]Sa T
( =0.5 s) [g]Sa T
PGA [g]
23
shown to be a useful tool within the context of performance-based earthquake engineering, as
demonstrated by the illustrative applications provided. DYANAS is freely distributed for research
purposes by the University of Naples Federico II under a GNU general public license v3.0, obtainable at
http://wpage.unina.it/georgios.baltzopoulos/software/dyanas.zip. The MATLAB and Tcl/Tk source
code is available at a dedicated GitHub repository at https://github.com/georgebaltz/SDOF-OSEES.
Acknowledgements
The work presented in this paper was partly developed within the activities of ReLUIS (Rete di
Laboratori Universitari d’Ingegneria Sismica) during the 2014-2018 research program, funded by the
Italian Civil Protection (DPC Dipartimento della Protezione Civile). Another part of the work was
supported via the Horizon 2020 MSKA-RISE research project EXCHANGE-RISK (Experimental &
Computational Hybrid Assessment of Natural Gas Pipelines Exposed to Seismic Risk) – Grant no.
691213.
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... In our case we have selected relatively well-behaved variables for the IM and EDP, i.e., AvgSa and θ max , with dispersions in the order of 30-40%. In similar EDP situations, Baltzopoulos et al. (2018) employed a minimum of 20 records per stripe. In our case, thanks to the good performance of AvgSa and to showcase a relatively frugal application, a single set of 9 ground motion records is adopted that are randomly selected out of the 44 records of FEMA P695. ...
... This is typically assumed to follow the lognormal distribution, characterizing the EDP|IM response by two parameters, i.e., the conditional mean and variance. Both parameters can be assessed either by subjecting the ESDOF to a number of dynamic analyses, e.g., via the open-source tools developed by Baltzopoulos et al. (2018), Elkady and Lignos (2018), or by employing advanced R-μ-T relationships, such as the ones proposed by Ruiz-García and Miranda (2007) for elastoplastic oscillators, or the SPO2IDA tool (Vamvatsikos and Cornell 2006;Baltzopoulos et al. 2016) for oscillators with complex quadrilinear backbones. ...
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A mixture model approach is presented for combining the results of different models or analysis methods into a single probabilistic demand model for seismic assessment. In general, a structure can be represented using models of different type or different number of degrees of freedom, each offering a distinct compromise in computational load versus accuracy; it may also be analysed via methods of different complexity, most notably static versus dynamic nonlinear approaches. Employing the highest fidelity options is theoretically desirable but practically infeasible, at best limiting their use to calibrating or validating lower fidelity approaches. Instead, a large sample of low fidelity results can be selectively combined with sparse results from higher fidelity models or methods to simultaneously capitalize on the frugal nature of the former and the low bias of the latter to deliver fidelity at an acceptable cost. By employing a minimal 5 parameter power-law-based surrogate model we offer two options for forming mixed probabilistic seismic demand models that (i) can combine different models with varying degree of fidelity at different ranges of structural response, or (ii) nonlinear static and dynamic results into a single output suitable for fragility assessment.
... For performing IDA, several software programs have been developed in recent years. Some of them are Static pushover to FRAGility (SPO2FRAG) [27], Interactive Interface for Incremental Dynamic Analysis Procedures (III-DAP), Dynamic analysis of single-degree-of-freedom systems (DYANAS) [28]. DYANAS is a graphical user interface for OpenSees [29]. ...
... It provides a platform to perform nonlinear dynamic analysis of an SDoF system. The interface offers ease in defining analysis parameters and seismic input [28]. ...
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The variability in source characteristics induces spatial variability in the characteristics of near-field ground motion intensities which in turn impacts the fragility behaviour of structures. The study aims to quantify such variability in fragility curves of low-rise structures with a natural period of 0.2 sec with change in nucleation asperity, a source parameter of earthquake rupture corresponding to (Mw ⋍ 7). Four rupture scenarios are considered which differ from each other based on the position of nucleation asperity. The rupture scenarios are termed as bilateral, partial bilateral, partial unilateral, and unilateral, respectively. For bilateral rupture, the nucleation asperity is placed exactly in the middle of the fault face. In contrast, in the unilateral case, the nucleation asperity is placed at the end of the fault. Additionally, fragility curves corresponding to each horizontal component (fault-parallel and fault-normal) of ground motions at stations with varying rupture distances are also highlighted. The study is also extended to evaluate structures’ fragility behaviour with a natural period of 0.6 sec, 1 sec and 4 sec. Additionally bi-directional response to the simulated ground motions were emphasized.
... State-of-the-art analytical estimation of structure-specific fragility involves the use of dynamic analysis of a numerical model of the structure; e.g., incremental dynamic analysis (IDA) [3,4]. For the evaluation of state-dependent fragility curves, an extended version of IDA has been suggested in several studies [5][6][7][8][9][10][11], referred to as back-to-back IDA. The main disadvantage of deriving fragility functions based on nonlinear dynamic analysis is the high computational cost involved, which includes both the time investment required for effectively modelling nonlinear structural behavior and computer time needed to run multitudes of analyses and post-process the results. ...
... Additionally, a twentyrecord IDA was used to estimate the scaled intensity causing a ductility demand of six, that is, incurring the condition B DS EDP edp  , for the intact structure; the latter analysis was therefore used for the derivation of the intact structure's traditional fragility,   | B P DS IM im = . These runs were performed using an OPENSees user interface developed to streamline the back-to-back IDA [11]. The resulting logarithmic means and standard Table 2; the corresponding cumulative probability functions are given in Fig. 7b. ...
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Earthquakes are clustered in space and time. This means that structures in seismically active regions can be subjected to multiple consecutive instances of base acceleration, with insufficient in-between time for repair operations to take place. In such situations, buildings may experience degradation of their lateral-force-resisting capacity due to damage accumulation. Consequently, the use of seismic fragility functions developed for the intact structure may not be enough, in the context of seismic risk assessment studies that consider the effect of seismic clusters. In these cases, one may employ state-dependent fragility curves, which are separate fragility functions assigned to the same structure, depending on distinct damage states that it may be brought to by prior shocks. State-of-the-art analytical estimation of structure-specific fragility entails the use of dynamic analysis of a numerical model of the structure, for example, incremental dynamic analysis (IDA), which can be computationally laborious, thus motivating the development of simplified, less time-consuming methods, often based on substituting the structural model by equivalent single-degree-of-freedom (SDOF) systems that can be defined via pushover analysis. In fact, existing procedures in the literature, such as back-to-back IDA, that can be used to estimate state-dependent fragility curves, tend to increase computational costs, rendering the development of simplified methodologies for this case a topical issue. In this context, the present paper presents a method for estimating state-dependent seismic fragility functions, based on pushover analysis and a predictive model for constant-ductility residual displacement ratio. This ratio is defined as the absolute value of the of residual-to-peak-transient seismic displacement ratio of an equivalent SDOF structure. The residual displacement model, which considers yielding SDOF systems that exhibit stiffness and strength degradation, with natural periods between 0.3 s and 2.0 s and post-yield hardening ratios from 0 % to 10%, is outlined first. The model also estimates the joint probability distribution of normalized elongated period and strength degradation, for a given ductility demand. This information allows for a probabilistic evaluation of the pushover curve characterizing a damaged structural system, which is then used to obtain state-dependent fragility, when damage states are defined via ductility demand thresholds. The state-dependent fragility curves are estimated via IDA of SDOF oscillators with pushovers that were previously determined from the model. An illustrative application showcases the ability of the proposed methodology to provide state-dependent fragility estimates in an expedient manner.
... The collapse capacity of a structure under a specified record is the intensity measure (IM) value at which the structure reaches the CP performance level. In this study, the 5% damping spectral acceleration corresponding to the fundamental period of vibration of the structure Sa(T1-5%) is used for the IM value (Baltzopoulos et al. 2018). In the IDA approach, the CP performance level for each record is reached when the local slope on the IDA curve reaches 20% of the elastic slope, or the EDP attains a predefined codebased rate (FEMA 355c 2000, Vamvatsikos and Cornell 2002). ...
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Moment-resisting frames (MRFs) are among the most conventional steel structures for mid-rise buildings in many earthquake-prone cities. Here, a simplified performance-based methodology is proposed for the seismic collapse capacity assessment of these buildings. This method employs a novel multi-mode pushover analysis to determine the engineering demand parameters (EDPs) of the regular steel MRFs up to the collapse prevention (CP) performance level. The modal combination coefficients used in the proposed pushover analysis, are obtained from two metaheuristic optimization algorithms and a fitting procedure. The design variables for the optimization process are the inter-story drift ratio profiles resulting from the multi-mode pushover analyses, and the objective values are the outcomes of the incremental dynamic analysis (IDA). Here, the collapse capacity of the structures is assessed in three to five steps, using a modified IDA procedure. A series of regular mid-rise steel MRFs are selected and analyzed to calculate the modal combination coefficients and to validate the proposed approach. The new methodology is verified against the current existing approaches. This comparison shows that the suggested method more accurately evaluates the EDPs and the collapse capacity of the regular MRFs in a robust and easy to implement way.
... At the present time, nonlinear time history analysis has also been widely considered due to developments in the processing power of computers, accessing ground motion record databases and the software industry. In the literature, the simpler single degree of freedom (SDOF) systems [2][3][4][5] and multi degree of freedom (MDOF) systems [6][7][8][9][10] were used as structural analysis models for nonlinear time history analysis. ...
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Nonlinear time history analysis is an analytical method generally used in performance-based seismic design. With this method, seismic responses are obtained more realistically. Selection of ground motion records for nonlinear time history analysis is an important step since it strongly affects the analysis results. Therefore, it has always been a matter of curiosity to investigate the effect of the characteristics, content and number of the records on the analysis results. In this study, seismic responses of regular and irregular RC frames were investigated by varying the number of real ground motion records in a set. For this purpose, 13 different groups that contain three to hundred real ground motion records in size in a set have been considered and ten different earthquake record sets are obtained for each group. Ground motion selection procedure of Eurocode-8 was considered and a total of 130 sets were used for nonlinear response analysis of RC frames. Global drift ratio, maximum floor acceleration, inter-story drift ratio and six different intensity measures (IMs) were used to investigate the effect of the number of records. According to analysis results, nonlinear responses of RC frames are more stable and might be sufficient when the number of real records in a set is higher than seven according to Eurocode-8. Results indicate that if the number of real records in a set are lower than seven, conservative seismic responses can be found since the maximum rather than mean response values are used. It is observed that dispersion of seismic demands and mean to median ratios can be increased if the number of real records in a set is higher than ten. In addition, the correlation between some of IMs and seismic demands increase when the number of records in a set increased from three to seven and it remains stable from seven to hundred records. Furthermore, 7, 8, 15 and 22 records show the lowest error terms of considered engineering demand parameters for regular and irregular RC frames.
... However, NDA of a 3D model is complex and time-consuming especially if the response of the structure is investigated by numerous seismic events. Thus, instead of constructing 3D models, many studies have focused on simpler models like equivalent single degree of freedom (SDOF) system (FEMA-440 2005, Priestley et al. 2007) to represent the behavior of an entire building (Liossatou and Fardis 2015, Baltzopoulos et al. 2018, Karimzadeh et al. 2020, Baigoly et al. 2020 Corresponding author, Professor E-mail: hkayhan@pau.edu.tr On the other hand, estimation of seismic response is extremely related to selected earthquake records which are used as seismic input for NDAs except from the assumptions about modeling and analysis (Macedo and Castro 2017, Strukar et al. 2019. ...
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Selection of appropriate ground motion records for dynamic analysis has uttermost importance since it significantly affects structural responses which are used for seismic performance assessment of buildings. This study focuses on probabilistic assessment of several record selection strategies that apply different level of constraints for spectrally matched real ground motion records. For this purpose, single degree of freedom (SDOF) systems with various lateral strength capacity ratios, vibration periods and hysteretic models were considered to cover broad type of structural systems and maximum displacement demands of SDOF systems were obtained by nonlinear dynamic analyses. Using the analysis results, central tendency of maximum displacement demands was evaluated. Confidence intervals of the demands were also estimated in probabilistic manner. In addition, non-exceedance probability curves of the displacement demands were constructed. Results indicate that using supplementary constraints about spectral matching, it is possible to control the variation of spectral accelerations and hence the variation of seismic displacement demands. In conclusion, displacement demands can be obtained for code- or probability-based design/performance assessment with appropriate selection approach considering desired variation which can be determined from either probabilistic or deterministic seismic hazard analysis.
... Formerly, various GUIs were developed for OpenSees which are able to define the numerical models and consequent running of analyses. Examples of these include STKO [3], Build-X [4], MDOF [5], OpenSees Navigator [6], GiD+OpenSEES [2], NextFEM Designer [7], PileGroupTool [8], MSBridge [9] and DYANAS [10] etc. STKO, GiD+OpenSees, NextFEM, and OpenSees Navigator are general-purpose pre-and post-processors. But most of them (STKO, GiD, NextFEM) are not free to use for researchers and industry. ...
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... In order to solve the problem, for all of the GMRs selected in the preselection phase, IDAs are conducted using DYANAS [28]. DYANAS is a MATHWORKS-MATLAB-based graphical user interface that uses OpenSees to conduct nonlinear dynamic analysis of SDF systems. ...
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In order to estimate the seismic performance of structures, performing incremental dynamic analysis, and obtaining fragility curves are essential. Since producing these curves is time-consuming due to performing numerous nonlinear dynamic analyses, the selection of appropriate earthquake ground motion records which give reliable responses with sufficient accuracy is important. Due to the lack of a solid framework, the selection of an appropriate ground motion recordset is still a challenging problem. In this paper, the primary goal is to select a suitable set of records from a general set of records in order to reach reliable limit-state capacities prediction of structures. To achieve this goal, incremental dynamic analysis is conducted for an equivalent single degree of freedom under a general set of records, and an optimization algorithm is employed to solve the problem by minimizing the error between the mean incremental dynamic analysis curves of each selected subset and the mean incremental dynamic analysis curve of a general set of records. The fragility curves obtained by all records and selected ones are compared and the results show that the fragility curves corresponding to the selected records estimate the target fragility curves appropriately.
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In this paper, a post-earthquake damage assessment method for 11 different structural systems is proposed. The method is mainly based on residual inter-story drift ratios of all structural elements in the building, an important response parameter due to its simplicity of measure after the earthquake. nearly 2000 test specimens representing structural elements characterized by different structural systems (walls, frames, etc.) and materials (masonry, reinforced concrete, and steel) which were tested under quasi-static cyclic loading provided more than 100,000 data points that were used to correlate the residual drift ratio and damage states in structural elements. The damage severity of each structural element was then determined based on the weighted average of each damage state's exceedance probabilities given a residual drift ratio. The method also considers the damage propagation among different structural elements by considering the weighted average overall damage scores in structural elements concerning their load-bearing contribution. The pre-earthquake condition of the building and the level of experienced earthquake intensity is the other input parameters for the proposed method, which finally leads to the building's overall damage score. To verify the accuracy of the proposed method's results, an example building hit by the Ezgeleh earthquake is evaluated, and the results are compared with those of other post-earthquake damage assessment methods. ARTICLE HISTORY
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Force‐based seismic design involves the reduction of elastic spectra by introducing a behavior factor, q. This approach is widespread in engineering practice; however, recent studies have shown that structures consistently designed at different sites will not share the same level of seismic risk, which can be defined as the annual rate of the structure failing to meet a seismic performance objective, despite seismic actions having the same exceedance return period at all sites. This paper investigates whether the definition of site‐specific q factors can lead to uniform risk across sites characterized by varying levels of seismic hazard, based on the pushover curves of bare frame reinforced concrete buildings. These pushover curves are used to establish the backbones of equivalent single degree of freedom systems with varying lateral resistance. These systems are fictitiously placed at several Italian sites and their seismic failure risk is computed by integrating their fragility, assessed by means of incremental dynamic analysis, with each site's hazard curve. By assuming an arbitrary risk threshold, the same for all sites, the corresponding lateral strength leading to said threshold is determined and the corresponding behavior factor is back calculated. As expected, risk‐targeted q factors tend to increase with decreasing seismic hazard and are highly sensitive to the shape of the hazard curve beyond the design return period. Coupled with the fact that at low hazard sites lateral strength is determined by detailing for gravity‐load design and minimum code requirements, rather than seismic design actions, the results suggest that q factor‐based design is unsuitable for warranting territorially uniform seismic safety, yet it may be suitable for setting an upper‐bound to the annual failure probability.
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II-DAP is a publicly-available standalone MATLAB-based program that was developed at the Resilient Steel Structures Laboratory (RESSLAB) in Ecole Polytechnique Federale de Lausanne (EPFL), Switzerland. With 10K+ lines of code, II-DAP is capable of performing a variety of dynamic analysis procedures, ranging from simple response history analysis to response spectrum and incremental dynamic analyses, on single degree-of-freedom systems (SDoF). IIDAP includes a wide-ranging library of SDoF system models ranging from simple non-deteriorating linear and bi-linear models to state-of-the-art deteriorating bi-linear, pinched, peak-oriented and self-centering responses. A seismic hazard module is also included for the quantification of collapse risk. II-DAP is highly beneficial for both educational and research purposes.
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Despite the breadth of the available finite element codes for seismic analysis and assessment, the associated complexity in use and the generality in orientation are likely to increase the epistemic uncertainty involved in the models, particularly in nonlinear analysis procedures. Thus, it is of interest to develop tools for improving the reliable use of existing structural engineering software. This paper aims to present the capabilities of Build-X, a recently developed knowledge-based system tailored to the prediction of the seismic response of 3D buildings. This expert system features a simple visual user interface that supports the structural engineer throughout the structural configuration of a building, providing expert suggestions as to critical modelling decisions, and automations that increase the reliability of the analysis and accelerate the pre-processing stage. Build-X is linked with OpenSees, a widely used script-based freeware for seismic analysis of structures, which is utilized to perform the core finite element analysis. Post-processing tasks are easy to handle through the graphical engine of the system developed. A verification study demonstrates the efficiency of the system and reliability of the results generated, pointing to the way in which Build-X may serve as a useful tool for the seismic analysis of newly designed buildings and the assessment of existing ones at reduced computational cost and modelling uncertainty.
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RINTC, which started in 2015, is a joint project of ReLUIS and EUCENTRE, two centers of competence for seismic risk assessment of the Italian civil protection. The goal of the project, which is still ongoing, is to assess in an explicit manner the seismic risk of structures designed according to the code currently enforced in Italy. To this aim five structural typologies were considered: masonry, reinforced concrete, precast reinforced concrete, steel, and seismically isolated buildings. In the framework of the project, multiple archetype structures have been designed for each typology according to standard practice at five sites across Italy, spanning a wide range of seismic hazard levels. The seismic vulnerability of the designed structures was assessed by subjecting 3D computer models to multi-stripe non-linear dynamic analysis. Integration of the probabilistic hazard and probabilistic vulnerability (i.e., fragility) yielded the annual failure rate, in terms of onset of non-structural damage and collapse, of each of the structures. Risk assessment takes into consideration record-to-record variability of non-structural response and, for selected cases, structural modeling uncertainty. Results preliminarily show that, for each structural typology, the collapse risk tends to increase with the hazard of the site and that risk is not uniform across typologies.
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SPO2FRAG (Static PushOver to FRAGility) is introduced, a MATLAB-coded software tool for estimating structure-specific seismic fragility curves of buildings, using the results of static pushover analysis. The SPO2FRAG tool (available online at http://wpage.unina.it/iuniervo/doc_en/SPO2FRAG.htm) eschews the need for computationally demanding dynamic analyses by simulating the results of incremental dynamic analysis via the SPO2IDA algorithm and an equivalent single-degree-of-freedom approximation of the structure. Subsequently, fragility functions may be calculated for multiple limit states, using the intensity-measure-based analytical approach. The damage thresholds may also be random variables and uncertainty in estimation of the fragility parameters may be explicitly accounted for. The research background underlying the various modules comprising SPO2FRAG is presented together with an operational description of how the various functions are integrated within the software’s graphical user interface. Two illustrative SPO2FRAG applications are also offered, using a steel and a reinforced concrete moment resisting frame. Finally, the software’s output is compared with the results of incremental dynamic analysis as validation of SPO2FRAG’s effectiveness.
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State-of-the-art approaches to probabilistic assessment of seismic structural reliability are based on simulation of structural behavior via nonlinear dynamic analysis of computer models. Simulations are carried out considering samples of ground motions supposedly drawn from specific populations of signals virtually recorded at the site of interest. This serves to produce samples of structural response to evaluate the failure rate, which in turn allows to compute the failure risk (probability) in a time interval of interest. This procedure alone implies that uncertainty of estimation affects the probabilistic results. The latter is seldom quantified in risk analyses, although it may be relevant. This short paper discusses some basic issues and some simple statistical tools, which can aid the analyst towards the assessment of the impact of sample variability on fragility functions and the resulting seismic structural risk. On the statistical inference side, the addressed strategies are based on consolidated results such as the well-known delta method and on some resampling plans belonging to the bootstrap family. On the structural side, they rely on assumptions and methods typical in performance-based earthquake engineering applications.
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This work discusses the simplified estimation of earthquake-induced nonlinear displacement demands as required by nonlinear static procedures, with particular attention on short-period masonry structures. The study focuses on systems with fundamental periods between 0.1 and 0.5 s, for which inelastic amplification of the elastic displacement demand is more pronounced; hysteretic force-displacement relationships characteristic of masonry structures are adopted, because these structures are more commonly found within the considered period range. Referring to the results of nonlinear dynamic analyses of single-degree-of-freedom oscillators, some limitations of the Eurocode 8 and Italian Building Code formulations are first discussed, then an improved equation is calibrated that relates inelastic and elastic displacement demands. Numerical values of the equation parameters are obtained, considering the amount of hysteretic energy dissipation associated with various damage mechanisms observed in masonry structures. Safety factors are also calculated to determine several percentiles of the displacement demand. It is shown that the proposed equation can be extended to more dissipative systems. Finally, the same formulation is adapted to the estimation of seismic displacements when elastic analysis procedures are employed.
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The determination of displacement demands for masonry buildings subjected to seismic action is a key issue in the performance-based assessment and design of such structures. A technique for the definition of single-degree-of-freedom (SDOF) nonlinear systems that approximates the global behaviour of multi-degree-of-freedom (MDOF) 3D structural models has been developed in order to provide useful information on the dependency of displacement demand on different seismic intensity measures. The definition of SDOF system properties is based on the dynamic equivalence of the elastic properties (vibration period and viscous damping) and on the comparability with nonlinear hysteretic behaviour obtained by cyclic pushover analysis on MDOF models. The MDOF systems are based on a nonlinear macroelement model that is able to reproduce the in-plane shear and flexural cyclic behaviour of pier and spandrel elements. For the complete MDOF models an equivalent frame modelling technique was used. The equivalent SDOF system was modelled using a suitable nonlinear spring comprised of two macroelements in parallel. This allows for a simple calibration of the hysteretic response of the SDOF by suitably proportioning the contributions of flexure-dominated and shear-dominated responses. The comparison of results in terms of maximum displacements obtained for the SDOF and MDOF systems demonstrates the feasibility and reliability of the proposed approach. The comparisons between MDOF and equivalent SDOF systems, carried out for several building prototypes, were based on the results of time-history analyses performed with a large database of natural records covering a wide range of magnitude, distance and local soil conditions. The use of unscaled natural accelerograms allowed the displacement demand to be expressed as a function of different ground motion parameters allowing for the study of their relative influence on the displacement demand for masonry structures.
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