Research
Projects Identification of Nonlinear Structural Systems
Research Team: Babak
Moaveni (PI), Eliyar Asgarieh (grad
student), Amin Nozari (grad student),
Mingming Song (grad student)
Funding: NSFCAREER
(1254338)
Duration: 9/2011 – present
Relevant publications to date:
Asgarieh, E., Moaveni, B., and Stavridis A.
(2014). "Nonlinear finite element model updating
of an infilled frame based on identified
timevarying modal parameters during an
earthquake." Journal of Sound and Vibration,
333(23), 60576073.
Moaveni, B., and Asgarieh, E. (2012).
"Deterministicstochastic subspace
identification method for identification of
nonlinear structures as timevarying linear
systems." Mechanical Systems and Signal
Processing, 31, 4055.
Asgarieh, E., Moaveni, B., Nozari, A.,
Barbosa, A.R., Chatzi, E. (2014). "Nonlinear
identification of a sevenstory shear wall
building based on numerically simulated
seismic data." Proc. of 32nd
International Conference on Modal Analysis
(IMACXXXII), Orlando, Florida, USA.
Asgarieh, E., Moaveni, B., and Stavridis,
A. (2013). "Nonlinear finite element model
updating of a largescale infilled frame
structure based on instantaneous modal
parameters." Proc. of 31st International
Modal Analysis Conference (IMACXXXI),
Garden Grove, California, USA.
Asgarieh, E., and Moaveni, B. (2012).
"Experimental modal analysis of a fullscale
sevenstory shear wall based on nonlinear
seismic response." Proc. of 30th
International Modal Analysis Conference
(IMACXXX), Jacksonville, Florida, USA.
Asgarieh, E., Moaveni, B., and Stavridis,
A. (2012). "Nonlinear structural
identification of a threestory infilled
frame using instantaneous modal parameters."
Proc. of 30th International Modal
Analysis Conference (IMACXXX),
Jacksonville, Florida, USA.
Asgarieh, E., Moaveni, B. (2011).
"Shorttime modal identification of
nonlinear systems using the
deterministicstochastic subspace
identification method." Proc. of the
2011 Engineering Mechanics Institute
Conference, Boston, Massachusetts, June
24.
This project includes the following
subtopics:
1. Modal Identification of Nonlinear
Structures as Timevarying Linear Systems
This study proposes to use a linear system
identification method to estimate the
instantaneous modal parameters of a
nonlinear structure. A windowed
deterministicstochastic subspace
identification methods (DSI) was proposed
for shorttime (instantaneous) system
identification of nonlinear systems when
subjected to nonstationary seismic base
excitations. Accuracy of this method is
compared to that of the wavelet transform
method when applied for identification of
SDOF as well as 7DOF systems with different
material hysteretic behavior. Effects of
several input factors on the accuracy of
system identification results are studied.
The considered input factors are: (1) type
of material nonlinearity (i.e., material
hysteretic behavior), (2) level of
nonlinearity, (3) input excitation, and (4)
length of the “shorttime” data windows used
in the identification. The contribution of
each input factor to the total variability
of two estimation error metrics is
quantified through analysisofvariance, an
effect screening method.
The proposed DSI is also used for shorttime
system identification of a fullscale
sevenstory shear wall structure and a
halfscale threestory RC frame structure
when subjected to seismic base excitation
through a shake table. The structures were
damaged progressively through earthquake
ground motions. This study highlighted the
effectiveness of the DSI method for
shorttime (instantaneous) modal
identification of nonlinear structural
systems. It is expected that accurate
estimates of instantaneous modal parameters
to be used for characterizing the hysteretic
behavior of structural components (e.g.,
substructures), which is the topic of an
ongoing research by the research team.
Figure 1 shows the time histories of the
first mode instantaneous natural frequency
of the test structure identified using 1
and 2second windows during the three
considered earthquake base excitation tests
as well as the square root of an effective
global stiffness estimate of the structure
during the base excitation time histories.
It can be observed that the identified
instantaneous natural frequencies match well
the trend of the square root of effective
global stiffness estimate. It is expected
that accurate estimate of instantaneous
modal parameters can be used for
characterizing the hysteretic behavior of
the structure at element levels.
Figure 1. Time histories of (a)
instantaneous fundamental natural frequency,
and (b) square root of an effective global
stiffness measure of the test structure,
estimated using 1 or 2second data windows
during the three considered earthquakes
2. Nonlinear Finite Element Model
updating
This study is focused on calibration of
nonlinear model parameters (hysteretic
material models) at considered finite
elements (or substructures, assuming all
elements in a substructure have similar
nonlinear behavior) to match the identified
timevarying modal parameters. The first
step in modeling the nonlinear material
behavior is to select the model class or
type of constitutive relations. This is
usually done based on prior information
about the material. For example, the
MenegottoPinto model for steel or the
Takeda model for concrete are common
hysteretic models for these two types of
materials. Another class of phenomenological
models includes the BoucWen models, which
are widely used to represent the hysteretic
behavior (as lumped or distributed
plasticity) of different structural
materials/components. The number of
parameters to be calibrated depends on the
selected model class. For example, a Takeda
model has five parameters while a BoucWen
model can take from 5 up to 13 parameters
depending on whether stiffness degradation,
strength degradation and pinching behavior
are considered in the model or not.
Following the selection of model class and
model parameters to be calibrated, the
nonlinear FE model updating will be
formulated as a nonlinear least squares. The
objective function will be defined as the
misfit between the identified modal
parameters and their counterparts from the
FE model. Figure 2 shows the nonlinear model
updating process employed in this
application example.
Figure 2. Nonlinear model updating process
The performance of the proposed updating
method is evaluated through numerical and
experimental applications on a largescale
threestory reinforced concrete frame with
masonry infills (Figure 3). The test
structure was subjected to seismic base
excitations of increasing amplitude at a
large outdoor shaketable. A nonlinear FE
model of the test structure has been
calibrated to match the timevarying modal
parameters of the test structure identified
from measured data during a seismic base
excitation. The accuracy of the proposed
nonlinear FE model updating procedure is
quantified in numerical and experimental
applications using different error metrics.
The calibrated models predict the exact
simulated response very accurately in the
numerical application, while the updated
models match the measured response
reasonably well in the experimental
application.
Figure 3. Test structure and the FE model to represent it
Figure 4. Comparison of model predicted roof acceleration (left) and first floor displacement (right)
with their measured counterparts for two cases of model updating
3. Online Nonlinear Identification Using Unscented Kalman Filter
Adaptive methods such as Kalman or Partcle
filters have the potential for realtime
nonlinear identification. In the application
these methods, nonlinearity is modeled using
analytical phenomenological material models
such as the BoucWen model. Although this
framework has been applied successfully in
many cases and promising results are
reported, its applicability is hindered by
problems such as large modeling errors in
the considered statespace models (such as
oversimplification of models, wrong boundary
conditions and constraint assumptions, and
incompatible nonlinear models) and
difficulty of applying these models to more
complex structural systems with large number
of degreesoffreedom. In this study, the
applicability of an Unscented Kalman filter
algorithm is investigated for identifying
two nonlinear statespace models of a test
structure relying on different modeling
assumptions. The effects of these modeling
assumptions on the performance of the
implemented Kalman filter are studied.
Figure 5 represent the predicted
acceleration and displacement responses at
floors 1, 3, 5, and 7 of a seven story shear
wall structure using the UKF. From this
figure, it can be seen that, the calibrated
model can accurately match the acceleration
data while in the absence of the
displacement measurements in the
identification, the model displacements at
the lower stories are estimated with some
error.
Figure 5. Predicted acceleration and
displacement responses of the model
calibrated using the UKF based on either
seven acceleration, seven acceleration and
one displacement, or seven acceleration and
two displacement channels of data
