Vertical System Identification of a 52-Story
High-Rise Building Using Seismic
Accelerations
VIVIANA VELA, ERTUGRUL
T
A
CIROGLU and MONICA KOHLE
R
ABSTRACT
In this study, various system id
entification approaches are utili
zed to estimate the dominant,
vertical-component modes of a 52-story, steel, mo
ment-and-braced frame build
ing in downtown Los
Angeles resulting from vertical seismic accelerat
ions. Tall buildings exhibit complex three-
dimensional responses during an earthquake due to
the varying material
and geometric properties
along the building’s height. For hi
gh-rise buildings, the dynamic re
sponse during shaking events is
often sensitive to multiple vibration modes, and mu
ltiple-mode structural be
havior unde
r horizontal
ground motion has been extensively studied. However,
the vertical component of ground motion can
also excite higher modes and vert
ical-polarity propagating seismic waves. Their effects are seldom
studied due to the scarcity of data. Still, they ar
e important because they ca
n provide information on
the axial loads on columns or stresses at floor slab
connections. The 52-story high-rise, with its dense
triaxial sensor array di
stributed vertically along
the height of the buildi
ng, provides a suitable basis
for examining vertical responses. System identifica
tion is performed using state-space methods with
low-amplitude earthquake data. Given the high spatia
l density of the building recordings, we show
how we can detect modal characteristics and identif
y the type of deformation that can occur when
considering the vertical com
ponent of seismic responses.
INTRODUCTION
An office building in downtown Los Angeles was
instrumented with an embedded 60-station (180
channels) accelerometer network. The 52-story high-ri
se office building has 60
triaxial sensors, one
per floor that starts at the lowest
basement level and extends to near
ly every floor alo
ng the building’s
height. The accelerometer sensors al
ong the floors of the building
are a component of
the Community
Seismic Network (CSN), which currently consists
of approximately 1000 stations densely spaced
around the Los Angeles, California area. The CSN sens
or array consists of
relatively inexpensive
Vela V. Department of Civil and Environmental Enginee
ring, University of California, Los Angeles, CA,
U.S.A.
Taciroglu, E. Department of Civil and Environmental
Engineering, University of California, Los Angeles,
CA, U.S.A.
Kohler, M. Department of Mechanical and Civil Engineer
ing, California Institute of Technology, Pasadena,
CA, U.S.A.
triaxial MEMS accelerometers, which record vibration data continuously to the cloud [1, 2, 3]. These
sensors have been deployed at several locations at ground level, as well as upper floors of mid
-
and
high
-
rise buildings [4, 5, 6].
It is known that struc
tures experience earthquake shaking in both the horizontal and vertical
direction during an earthquake. Most structural analysis involves horizontal ground motions and little
attention has been given to the vertical response of structures caused by the ver
tical component of
ground motion
. Large vertical amplitudes are not as common as the large horizontal amplitudes seen
in strong
-
motion earthquakes, nonetheless, relatively large vertical amplitudes can be seen even for
smaller magnitude earthquakes. The em
bedded sensor network from CSN provides a unique platform
that allows for the use of small
-
magnitude earthquake vibrations, particularly focusing on the vertical
component in this paper, for system identification of
structural
modal properties.
Since syst
em identification uses data to identify modal properties such as natural frequencies,
damping
,
and mode shapes of a system, it has been adapted for structural engineering applications to
estimate structural parameters, such as stiffness
and damping
, which
have been used to perform model
updating for structures such as bridges, roads,
and
buildings [7, 8, 9]. System identification models
have also been developed for structural health monitoring of structures to identify damage [10, 11].
With the increased co
mplexity of multivariable structural systems, it has become particularly
appealing
to use state
-
space model representations of systems for system identification.
Of the many
existing
subspace
system identification algorithms, subspace state
-
space system id
entification
(
N4SID
)
is
utilized in this paper. N4SID
is numerically reliable in the estimation of state
-
space
matrices using measured input and output data.
The objective of this study is to use
subspace state
-
space
system identification
to investigate th
e
structural response of a high
-
rise building to ground motions with large vertical amplitudes.
Given
the high spatial density of the building recordings, we show how we can detect modal characteristics
and identify
the type of deformation that can occur
i
n this tall building.
52
-
STORY BUILDING AND ITS MONITORING SYSTEM
52
-
Story High
-
Rise Building
Constructed in 1988, the 52
-
story, 717
-
foot
-
height office building is one of the few high
-
rise
buildings in downtown Los Angeles exceeding 700
-
ft [12]. The bu
ilding comprises of a dual lateral
force
-
resisting system that consists of an interior steel braced frame core with outrigger beams at
various levels to tie the core to the
perimeter
structural
column
framework
[4]. The beams act as
moment
-
resisting beams to
reduce the lateral deflection and base moment
. In addition, there are five
basement levels.
Figure 1 shows
a typical floor plan and the orientation of the seismic framing system.
The floor and roof diaphragm
s consist of cast
-
in
-
place concrete slabs over metal deck. Additional
building details are available in [4].
Figure 1.
(A) Typical floor plan (floor 27) and (B) 3D image of the 52
-
story high
-
rise building in downtown Los Angeles
(image from Google
Earth)
Earthquake Data
T
he CSN sensor network is continuously recording acceleration data and has recorded several
earthquakes, including the M4.2 earthquake in Pacoima, California that took place on July 30, 2020,
at 4:29 p.m. local time. As reported b
y USGS Earthquake Hazards Program (2021), the epicenter was
located approximately 20 miles from the 52
-
story high rise
building [
13
].
CSN raw data collected for
this earthquake show a maximum absolute acceleration
of
1
%g
recorded at level two.
Figure 2
sho
ws
the acceleration time series for the Pacoima earthquake along the
building
height after detrending to
remove any signal bias and arbitrarily normalizing each acceleration time series by the same constant
in order to show comparable relative amplitudes.
Figure 2.
Story accelerations from the vertical component of CSN sensors
i
n
a
52
-
story building
in downtown Los
Angeles. Acceleration amplitudes are normalized by the same arbitrary constant to show amplitudes as a function of
height (in feet).
SYSTEM IDENTIFICATION AND MODAL PROPERTIES
Subspace State
-
Space System Identification
System identific
ation is a methodology that aims to approximate dynamic models of systems
based on observed input and output data. Subspace methods of system identification build upon
realization theory and focus on estimating the state vectors by making projections of ce
rtain subspaces
generated from input
-
output data and solving the least squares problem for state
-
space realizations. It
is known that a
linear time
-
invariant (LTI)
system can be represented by a set of first
-
order differential
equations. Assuming the dynam
ical system’s model structure is a discrete
-
time
LTI
system, the state
-
space formulation is given by
푥
(
푡
+
1
)
=
퐴푥
(
푡
)
+
퐵푢
(
푡
)
(1)
푦
(
푡
)
=
퐶푥
(
푡
)
+
퐷푢
(
푡
)
where
퐴
is the state matrix,
퐵
is the input matrix,
퐶
is the output matrix, and
퐷
is the throughput
matrix. Using (1), the matrix input
-
output equations can be written as
[
푦
(
푡
)
푦
(
푡
+
1
)
⋮
푦
(
푡
+
푘
−
1
)
]
=
[
퐶
퐶퐴
⋮
퐶
퐴
푘
−
1
]
푥
(
푡
)
+
[
퐷
퐶퐵
...
퐶퐵
퐷
⋮
⋱
⋱
퐶
퐴
푘
−
2
퐵
...
퐶퐵
퐷
]
[
푢
(
푡
)
푢
(
푡
+
1
)
⋮
푢
(
푡
+
푘
−
1
)
]
(2)
Constructing block Hankel matrices for the input data
푢
(
푡
)
and
output data
푦
(
푡
)
allow
s
for the
us
e of
linear algebra techniques
to solve the differential
equations
of (2)
[14]
.
In particular, the
subspace state
-
space system identification (N4SID) algorithm developed by Van Overschee and
DeMoor shows that through oblique projection, the state vector is
a basis of t
he past and future input
and output matrices
[15].
Therefore, using
block
Hankel
and
shifted
block
Hankel matrices
for
represent
ation of
past and future input and output
, the
state vector
푥
(
푡
)
can be computed
using
singular value decomposition (SVD)
of
past and future subspace
co
nfiguration
.
After computing the
state vector
푥
(
푡
)
, matrices
퐴
,
퐵
,
퐶
, and
퐷
are solved by using the least
-
squares technique.
More
details
on the mathematical f
ormulation
of N4SID algorithm
can be found in
[
14,
15
]
.
Modal
Properties
of 52
-
Story High
-
Rise Building
T
he
state
-
space model
obtained from
N4
SID is effective
in describing
the
dynamic behavior
of
the structure. With the system matrices, modal parameters such as natural frequencies
(
푓
)
, damping
ratios
(
휁
)
, and mode
shapes
(
휙
)
can be obtained. Using the complex eigenvalues
(
휆
)
and
eigenvectors
(
휈
)
of the system matrix
퐴
, and assuming small and classical damping, the modal
properties of the undamped system can be approximated
as
in
(5)
[1
6
, 1
7
].
푓
푛
=
|
휆
푛
|
/
2
휋
,
휁
푛
=
푅푒
(
휆
푛
)
/
2
휋
푓
푛
휙
푛
=
|
퐶
휈
푛
|
∙
푠푔푛
(
푅푒
(
퐶
휈
푛
)
)
(5)
where
푛
is the
푛
-
th mode,
푅푒
(
∙
)
is the real part, and
푠푔푛
(
∙
)
is the algebraic sign.
The N4SID algorithm has been implemente
d in the System Identification Toolbox of MATLAB
(202
3
)
and is used for results of modal identification of the 52
-
story building
.
To simplify analysis, it
is assumed the building is
a
LTI
system
.
For data pertaining to the Pacoima earthquake, the excitation
signal used as
the
known input
vector
is
set to be
the acceleration
measurement
closest to the ground
level on the second floor.
Fourier spectra were obtained by computing the Fast Fourier
t
rans
form
(FFT)
of the
120 second
acceleration time
-
series containing
the
seismic signal. Figure
3
shows the
smoothed
Fourier
spectra of the three
-
component acceleration records after removing individual
means from the recording. The spectra show the
excitation
of
lower frequencies
particularly in the
vertical (HNZ) direction during this earthquake
.
Figure 3.
Smoothed
Fourier spectra
of the
three
component
s, horizontal directions HNE and HNN and vertical
direction HNZ,
of floor two
for the
2020
Pacoima earth
quake
.
For response or output
vectors
, the remaining
upper
floor acceleration measurements are used.
Basement level measurements are not used since the focus is on structural behavior above ground
level
. For implementation of the N4SID algorithm, the orde
r of states to consider must be determined,
which can be a challenging problem the more degrees of freedom you have. Only focusing on the
vertical degrees of freedom (DOF), for floors two to fifty
-
two, the 52
-
story building is considered a
50
-
DOF system fo
r our purposes. At minimum, model order greater than the number of degrees of
freedom is needed. Due to inherent measurement noise, higher model order is typically needed to
extract as many stable modal parameters as possible. To distinguish stable modes f
rom modes that
are not consistent with increasing model order, stability criteria are set to focus on changes in
estimated frequencies, damping ratios and modal assurance criterion (MAC)
with increasing model
order
. Stability plots are created using tolera
nces:
Δ
푓
<
1%
,
Δ
휁
<
50%
,
푀퐴퐶
>
99%
(6)
Figure 4 displays
the stability plot for model identification using data from the Pacoima earthquake
with red circles for stable modes determined by tolerance criteria from (6)
.
Figure 4.
Stability plot for system identification results
from
recorded responses to
the
Pa
coima earthquake
with FFT
overlay
of floor 2. The y
-
axis represents the selected model order
.
There is a clear
dominant coherent pattern
of stable natural frequency at around 1.86 Hz.
It is
important to note that the
identified mode shape
vectors
that can
be determined from the SID results
are only identifiable at measured
DOF. Figure 5 shows the resulting mode shape, arbitrarily scaled,
associated with the first fundamental natural frequencies detected of 1.86 Hz. This is a simple 2
-
D
elevation view to ca
pture the basic overall vertical deformation behavior of the floor levels. The type
of deformation seen is either extending vertically by a positive amount or compressing vertically by
a negative amount. This type of mode can be considered a compressional
vertical mode. For other
modes detected like the one at about 2.1 Hz, the behavior is not all positively/negatively extending
and compressing at each floor level. This can be due to
contamination from
a
horizontal mode
or a
combination of
horizontal and fl
exur
al motion
.
A
fifth
translational mode
in the
North
-
South direction
at around this frequency
was observed for the
52
-
story building
in
[
4
].
Figure 5.
Mode shape results from
system identification results for natural frequency of 1.86 Hz.
CONCLUSION
Subspace state
-
space system identification
(N4SID)
is tested for its effectiveness in capturing
dominant modes excited
using the
vertical
component earthquake data
for a 52
-
story high
-
rise
building in downtown Los Angeles
.
This mode
consists of
compressional
and extensional motions
in
the
vertical direction,
caused by overall axial
extension of columns
.
Axial loads have significant
effects on the ductility of columns.
For lateral systems
such as
r
einforced concrete columns
that
typically do well under confinement,
the
extension of columns
can cause
reduction in
ductility
and
overall performance of columns
.
In addition, d
ifferences
in vertical
stiffness of inner core and
outrigger beams
on joints ca
used by the extension and compression of columns
can cause increased
stress
among
beam
-
column or slab
-
column critical points
which
can be damage
-
inducing
for some
types of earthquake scenarios
.
Further studies using data from other earthquake events that e
xhibit large vertical amplitudes are
needed to fully capture other dominant modes that can arise
for the high
-
rise building
due to the
vertical components of ground motion. Future work will concentrate on using SID results and finite
element modeling of the
tall building
to understand the type of deformations
and stress transfers
that
can occur for higher modes
in the vertical
direction
.
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