Ⓔ
Detection of Building Damage Using Helmholtz Tomography
by M. D. Kohler, A. Allam, A. Massari, and F.-C. Lin
Abstract
High-rise buildings with dense permanent installations of continuously
recording accelerometers offer a unique opportunity to observe temporal and spatial
variations in the propagation properties of seismic waves. When precise, floor-by-floor
measurements of frequency-dependent travel times can be made, accurate models of
material properties (e.g., stiffness or rigidity) can be determined using seismic tomo-
graphic imaging techniques. By measuring changes in the material properties, damage
to the structure can be detected and localized after shaking events such as earthquakes.
Here, seismic Helmholtz tomography is applied to simulated waveform data from a
high-rise building, and its feasibility is
demonstrated. A 52-story dual system
building
—
braced-frame core surrounded by an outrigger steel moment frame
—
in
downtown Los Angeles is used for the computational basis. It is part of the Community
Seismic Network and has a three-component accelerometer installed on every floor. A
finite-element model of the building based on structural drawings is used for the
computation of synthetic seismograms for
60 damage scenarios in which the stiffness
of the building is perturbed in different locations across both adjacent and distributed
floors and to varying degrees. The dynamic analysis loading function is a Gaussian pulse
applied to the lowest level fixed boundary condition, producing a broadband response on
all floors. After narrowband filtering the synthetic seismograms and measuring the maxi-
mum amplitude, the frequency-dependent travel times and differential travel times are
computed. The travel-time and amplitude measurements are converted to shear-wave
velocity at each floor via the Helmholtz wave equation whose solutions can be used
to track perturbations to wavefronts through densely sampled wavefields. These results
provide validation of the method
’
s application to recorded data from real buildings to
detect and locate structural damage using earthquake, explosion, or ambient seismic
noise data in near-real time.
Electronic Supplement:
Table and figures describing nine additional velocity
imaging tests that were run using the same procedure described in the main article.
Introduction
We show how approximating a building as a continuum
allows us to image temporally varying gradients in building
material properties using Helmholtz tomography. Represent-
ing a building as a continuum leads to convenient theory us-
ing the steady-state wave equation, or Helmholtz equation, to
image variations in elastic properties. This allows us to solve
the wave equation for heterogeneous spatial properties to
investigate anomalous gradients in an objective approach.
It also precludes the dependence of building damage detec-
tion on parameterized (modal) systems (
Farrar and Worden,
2012
, and references therein) that could bias the estimation
of the damage locations and spatial wavelengths. Solutions
to the wave equation for heterogeneity in material properties
in theory lead to better spatial resolution of damage locali-
zation. Moreover, dispersion properties can be measured in
a flexible way that does not depend on a parameterized or
analytical model of the building.
This study leverages the success of high-resolution
imaging techniques applied to seismic array data to develop
a propagating wave approach to detect and map damage in
buildings via waveform feature extraction immediately after
a major earthquake. We test the application of a waveform
time-series
–
based damage detection technique not applied
to buildings previously because before now, long-time-opera-
tional (order of years) seismic arrays in buildings have not
provided the spatial sampling resolution to detect spatial and
temporal variations in building response on a floor-by-floor
scale. For example, permanently installed seismic arrays in
structures in the United States operated by the California
Strong Motion Instrumentation Program in cooperation with
1
Bulletin of the Seismological Society of America, Vol. , No. , pp.
–
, , doi: 10.1785/0120170322
the U.S. Geological Survey usually provide single-channel
data from multiple floors but not continuously or in real time
(Center for Engineering Strong Motion Data, 2018, see
Data
and Resources
). The few exceptions include the seismic arrays
in the Atwood Building in Anchorage, Alaska (
Celebi, 2006
),
Factor Building in Los Angeles, California (
Kohler
et al.
,
2007
), Veterans Administration hospital buildings (
Ulusoy
et al.
, 2013
), and Green Building in Cambridge, Massachu-
setts (
Mordret
et al.
, 2017
), which are (or were) real time
and dense, in at least one channel.
We are motivated by the 1994 Northridge earthquake,
which highlighted a common type of unanticipated structural
failure
—
a brittle fractured weld in beam-column connections
—
that is difficult to identify either visually or through local-
ized ultrasonic testing (
Youssef
et al.
, 1995
;
Updike, 1996
;
Mahin, 1998
;
Roeder, 2000
;
Rodgers and Mahin, 2004
,
2009
). Post-Northridge analysis showed that weld fracture
significantly decreases the ductility of tall steel buildings (
Hall
et al.
,1995
). The prevalence of fractured welds in mid- and
high-rise structures shows how new computational tools that
take advantage of next-generation seismic arrays can identify
their occurrence after a large earthquake. This study numeri-
cally tests a damage detection method on computational finite-
element damaged model responses of a building that is also
currently instrumented by a strong-motion network.
A large bulk of mechanical structure damage detection
work uses analogous phenomena associated with guided elas-
tic waves propagating along the structure
’
s free surface boun-
daries to extract features (
Farrar and Worden, 2007
,
2012
).
One of the most predominant of these is the experimental gen-
eration and recording of Lamb waves, which has led to useful
techniques to locate damage features in plates, shells, frames,
and structural components associated with moving or rotating
machinery (
Farrar and Worden, 2012
). Like the approach pre-
sented here, Lamb waves have wavelengths of the same order
as the smaller dimension of the structure and wavespeeds that
are dispersive. Lamb wave generation is usually produced by
driving an actuator signal at specific frequencies and recording
the response at an array of proximal receivers; thus, variations
in propagating wave properties such as travel time, phase, am-
plitude, and reflections can be used to identify and locate dam-
age-related features in the structure. However, Lamb waves
are a combination of longitudinal and shear waves, but the
approach presented here has its analogies in shear-wave (as
opposed to compressional or longitudinal wave) seismic body
waves; it furthermore does not depend on knowing or meas-
uring the modal properties of the structure.
This study is a postevent, state-of-health approach that
is optimally applied in the presence of dense array data and
knowing pre-event wave propagation characteristics such as
phase velocities and amplitudes. It does not depend on record-
ing the transient signal during the damage event itself. This is
different from the few unique exceptions such as the studies
done by
Dunand
et al.
(2004)
and
Rodgers and Celebi (2005)
,
which involved analysis of transient signals recorded during
damaging shaking caused by the 1994 Northridge earthquake.
Low-cost microelectromechanical systems (MEMS) technol-
ogy sensors are now making it possible to instrument build-
ings on a floor-by-floor scale and to record continuous
vibration data at high sampling frequencies of 200 Hz or
higher. The Community Seismic Network (CSN) is one of
these low-cost networks, expanding at a steady rate and now
consisting of 900 active accelerometer sensors deployed in the
urban region of the Los Angeles basin, including on multiple
floors of buildings (
Clayton
et al.
, 2011
,
2015
;
Kohler
et al.
,
2013
,
2014
). Because the cost of the sensors is low, making
them easy to install at high densities over small areas, it is
critical to be able to interpret data recorded by the sensors in
the face of continuously changing environments subjected to
shaking and damage from earthquakes, natural hazards such
as wind storms, and anthropogenic hazards such as explosions
(
Kohler
et al.
,2016
). Community-hosted seismic arrays such
as CSN are now making it possible to analyze vibration data
on a small spatial scale because of the deployment of up to
three triaxial accelerometers per floor. CSN instrumented
more than 15 mid- and high-rise buildings that represent criti-
cal types of construction that are prone to damage during
strong earthquake shaking. They include a 52-story steel mo-
ment and braced-frame, a 9-story steel moment frame with
trusses and girders, a 15-story moment steel frame with con-
crete shear walls in the core, and a 12-story reinforced con-
crete with moment frames.
Seismic tomography techniques applied to arrays of seis-
mometers deployed on the surface of the Earth have been
successful for decades at imaging seismic velocities within
the solid Earth, including large-scale mantle structure (e.g.,
French
et al.
, 2013
), magma chambers (e.g.,
Farrell
et al.
,
2014
), fault zones (e.g.,
Allam
et al.
, 2014
), and individual
volcanoes (e.g.,
Lin
et al.
, 2014
). Recent instrumental and
methodological advances have led to the application of seismic
imaging techniques to spatially dense seismometer arrays (e.g.,
Lin
et al.
, 2013
;
Ben-Zion
et al.
, 2015
;
Hillers
et al.
, 2016
). In
particular, ambient seismic noise (
Shapiro
et al.
, 2005
)isnow
routinely used to construct surface waves and measure travel
times between pairs of seismic stations via cross correlation
(e.g.,
Lin
et al.
, 2009
;
Porritt
et al.
, 2011
;
Saygin and Kennett,
2012
;
Ward
et al.
, 2013
). These methods have also been
applied to ambient vibrations from instrumented buildings
(
Prieto
et al.
, 2010
;
Nakata and Snieder, 2014
;
Sun
et al.
,
2017
). Our study takes this one large critical step forward by
presenting here a method that identifies and locates seismic-
velocity changes in buildings; furthermore, we show how these
velocity changes are related to small-scale (localized) material
property changes (stiffness) due to damage.
A growing number of seismological studies demonstrate
that interferometric methods applied to seismic array data
show large-scale correlations between time-varying changes
in seismic velocity and rock fracturing or coseismic (or post-
seismic) stress redistribution. For example, continuous seis-
mic noise data recorded on the San Andreas fault have been
used to monitor small variations in seismic velocity (
Bren-
guier
et al.
, 2008
), and earthquake tomography has been
2
M. D. Kohler, A. Allam, A. Massari, and F.-C. Lin
applied to relate time-dependent variations in
V
P
=V
S
to
abrupt changes in stress, fracturing, and active transport of
fluids (
Koulakov
et al.
, 2013
). Correlations have been found
with precursory volcanic activity (
Lecocq
et al.
, 2014
),
annual oscillations in volcanic activity causing deformation
(
Hirose
et al.
, 2017
), and stress changes in rocks associated
with geothermal reservoirs (
Taira
et al.
, 2015
). In these stud-
ies, however, although small velocity changes can be detected,
their sources are not precisely located and are only generally
attributed to fault zone damage or stress change in rocks
undergoing deformation.
Similar methods are now also being applied to seismic
array datasets containing dynamic response of mid- and high-
rise structures to earthquakes, environmental conditions, and
ongoing sources of low-level vibrations. Building on earlier
work demonstrating the application of interferometric methods
to mid-rise structures (
Snieder and Safak, 2006
;
Kohler
et al.
,
2007
;
Prieto
et al.
, 2010
),
Nakata
et al.
(2013)
show that de-
convolution interferometry applied to earthquake data can be
used to compute time-lapse changes in velocities, resulting in
negative correlations between the maximum accelerations and
wavespeeds in the building; they furthermore demonstrate that
this method can separate the response of the building from
soil
–
structure coupling effects.
Nakata
et al.
(2015)
found,
from application of interferometric methods to 300 earth-
quakes before and after the 2011 Tohoku earthquake, that re-
ductions in seismic velocities in a 10-story reinforced concrete
building in Tokyo could be ascribed to damage in the building
because no velocity reductions were found in the underlying
near-surface soil layers. Most recently,
Mordret
et al.
(2017)
applied deconvolution interferometry to two weeks of ambient
vibration data from a 20-story reinforced concrete building,
observing correlations between small wavespeed changes (in-
cluding recoveries) and air humidity (as well as temperature to
a lesser extent). Although it has been established that useful
time-domain impulse response functions (or, more generally,
transfer functions) can be obtained and monitored for time-
varying changes of seismic velocities in buildings, techniques
to compute precise locations of these changes for use in dam-
age detection in buildings have remained elusive.
Approach
Theory
We show how solutions to the wave equation for mid- and
high-rise buildings enable damage detection. The goal is to
develop an approach for imaging structural defects in a system
for which the eigenfunctions and eigenfrequencies either are
not known or cannot be measured; if they are known, they can
provide complementary information. Solutions to the relevant
equations of motion describe longitudinal and shear waves
that are body waves propagating throughout a continuum. The
body-wave solutions describe wavefronts that travel through-
out the structure and whose properties can be used to conduct
nonparameterized system identification of the structure, as
well as identify locations of perturbations to its properties
(i.e., damage detection and localization). In this study, we fo-
cus in particular on vertical variations of elastic properties of
buildings and their dynamic responses to external forces. We
show that the assumption of a continuum is useful and appro-
priate for damage detection even if it is not correct on a struc-
tural element spatial scale (i.e., beam-column connection,
section of column). Through this assumption, we will show
how this leads to a second-order Helmholtz equation. The
Helmholtz equation provides a straightforward way to solve
for seismic-velocity variations in a building whose pre- and
postdamage states have been determined by vibration time
series recorded on seismic arrays.
To illustrate the parallels between a system approximated
as lumped masses with massless supporting structures and a
system represented as a continuum, we begin with the tradi-
tional force balance equation of motion used in structural
dynamics. Consider the equation of motion for deformation
of an idealized structure, for example, a steel moment
–
frame
high-rise, subjected to external dynamic forces. External
forces such as earthquake excitations
P
t
, are balanced with
resisting forces:
EQ-TARGET;temp:intralink-;df1;313;469
m
∂
2
u
i
∂
t
2
c
∂
u
i
∂
t
k
u
i
P
t
;
1
whose individual components comprise an inertial force
m
∂
2
u
i
∂
t
2
,
in which
m
is the mass matrix; damping forces
c
∂
u
i
∂
t
,inwhich
c
is the viscous damping coefficient; and the elastic (or inelastic)
resisting forces (force
–
displacement rela
tion, i.e., Hooke
’
slaw)
k
u
i
,inwhich
k
is the lateral stiffness matrix dependent on the
dimensions of the structure and elastic modulus;
∂
2
u
i
∂
t
2
,
∂
u
i
∂
t
,and
u
i
are acceleration, velocity, and
displacement responses as a
function of time
t
, with spatial coordinate index
i
1
–
3
(e.g.,
Chopra, 2001
). The equations of motion are typically ex-
pressed for single- or multiple-degree-of-freedom (MDOF)
systemsandaresolvedbyassumingalumpedmassandmass-
less supporting system, lumped stiffness, and an empirical
form of damping such as Rayleigh damping.
When a nonparameterized method of measuring dynamic
response is desired, it can be appropriate and useful to re-
present building structures as a continuum. This is possible
because the low-frequency shear waves traveling through a
building are influenced by the average properties of the build-
ing
’
s structural and nonstructural elements, free surfaces, and
empty volume (i.e., open air). They are not guided waves trav-
eling solely along the surface or interior elements, such as the
columns of the structure, and are not sensitive solely to each
individual element. This approximation holds for systems in
which the elastic moduli and density, that is, seismic veloc-
ities, vary smoothly in space (spatial variations in
λ
,
μ
,and
ρ
are small and gradual), and a solution to the wave equation can
be found by plane waves that describe propagating wave-
fronts. Consider data-based observations that are made on a
floor-by-floor spatial scale such as that made possible by com-
munity-hosted, triaxial, MEMS accelerometers deployed at a
Detection of Building Damage Using Helmholtz Tomography
3
density of at least one per floor. When we consider small
deformations (e.g., beam-column fractures or broken brace
frames) in the path of propagating waves with wavelengths
that are of the same order of magnitude as the spatial sampling
scale (one floor or
∼
5
m), then treating a structure as a con-
tinuous medium leads to methods that are complementary to
modal coordinate solutions of an MDOF system. We thus
focus on the equations of motion in terms of stresses and
strains for an elastic body acted on by internal and external
forces. The corresponding equilibrium equation analogous
to equation
(1)
is
EQ-TARGET;temp:intralink-;df2;55;601
ρ
∂
2
u
i
∂
t
2
f
i
∂
σ
ij
∂
x
j
;
2
which defines the relationship between the inertial forces
ρ
∂
2
u
i
∂
t
2
with density
ρ
,bodyforces
f
i
, and stress gradients
∂
σ
ij
∂
x
j
in
spatial coordinates
x
j
(e.g.,
Lay and Wallace, 1995
), with sum-
mation convention
j
1
–
3
. Using stress
–
displacement rela-
tionships described by the continuum version of the elastic
resisting force (Hooke
’
slaw)
EQ-TARGET;temp:intralink-;df3;55;487
σ
ij
C
ijkl
ε
kl
;
3
in which
σ
ij
is the second-order stress tensor,
C
ijkl
is the
fourth-order elastic modulus tensor, and
ε
kl
is the second-order
strain tensor and considering the strain
–
displacement relations
relating stress to the elastic parameters, the 3D homogeneous
vector equation of motion for a uniform, isotropic, linear elastic
medium becomes
EQ-TARGET;temp:intralink-;df4;55;385
ρ
u
λ
μ
∇
∇
·
u
μ
∇
2
u
4
(
Lay and Wallace, 1995
). This is the 3D partial differential
equation for displacements produced by an external force, with
λ
= bulk modulus,
μ
= shear modulus,
u
∂
2
u
i
∂
t
2
,
∇
=3Dpartial
derivative operator
∂
∂
x
1
;
∂
∂
x
2
;
∂
∂
x
3
, and the dot indicates dot
product. We will focus on solutions in orthogonal horizontal
directions for a test bed structure consisting of a 52-story
building.
The solutions to the 3D wave equation applied to build-
ing structures lead to seismic body waves with ray behavior
that is analogous to seismic body waves propagating through
the Earth
’
s interior. Waves propagate along paths throughout
the structure with propagation direction normal to the wave-
fronts. The physical and mathematical approximations that
can be made for body-wave behavior are given by geometric
ray theory in which waves in our building test bed are as-
sumed to follow minimum-time paths, and perturbations in
a building
’
s elastic properties lead to changes in travel time.
We take advantage of this property of geometric rays to show
how solutions to an approximate wave equation might be used
to identify and locate failure defects in buildings through the
mapping of seismic-velocity perturbations that result from
changes in travel time. Because we focus on low-frequency
response as already described, this approach allows us to
neglect the anisotropy that is certainly present in the high-
frequency response of wave propagation on a structural
element spatial scale.
The 3D wave equations as they apply to geometric ray
theory for waves traveling at compressional or shear velocity
c
p
x
or
c
s
x
are
EQ-TARGET;temp:intralink-;df5;313;673
∇
2
φ
1
c
2
P
x
∂
2
φ
∂
t
2
5
and
EQ-TARGET;temp:intralink-;df6;313;614
∇
2
Ψ
1
c
s
2
∂
2
Ψ
∂
t
2
;
6
with scalar potential
φ
, vector potential
Ψ
, and total displace-
ment
u
∇
φ
∇
×
Ψ
; these are an approximation to the
equation of motion for heterogeneous media (
Aki and Ri-
chards, 1980
;
Lay and Wallace, 1995
;
Shearer, 1999
). We next
consider a general functional form of displacement,
φ
x
;t
given by plane-wave solutions
EQ-TARGET;temp:intralink-;df7;313;495
φ
x
;t
A
x
e
i
k
·
x
ω
t
;
7
in which
A
x
is wave amplitude,
ω
is frequency,
k
is a vector
pointing in the direction of wave propagation, and
x
is the 3D
spatial coordinate vector (
Lay and Wallace, 1995
;
Shearer,
1999
). For the single-direction propagating wave of interest,
for example, a shear-wave traveling vertically up a building, as
considered in this study, this will be
EQ-TARGET;temp:intralink-;df8;313;389
φ
x
;t
A
x
e
i
k
·
x
−
ω
t
8
or
EQ-TARGET;temp:intralink-;df9;313;343
φ
x
;t
Ω
x
e
−
i
ω
t
;
9
in which
EQ-TARGET;temp:intralink-;df10;313;297
Ω
x
A
x
e
i
k
·
x
;
10
and
x
defines the 3D position coordinates. Within a building,
this would be the vertical, and two orthogonal horizontal direc-
tions;
j
k
j
ω
c
are the wavenumbers. For a constant frequency,
φ
x
;t
represents a complex, monochromatic wavetrain de-
fined by amplitude and phase.
For heterogeneous media, it is more useful to describe
the wave-position vector expression
k
·
x
in terms of a travel-
time function that defines wavefront surfaces propagating
with local slowness when this function is a constant. The
travel times are often measured directly from the data, as is
done in this study, to map seismic velocities. Replacing
k
·
x
with
ωτ
x
, in which
τ
x
is the travel time as a function of
position, we obtain a similar expression to equation
(10)
:
EQ-TARGET;temp:intralink-;df11;313;107
Ω
x
A
x
e
i
ωτ
x
:
11
4
M. D. Kohler, A. Allam, A. Massari, and F.-C. Lin
Using this form for
φ
in equation
(9)
, the spatial and tem-
poral parts of
φ
can be separated upon insertion into the wave
equation. Substituting equation
(9)
into equation
(5)
, we ob-
tain the time-independent (steady-state) wave equation with
EQ-TARGET;temp:intralink-;df12;55;685
∇
2
Ω
x
−
ω
2
c
2
x
Ω
x
;
12
which is also referred to as the Helmholtz equation. Equating
the real terms of each side of equation
(12)
using equa-
tion
(11)
yields
EQ-TARGET;temp:intralink-;df13;55;602
1
c
2
x
j
∇
τ
x
j
2
−
∇
2
A
x
A
x
ω
2
13
(
Lay and Wallace, 1995
). Because small-scale properties are
being averaged in the solution, we focus only on isotropic
properties. If much higher frequency signals were resolvable
in seismic array observations, then equations
(5)
–
(13)
would
need to be modified to account for shear-wave anisotropy. A
similar derivation can be carried out for shear waves using
equation
(6)
; thus,
c
x
represents either compressional or
shear-wave velocity. This is the basis of the gradient calcu-
lations used to compute narrowband-filtered shear-wave
velocities in this article.
This derivation is the same as the elastic Earth derivation
applied to 2D surface waves (
Lin and Ritzwoller, 2011
). The
right side of this equation containing the spatial Laplacian of
the amplitude is referred to as the amplitude term and will be
discussed in further detail later in the
Results
section. If we
assume that the amplitude term is small and negligible (when
ω
is large), then equation
(13)
reduces to equation
(14)
,a
partial differential equation relating the seismic-wave travel
times to phase velocity distribution, which is the eikonal
equation used in seismology
EQ-TARGET;temp:intralink-;df14;55;315
1
c
x
j
∇
τ
x
j
14
(
Lay and Wallace, 1995
;
Shearer, 1999
). Waves propagating
vertically up a building may be finite frequency in nature.
Finite-frequency effects such as large lateral stiffness gradients
and wave interference can produce nonnegligible bias on wave
velocity estimates based on the eikonal equation (equation
14
).
However, if the wave amplitude varies smoothly on the single
wavelength scale, then the second term on the right side of
equation
(13)
may be insignificant.
Numerical Test Bed.
Damage-detection calculations using
the theory and approach described earlier were carried out
with the aid of linear-elastic finite-element models of an
existing building, developed based on detailed information
contained in structural drawings provided by the building
owner. The major structural and connection elements ob-
tained from the drawings were modeled using object-based
physical-member modeling, such as built-in steel sections
and braces, to represent each component
’
s effective level of
stiffness and mass (ETABS, Computers and Structures, Inc.).
The dynamic modeling software allows for static and
dynamic linear simulations, as well as nonlinear analysis
through insertion of nonlinear elements at locations of inter-
est. The models were further refined by comparing recorded
data from the building with simulated building response.
Validation of the pre-event response baseline was also
provided by the continuous ambient vibration sensing.
The test bed building consists of a 52-story (
5
basement
levels) high-rise building located in downtown Los Angeles
(Fig.
1
). This building
’
s lateral dual system consists of a
braced-frame core surrounded by a steel moment frame. The
floor plans contain various setbacks and notches along the
building
’
s vertical profile. The building was constructed in
1988 and is used exclusively as an office building. The struc-
tural system consists of three major components: an interior
concentrically braced core, outrigger beams spanning
∼
12
m
from the core to the building perimeter, and eight exterior
outrigger columns (Fig.
1
). The beams perform three primary
functions: They support gravity loads, act as ductile moment-
resisting beams between the core and exterior frame columns,
and enhance the overturning resistance of the building by en-
gaging the perimeter columns to the core columns (
Taranath,
1997
). CSN has instrumented this building with one, and on a
few floors two, triaxial sensors per floor, recording continuous
acceleration waveform data. Several small- and moderate-size
earthquake recordings made by CSN from this building, for
example, the earthquake records shown in Figure
2
,were
used for model validation. The first few translational resonant
frequencies of this building in the horizontal directions are
∼
0
:
2
,0.6,and1.2Hz(
Kohler
et al.
,2016
).
Damage Scenario Generation Method.
To generate the
multiple building damage scenarios, we worked with a cus-
tomized user interface to ETABS that computes modified lin-
ear finite-element models that contain user-specified damage
scenarios. We used our undamaged finite-element model as
the starting points with this interface to generate on the order
of 60 new models of each building, each of which incorpo-
rates a realistic damage state that would be expected after
strong shaking but that has not resulted in collapse or near
collapse (Table
1
).
The structure featured in this study offers different types
of damage scenarios to explore in detail. The 52-story build-
ing
’
s dual system provided an opportunity to test the interplay
between beam-column connection failures and global buck-
ling and tension rupture of the braced-frame components.
In this study, brace frame damage is imposed on single and
multiple floors distributed throughout the model (Table
1
).
Travel Times.
For each damage-state scenario listed in
Table
1
, displacements are computed by solving the linear
equation of motion for the modified finite-element model
with imposed damage. The external force applied at the
base of the structure (lowest basement
“
0E
”
level) is actually
Detection of Building Damage Using Helmholtz Tomography
5
acceleration derived from displacement represented by a
Gaussian function with width equal to
T
0
=
15
, in which
T
0
is the fundamental period (
∼
5
:
5
s). The reason for using a
relatively narrow Gaussian is to produce a broadband re-
sponse similar to an earthquake (e.g.,
Aki, 1972
), and it is a
useful representation of the impulse response function of the
building that could be computed from recorded data.
The simulated displacements consisting of each floor re-
sponse (from the lowest basement 0E level to the 50th floor)
are filtered using 39 separate Butterworth filters with center
periods ranging from 1.0 to 20.0 Hz, incremented at 0.5-Hz
intervals. A sigma (half-width) value equal to 10% of the
central period is applied to obtain the narrowband-filtered
numerical (synthetic) waveforms. The amplitude at each fre-
quency is measured as the maximum of the real part of the
Hilbert transform of the filtered waveform.
To measure the travel time as a function of propagation
distance and frequency, we present two different methods
representing different levels of prior knowledge about the
system. The first method assumes that no characterization of
the building
’
s travel-time properties exists prior to the propa-
gating wave, and the travel time is calculated by assuming
the base floor as a reference waveform. The second method
assumes some earlier undamaged characterization of the
structure before the propagating wave, allowing the travel
time to be calculated by comparison of damaged and undam-
aged cases. For the first method, the travel time of the lowest
floor (lowest basement 0E level) is treated as a reference
(b)
(a)
(c)
Figure
1.
Finite-element model of the 52-story dual system building used in the dynamic analysis. (a) 3D frame image. Outrigger moment
frames and diagonal braced core elements are shown in gray. Interior core is also shown in gray. (b) 2D profile showing frame and brace
elements. (c) Plan view showing outrigger moment frame configuration typical of most floors. The color version of this figure is available
only in the electronic edition.
6
M. D. Kohler, A. Allam, A. Massari, and F.-C. Lin