of 9
DETECTING FAILURE EVENTS IN BUILDINGS: A NUMERICAL AND
EXPERIMENTAL ANALYSIS
V. M. Heckman
1
M. D. Kohler
2
and T. H. Heaton
3
ABSTRACT
A numerical method is used to investigat
e an approach for detecting the brittle
fracture of welds associated with beam
-column connections in instrumented
buildings in real time through the use of
time-reversed Green’s functions and
wave propagation reciprocit
y. The approach makes use of a prerecorded catalog
of Green’s functions for an instrumented building to detect failure events in the
building during a later seismic event
by screening continuous data for the
presence of waveform similarities to one
of the prerecorded events. This study
addresses whether a set of Green’s functions in response to an impulsive force
load can be used to approxi
mate the response of the stru
cture to a localized failure
event such as a brittle weld fracture. Specifically, we investigate whether
prerecorded Green’s functions
can be used to determine the absolute time and
location of a localized
failure event in a
building. We also seek to differentiate
between sources such as a weld fractur
e that are structurally damaging and
sources such as falling or colliding furn
iture and other non-st
ructural elements
that do not contribute to structural fa
ilure. This is explored numerically by
comparing the dynamic response of a fin
ite-element cantilevered beam model
structure to a variety of loading m
echanisms. A finite-element method is
employed to determine the behavior of th
e resulting elastic waves and to obtain a
general understanding of th
e structural response.
Introduction
The 1994 Northridge earthquak
e caused structural failure,
notably fracture of steel
frames, in more than one hundred buildings that
were designed for the ground motions produced
by the earthquake. As a result, civil engineers recognized the potential seismic hazard of brittle
fracture of welded beam-column connections
in steel frame buildings (Updike 1996). Post-
Northridge analysis has shown that weld fracture
significantly decreases the ductility of tall steel
buildings (Hall 1995). Olsen (2008) report that simula
tions of 20-story steel
buildings with brittle
1
Graduate Student, Department of Civil Engineering, Ca
lifornia Institute of Technology, Pasadena, CA 91125
2
Professional Researcher, Center for Embedded Networked Sensing, University of California at Los Angeles, Los
Angeles, CA 90095
3
Professor of Engineering Seismology, Department of Ci
vil Engineering, California Institute of Technology,
Pasadena, CA 91125
welds are three to five times as likely to collapse
in near-source shaking than identical buildings
with perfect welds.
According to Rodgers and Celebi (2005), sudden
structural failure (suc
h as brittle failure
of welded connections) should produce both near
and far-field dynamic response that is
observable over seismic loading. Ro
dgers and Celebi further suggest
that the frequency content
of this dynamic response should
be higher than that
of the predominant
building response for
several reasons: 1) the sudden chan
ge in the stiffness and streng
th of damaged members results
in sudden changes in global accelerations to satisfy the equations of motion; 2) weld fracture
occurs over the span of milliseconds and results in
excitation of local member vibration modes;
and 3) the sudden release of strain energy create
s elastic waves that propagate away from the
fracture in connected members.
A sophisticated damage detection system th
at monitors data recorded by a structure’s
vibration monitoring array will take into account locations of relatively high probability of
failure, such as welded connections, and use wa
veform data to detect locations of dynamic
failure. If it is possible to assemble a database of
expected building response
s to failure at each of
these locations (by using the building response to an impulse load applied at each location, e.g.
the Green’s function), then data
recorded by the structure’s se
ismic array during an earthquake
could be screened continuously in
real time for the presence of one
or more failure events at each
of the suspect locations. This
would be accomplished by performing a series of cross correlations
using both the preexisting database of building
responses as well as by running a time-window of
seismic data. Application of a high-frequency filter
further distinguishes the transient response to
dynamic failure from the predominant building response.
Such a damage detection system
would be able to sort throu
gh a wealth of seismic data in
real time as well as store detected locations of
probable damage for later use. This information
could be used to help guide physical inspecti
on of welded connections after an earthquake.
Inspections can be expensive as weld fracture often occurs wit
hout accompanying distress to
architectural finishing and clad
ding. The level of spatial cove
rage and tempor
al resolution
needed by a seismic array to a
ttain a high level of spatial reso
lution required
to distinguish
between failure events at different beam-column
connections remains to be seen. However, the
increasing prevalence of new inexpensive smaller sensors (e.g. MEMS-based USB sensors
designed for use at desktop com
puter sites such as the Quake-
Catcher Network; Cochran 2009),
and the plans for network installation during ne
w high-rise construction (Los Angeles Tall
Buildings Structural Design Council 2008) ma
kes the method proposed here application-
realistic.
Methodology
Kohler et al. (2009) show that wavefield
properties of linear elas
tic media provide the
basis for a method to determine the location a
nd time of the occurrence of a high-frequency
fracture event in a civil structure. First, th
e structural displacement from a hammer blow
(impulse point force load) is expressed in the
Green’s function for the building. A weld fracture
is a more complicated source signal and can be
approximated two ways. The first approximation
is the one used is this study; it consists of an elastic model with a tensile crack that experiences a
step change in normal traction on the crack surf
ace. An alternative way to approximate a weld
fracture is a method commonly used in seismol
ogy to model an earthquake source. The weld
fracture is approximated as a localized region
that experiences large elastic tensile strains
resulting in finite opening across the strain
ed region. A body-force e
quivalent source is
characterized by a seismic moment
tensor that consists of a comb
ination of linear force couples
and shear force couples. The response of the me
dium to these force couples is the spatial
derivative of the point force Green’s functions.
The response of the medium can be decompos
ed into near-field and far-field terms. The
near-field terms are necessary to describe the steady-state solution. The far-field terms are
solutions to the homogeneous equation of motion and
generally have a time history that is a time
derivative of the near-field terms. Kohler et al
. (2009) show that if the far-field portion of the
Green’s function is approximated by a sum of tr
aveling rays, each trav
eling at a different
velocity, then for each ray in the sum the exp
ected similarities between seismograms produced
by a hammer blow point source and the seism
ograms produced by a weld fracture differ only by
a constant ratio; this ratio is different for each
ray and is a function of the radiation patterns of
the different sources. Given these assumptions, th
e correct location and time of each subsequent
fracture source can be inferred by identifying whic
h Green’s function has a high cross correlation
value when it is cross correlated with a record
from an individual station. If the signal from a
weld fracture is recorded on several stations, it
should produce an impulse at the same time for
the cross correlation at each station.
Numerical Method
Finite-element analysis is used to simulate the response of a model structure to various
loading mechanisms in order to determine conditions for which a set of Green’s functions
approximates the response of the structure to a
complex failure source such as a weld fracture.
Numerical simulations are conducted on a cantilevered beam finite-element model. The goal is to
compare a cantilevered beam’s response to an im
pulsive load to that of an opening crack.
Numerical analysis is carried
out using Pylith, a finite-ele
ment code for the solution of
dynamic and quasi-static tectonic defo
rmation problems that runs in ei
ther serial or parallel mode
(Aagaard 2007). It has the flexibility to work on large and small spatial and temporal scales, and
it is well-suited for our problem as it is capab
le of simulating wave propagation through solids.
Material properties (i.e., density, shear-wave and compressional-wave velocities) and parameters
for boundary and fault conditions are user-sp
ecified through a spatial database. Boundary
conditions include the option of applying a force to
selected nodes, with force-time histories that
can be specified by the user. Nodes are chosen to
satisfy a desired boundar
y condition in Cubit, a
full-featured software toolkit fo
r robust generation of 3D finite
-element meshes (Benzley 1997).
The cantilevered beam models and accompanyi
ng force-time histories, shown in Fig. 1,
simulate a horizontal impulsive force (i.e., hammer
blow) and a crack subjected to a step change
in tensile stress (i.e., weld fracture). The material has the properties
of A36 structural steel, with
a Young’s modulus of 200 GPa, shear m
odulus of 80 GPa, density of 7850 kg/m
3
, and shear-
wave and compressional-wave speeds of 3.2
and 5.6 km/s, respectively. We first test our
methodology on a simple case; we constrain our me
sh to experience displacement only along the
a)
0.05m
F
F
F
0.05m
0.05m
0.5m
5m
b)
Figure 1. Cantilevered beam m
odel. a) A horizontal impulsive fo
rce is applied to an unnotched
beam (top). A crack (square notch) is subjec
ted to a step change in tensile stress
(bottom). A model mesh size of 2.5 cm and a time step of 1
μ
s are used. Beam
dimensions are 5.0 m
×
0.5 m
×
0.25 m. The mesh is constrained to experience
displacement only along the longitudinal axis
. The longitudinal force is distributed
over nodes located in each highlighted cro
ss section of the beam. The value of the
force applied to each node scales with th
e amount of surface area contained by the
node. b) Force-time histories applied to node
s. The red (dashed)
curve simulates an
impulsive hammer blow. The black (solid)
curve simulates a weld fracture. The
maximum force (F = 2 kN) is multiplied
by a time-dependent scale factor.
R
1
R
2
R
4
R
3
0.25m
1.25m
1.25m
1.25m
1.25m
Figure 2. Receiver locations
. Receivers are equally spaced along the beam’s center axis.
longitudinal axis. This effectively reduces th
e problem to two dimensions and removes the
possibility of dispersion. The resu
lting displacements, velocities
and strains are recorded at the
four receiver locations shown in Fig. 2.
Experimental Results
The numerical solution is consistent with
the traveling wave solution to the beam
equation (Timoshenko 1951). According to Timoshe
nko, a body’s reaction to
a suddenly applied
force is not present at all pa
rts of the body at once. The remo
te portions of the body remain
unaffected during early times. Deformation propaga
tes through the body in the form of elastic
waves. This can be observed in Fig. 3 which shows an elastic wave propagating away from the
source. At first the beam reaction is present only
in the immediate vicinity of the source. Elastic
waves reach the ends of the beam at 450
μ
s. Reflected waves interfere in a manner consistent
with the traveling wave solution for long
itudinal vibration of a prismatic beam.
The numerical solution for the unnotched beam
is similar to the analytic solution to the
longitudinal vibration of a beam with initial
conditions of zero displacement and a positive
horizontal pulse in velocity along the vertical axis that passes through R
2
. The analytic solution
consists of the superposition of two waves: a comp
ression wave traveling to the right and a tension
wave traveling to the left. If the beam is constr
ained to move only along the longitudinal axis, then
the wave speed will equal the
compressional-wave speed,
c
p
=
μ
(
E
4
μ
)
ρ
(
E
3
μ
)
(1)
where
E
,
μ
, and
ρ
are the Young’s modulus, shear modulus
, and density, respectively. If this
constraint is not present, the wave speed will be the longitudinal
wave speed in a prismatic beam,
c
l
=
E
ρ
(2)
A wave that reflects off the free
end of the beam has the same polarity and magnitude as the
original forward traveling wave,
since the sum of the combined st
rains must equal zero. A wave
that reflects off the fixed end has the same ma
gnitude but opposite polarity as the original
backward traveling wave, since
the sum of the displ
acements must equal zero (Timoshenko 1951).
Longitudinal Displacement
Unnotched beam
Time (
μ
s)
Notched beam
100
500
900
200
300
400
600
700
800
Longitudinal Displacement (m)
-10
-
8
10
-
8
0
Figure 3. Beam displacements for two source loadi
ng cases. Left panels show the response of the
unnotched beam to an impulsive horizontal force. Right panels show the response of
the square notched beam to a step source applied to open the crack. In both cases, the
left end of the beam is fixed and the right
end is free. Time is listed along the left.
Waves interfere either constructively or deconstructively after reflecting off the right
and left ends of the beam in a manner consistent with th
e traveling wave solution for
longitudinal vibration of prismatic beams.
In the left column panels of Fig. 3, an elas
tic wave begin to propagate away from the
source, with compression to the right of the
source and tension to its left. The forward
propagating wave reflects off the fr
ee end of the beam at a time of 450
μ
s. The average
combined displacement of the original and newly reflected waves is twice the displacement of
the original wave. At the fixed end of the b
eam, the resulting averag
e displacement from the
original backward propagating wave
and its reflected wave is ze
ro. The asymmetric application
of the load results in additional co
mplexity in the numerical solution,
due to vertical reflections of
waves off the top and bottom of the beam. A Ga
ussian FIR filter is applied to recorded
displacements to remove frequencies of high mo
des of oscillation and reveal the underlying
Dis
p
lacement
(
m
)
Velocity (m/s)
R
1
R
2
R
3
R
4
Figure 4. Seismograms for an impulsive force (red) and a tensile crack (blue); each seismogram
displays either the filtered di
splacement (left) or velocity (right) recorded at a given
receiver location (labeled along th
e left). The period is 3.6 ms.
planar wave solution. The filtered displacements
and velocities, shown in Fig. 4, are similar to
the one-dimensional solution to the
longitudinal vibration of a beam
subjected to an initial pulse
in velocity.
The velocity records in Fig. 4 illustrate how
they are better suited for performing cross
correlation than displacement records due to their pulse-like nature. The displacement records
resulting from the impulsive load do not always gi
ve a good approximation to the displacement
records resulting from the opening crack. This is due
to the fact that the crack source double couple
initially leads to compression in
both sides of beam, whereas the
single impulsive force initially
leads to compression in the right
half of the beam and tension in the left half of the beam. The
pulses in velocity often only differ in polarity, su
ggesting that the absolute
values of the recorded
velocities at each receiver location in response to an
impulse load can be used to approximate the
absolute velocities recorded at a
number of locations to a crack in
the structure at that receiver
location. The reason for the discrepanc
y between velocity records at R
2
for the two different source
loads is that R
2
lies at a node where the two waves orig
inating from the notch deconstructively
interfere. For the notched case, R
2
,
R
3
, and R
4
displacements oscillate about a nonzero offset due to
tensile deformation across the notch.
Conclusions
We performed a numerical study to gain in
sight into applying a
novel method to detect
high-frequency dynamic failure in buildings. A 2D
finite-element model of a cantilevered beam
that is discretized on a spatial scale of centimeter
s provides the basis for our numerical tests. The
beam’s elastic, longitudinal response to two lo
ading cases, impulsive force and opening crack
tensile stress, is computed on a temporal scal
e of microseconds. We find that the velocity
waveform of a tensile crack can be approxima
ted by the velocity waveform of a horizontal
impulsive force load applied at
the proper location. These results
support the idea of using an
impulse force to approximate a weld fracture
damage source to characterize a high-frequency
fracture event in the far field. The similarities be
tween impulsive force and tensile crack velocity
waveforms is further underscored by
the fact that, for traveling waves of either type of source,
the strains are proportional to their associated part
icle velocities. The results pave the way for the
use of waveform cross correlation using a pre-ev
ent catalog of Green’s functions to determine
the location and time of occurrence
of a subsequent fracture reco
rded on a network of vibration
sensors. It remains to be seen what types of
waves, reflections, and in
terfering wave behavior
will occur in three dimensions because of the added complexity due to dispersion and surface
waves.
Acknowledgments
We would like to thank donors to the
Hartley Fellowship for their support.
References
Aagaard, B., C. Williams, M. Knepley, 2007. PyLith:
A finite-element code for modeling quasi-static
and dynamic crustal deformation,
Eos Trans. AGU
, 8 8(52), Fall Meet. Suppl., Abstract
T21B-0592.
Benzley, S., R. Kerr, S. R. Jankovich, and D. B.
Mcrae, 1997. CUBIT Mesh Generation Environment.
Sandia National Laboratories. Albuquerque NM.
Cochran, E. S., J. F. Lawrence, C. Christensen,
and R. S. Jakka, 2009. The Quake-Catcher Network:
Citizen Science Expanding Seismic Horizons,
Seis. Res. Lett., 80
, 26-30.
Hall, J., T. Heaton, M. Halling, and D. Wald, 199
5. Near-source ground motions and its effects on
flexible buildings,
Earthquake Spectra
,
11
, 569-605.
Kohler, M. D., T. H. Heaton, and V. Heckman, 2009. A time-reversed reciprocal method for detecting
high-frequency events in civil structures with accelerometer arrays,
Proceedings of the 5
th
International Workshop on Advanced Smart Materials and Smart Structures Technology,
Boston,
MA, July 30-31.
Los Angeles Tall Buildings Structural Design Council (LATBSDC 2008),
An Alternative Procedure for
Seismic Analysis and Design of Tall Buildings Located in the Los Angeles Region
, Los Angeles,
CA.
Olsen, A., B. Aagaard, and T. Heaton, 2008. Long-period building response to earthquakes in the San
Francisco Bay area,
Bull. Seis. Soc. Am., 98
, 1047-1065.
Rodgers, J. E., and M. Celebi, 2005. A proposed met
hod for the detection of steel moment connection
fractures using high-frequency, transient accelerations,
USGS Open-File Report 2005-1375
.
Timoshenko, S., and J. N. Goodier, 1951.
Theory of Elasticity, 2nd Ed.,
McGraw-Hill, New York, New
York.
Updike, R. E., 1996. USGS Response to an Urban Earthquake - Northridge '94,
USGS Open-File Report
96-263
.