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Damage Detection in a Simple Shear Beam Model:
Modal Analysis and Seismic Interferometry
Vanessa Heckman
Monica Kohler
Tom Heaton
May 25, 2012
University of Memphis
Overview
• Modeling a building as a continuous beam
• Seismic Interferometry of UCLA Factor Building
• Experimental Analysis Using Uniform Shear Model
Flexible Building as a Continuous
Cantilevered Beam
• Dynamic motions of a continuous beam can provide
insight into the deformation of buildings.
• Example: MRF
• If the building is considered as a continuum, it would often be
•
•
•
•
anisotropic.
Stiffness associated with inter-story drift (ε13 and ε23) much less
than the stiffness associated with shearing the actual floor slabs
(ε12).
Stiffness associated with extension along columns (ε33) is different
than for extension along the floor slabs (ε11 and ε22).
Assume that only parts of the strain tensor which are important to
describe the deformation of the building are inter-story shear strain
and extensional strain along the columns.
Can thus approximate the building as being isotropic, since the
other elastic moduli are not important.
Heaton, T. CE 151 Class Material, 2010.
Shear Beam
• When the ground beneath the building moves
horizontally, this is identical to the problem of
having an SH wave propagate vertically in a
layer of building; the bending is approximately
zero in this case.
• Total stiffness against shear is equal to the shear
modulus times the cross sectional area.
• While actual buildings are neither a true bending
beam nor a shear beam, we can gain some
useful insight by looking at these approximate
modes of deformation.
• It is identical to the problem of a vertically
propagating SH wave in a plate with a rigid
boundary at the bottom and a free boundary at
the top.
Heaton, T. CE 151 Class Material, 2010.
Shear Beam
• Solution to this problem can be written as a sum of reflecting pulses. The motion
in the building is given by
u1g(t)
c=β=
• Drift is given by
m/r
Horizontal motion of the ground at the base of the building
Shear wave velocity in the building
• Base of the building: zero displacement, double the drift
• Top of the building: zero drift, double the displacement
• Sequence repeats with periodicity 4h/c, which is the fundamental period of the
building oscillation.
Heaton, T. CE 151 Class Material, 2010.
Shear Beam
• Mode Shapes for the first four modes
• Natural Frequencies
Background: UCLA Factor Building
• UCLA Factor Building
• 17-story, moment-resisting steel-frame structure
• Embedded 72-channel accelerometer array
• N-S modal frequencies: 0.59 Hz, 1.8 Hz, 3.1 Hz (1:3:5 ratio for shear beam)
• E-W modal frequencies: 0.55 Hz, 1.6 Hz, 2.8 Hz (1:3:5 ratio for shear beam)
• Experimental data and numerical model (ETABS)
Kohler, M. Heaton, T., Bradford, C. 2007. BSSA.
Background: UCLA Factor Building
• Impulse Response Function
•
•
•
Experimental Hammer Test
Deconvolution is used to extract the transfer function
Bandpass filtered between 0.5 Hz and 10 Hz
Stack over small EQs to stay in linear response
• Shear beam: waves travel nondispersively throughout the
lower floors of the building (v = ~160 m/sec)
• For bending beams, the waves would disperse with the wave
velocity increasing as the square root of the frequency
Impulse Response Functions
Kohler, M., Heaton, T., Bradford, C. 2007. BSSA.
Laboratory Example: Uniform Shear Beam
• Experimental Setup
• Five-story uniform shear model
• Piezoelectric accelerometers, DAQ
• Force transducer hammer
• Shake table, signal generator
• Damaged/undamaged configuration
Laboratory Example: Uniform Shear Beam
• Consider the structure modeled by a multi-degree-of-freedom system
æ
ç
ç
ç
x(t) = ç
ç
ç
ç
è
x1 (t) ö
÷
x2 (t) ÷
÷
x3 (t) ÷
x4 (t) ÷
÷
x5 (t) ÷ø
Mx(t)+Cx(t)+ Kx(t) = f (t)
æ
ç
ç
M = mç
ç
ç
ç
è
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
ö
÷
÷
÷
÷
÷
÷
ø
æ
ç
ç
K = kç
ç
ç
ç
è
2 -1 0 0 0 ö
÷
-1 2 -1 0 0 ÷
0 -1 2 -1 0 ÷
0 0 -1 2 -1 ÷
÷
0 0 0 -1 1 ÷ø
• Compute constants m, k from mass, geometry, and material properties
• m: Mass of each floor
• k: Stiffness due to bending plate (column)
• Determine the mode shapes and frequencies of the system from the
eigenvalues and eigenvectors of M-1K
• Damping is found experimentally
• 1st mode: logarithmic decrement method
• Higher modes: half-power bandwidth method
• Neglected: ζn< 0.005, n = 1,…,5
Kohler, M., CE 181 Class Notes, 2010.
MDOF System
• Transfer function used to find mode shapes
• Relates the response at DOF k to simple harmonic excitation at DOF m
frkfrm
H km (wr ) = å2
2
M
(w
+
2i
z
w
+
w
rr
r r
r )
r=1
N
frk
zr
wr
kth component of the rth mode shape
modal damping
frequency of the rth mode
• Consider the transfer function at the rth modal frequency
frkfrm
H km (wr ) = i+0
2
M (2z rwr )
H km (w r ) frkfrm frk
= m m= m
H mm (w r ) fr fr fr
Kohler, M., CE 181 Class Notes, 2010.
Damping Coefficient
Logarithmic Decrement Method
• 1st Mode damping ratio
• Free vibration
Half-Power Bandwidth Method
• Higher Mode damping ratio
• Based on the response of a SDOF
system, but can be used for MDOF
system as long as there is no
coupling between the modes
• Can be problems obtaining low
damping ratio due to freq resolution
Mode Shapes, Frequencies, and Wave Propagation
• Calculated vs. Observed: Good
agreement
• Damage: Introduced at the 4th floor
• Damaged vs. Undamaged
Calculate
d
Observed
Damaged
ω1
3.2
3.3
2.6
ω2
9.5
9.6
6.2
• Significant decrease in frequencies
ω3
14.9
15.2
10.8
• Changes in mode shapes
ω4
19.2
20.0
14.6
ω5
21.9
22.4
21.2
• Changes in wave propagation
SH Plane Waves
• Ray Path Diagram
• Continuous displacements
transmitted
θT
β2< β1
β2=sqrt(μ2/ρ2)
• Continuous tractions
• Reflection/Transmission Coefficients
x1
β1=sqrt(μ1/ρ1)
incident
• Vertical incidence
θR
θI
x3
reflected
• Fixed Surface: Rss = -1
• Free Surface: Rss = 1
• Higher to Lower Velocity/Stiffness: Rss > 0; Tss > 1
Heaton, T. CE 151 Class Material, 2010.
Experimental Results: 1st Floor
Experimental Results: 2nd Floor
Experimental Results: 3rd Floor
Experimental Results: 4th Floor
Experimental Results: 5th Floor
Seismic Interferometry
• Seismic interferometry may aid in damage
detection by comparing post-event
waveforms with pre-event waveforms
• Changes in wave speed
• Floor-to-floor reflections
• Forward modeling using a numerical model
(ETABS) will be used to test the application
to civil structures
Impulse Response Function
Kohler, M., Heaton, T., Bradford, C. 2007. BSSA.
Hayashi, Y., Sugino, M., Yamada, M., Takiyama, N., Onisha, Y. 2012. STESSA
Acknowledgements
• Thomas Heaton, Professor of Seismology, Caltech
• Dr. Monica Kohler, Researcher, Caltech
• Ming Hei Cheng, PhD Candidate, Caltech
• Rob Clayton, Professor of Geophysics, Caltech
• Hartley Fellowship
• Housner Fund
THANK YOU!
Questions?
Seismic Interferometry
• Cross-Correlation of Ambient Noise vs Deconvolution
• Synthetic Example: Simple Shear Beam
• D’Alembert Solution
• Cross-Correlation
CrossCorrelation
IRF
Floor #
• Deconvolution
Deconvolution
12
12
12
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
0
0
0
0.5
1
0
0.5
Time (s)
1
0
0.5
1
Seismic Interferometry
• IRF from Cross-correlation of Ambient Noise
Prieto, G., Lawrence, J..Chung, A., Kohler, M. 2010

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