Real-Time Integral Based Structural Health Monitoring
I. Singh-Levett, J.G. Chase & C.E. Hann
Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
B.L. Deam
Department of Civil Engineering, University of Canterbury, Christchurch, New Zealand
ABSTRACT: An algorithm has been developed to provide real-time structural health monitoring during
earthquake events. For a given input ground acceleration the algorithm matches the Bouc-Wen hysteresis
model to structural response data using piecewise least squares fitting. The methodology identifies pre-yield
and post-yield stiffness, elastic and plastic components of displacement and final residual displacement. This
approach is particularly useful for rapid assessment of structural safety by owners or civil defense authorities.
The algorithm is tested with simulated response data using the El Centro and Kobe earthquake records. Using
simulated data for a two degree of freedom shear building model, the algorithm captures stiffness to within
2% of the real value and permanent deflection to within 5% when significant non-linear response occurs. This
is achieved with acceleration data sampled at 1KHz and displacement data sampled at 10Hz.
1 INTRODUCTION
Structural Health Monitoring (SHM) is the process of comparing the current state of a structure’s
condition relative to a baseline state and determining
the existence, location, and degree of damage that
may exist, particularly after a damaging input, such
as an earthquake or other large environmental load.
Many current vibration-based SHM methods, particularly for large civil structures, are based on modal
parameter damage detection in both the time series
and frequency domain. Changes in modal parameters, such as frequencies, mode shapes and modal
damping, are a result of changes in the physical
mass, damping and stiffness properties of the structure (Doebling et al, 1996). SHM can simplify typical procedures of visual or localized experimental
methods, such as acoustic or ultrasonic methods,
magnetic field methods, radiography, eddy-current
methods or thermal field methods (Doherty, 1997),
as it does not require visual inspection of the structure and its connections or components.
SHM in Civil structures is useful for determining the
damage state of a structure. In particular, the ability
to assess damage in real-time or immediately after a
catastrophic event, such as an earthquake or terrorist
bomb blast, would allow Civil Defence authorities to
determine which structures were safe. Current
methods relying on the identification of modal parameters are more applicable to steel-frame and
bridge structures where vibration response is more
linear. Modal-based methods can also be insensitive
to localised damage and non-robust in the presence
of noise. Another drawback of current methods is
the inability to be implemented in real-time, as the
event occurs. For example, current wavelet and
ERA (Eigensystem Realisation Algorithm) methods
(Lus et al ,2004 and Caicedo et al, 2004) require the
entire measured response to process and identify
damage.
Other identification methods with potential near real-time SHM have been employed to identify modal
parameters by using the adaptive fading Kalman filter technique (Loh et al, 2000), and an Adaptive H
Filter (Sato and Qi, 1998). However these methods
involve significant computational complexity.
The approach presented in this paper uses an integral-based linear least squares method to identify
changes in structural stiffness and permanent displacement. This is achieved by matching the BoucWen Hysteresis model (Bouc, 1967) to ground acceleration and structural response data. This approach can be easily implemented in real time, is robust in the presence of noise and is shown to
accurately identify localized damage by simulation.
2 METHODOLOGY
tem is over determined, and the required unknowns
ki , i  1..N can be found using linear least squares.
2.1 Bi-Linear Stiffness Model and Identification
The motion of a structure undergoing earthquake
acceleration is defined:
M x  Cx  K (t )x   M xg 
(1)
where M is the mass matrix, C is the viscous damping matrix, K(t) is the time varying stiffness matrix,
x is the displacement vector, x is the velocity
vector, x is the acceleration vector and xg is the
ground acceleration vector.
For a single degree of freedom, equation (1) reduces
to the form:
mx  cx  k (t ) x  mxg
(2)
where m, c, and k(t) are scalar quantities.
In order to estimate a time-varying stiffness using
only the structural and ground acceleration data, the
displacement and velocity terms in Equation (2) are
replaced by integral approximations defined:.
t
x   xdt  
0
t
x
t
 xdt  t  
0 0
(3)
(4)
2.2 Bouc-Wen Hysteresis Model and Identification
The motion of a structure undergoing earthquake
acceleration with Bouc-Wen hysteresis is given by
the matrix equation defined:
M x  Cx  K e x(t )  K h z (t )   M xg 
(9)
where Ke is the pre-yield stiffness matrix and Kh is
the post-yield stiffness matrix. The vector z(t ) represents hysteretic displacement and is governed by
the equation:
ni

zi 
(10)
zi  ri 1  0.5(1  sgn( ri zi ))
 , i  1..N
Yi 


(2) of degrees
where N is the number
of freedom, Y is
the yield displacement and n is a shaping parameter.
2.2.1 Displacement and Velocity Estimation
Estimation of the displacement and velocity by
integration of measured acceleration is naturally subject to drift and numerical error. This error is corrected using the 10Hz measured displacement data
which is assumed to be a 100pt backward moving
average from 1KHz sampled data. Thus an integration displacement error at 0.1s intervals is defined:
where δ and σ take account of initial conditions and
errors due to noise.
ei  xm (0.1i)  x (0.1i) , i  1..N
Substituting Equations (3) and (4) into Equation (2)
a revised equation of motion is developed:
where ei is the error at the time t  0.1i , xm (t ) is the
10Hz measured displacement and x (t ) is a 100pt
backward moving average of the integrated displacement x(t ) defined:
t
t t
mx  c   xdt     k (t )    xdt   mxg
(5)
 0

 0 0

Assuming that k(t) is a piecewise constant function
over N fixed time intervals Δt, k(t) is defined:
k (t )  ki , (i  1)t  t  it , i  1..N
(6)
t t
x(t )    xdt  dt
(12)
0
 0 
An integration velocity error is calculated by numerical differentiation:
ci  (ei  ei 1 ) / 0.1
Equation (5) can then be reformulated:
t
t
t
k i    xdt dt   i t   i  mxg  mx  c   xdt  ,
 0 
0
 0 
t i  t  t i  t , i  1..N
(7)
(11)
(13)
The time-varying corrected velocity is then calculated by integrating acceleration and adding the velocity error:
t
v(t )   xdt  ci , 0.1(i  1)  t  0.1(i ) , i  1..N (14)
where
 i  k i ,
0
 i  k i
(8)
Choosing for example 10 values of t in the interval
t i  t  t i  t will give in total, 10N linear equations
in 3N unknowns ki ,  and  i . Thus the linear sys-
where 0.1N is the total time interval of interest
The time-varying corrected displacement is calculated by twice integrating acceleration, adding the interpolated velocity error and adding the displacement
error:
t
t
d (t )     xdt dt  ci [t  0.1(i)]  ei

0
0

0.1(i  1)  t  0.1(i ) , i  1..N
Equation (16) can be rewritten:
(15)
Figures 1 and 2 show an example where the real velocity and displacement are compared with those estimated using equations (18) and (19). Acceleration
data used in estimation had 10% uniformly distributed noise applied. Displacement data used had 3%
normally distributed noise applied.
k p (t )d (t )  Z (t )  mxg  mx  cv (t )
(17)
where k p  k e  k h and Z (t )  k h ( z(t )  x(t ))
The time varying term Z (t ) is then represented by
a piecewise constant function over N fixed time intervals Δt, defined:
Z (t ) is defined:
Real Velocity
Adjusted Velocity
0.06
Z (t )  Z i , (i  1)t  t  it , i  1..N
(19)
0.04
Equation (11) can then be reformulated:
0.02
k p d (t )  Z i  mxg  mx  cv (t ) ,
0
(i  1)t  t  it , i  1..N
(20)
-0.02
-0.04
-0.06
35
35.5
36
36.5
37
37.5
38
38.5
k p (t )  k p , j , ( j  1) tkp  t  jtkp , j  1..M
Figure 1. Comparison of Real and Estimated Velocities
-0.012
Real Displacement
Adjusted Displacement
-0.014
(21)
for some chosen interval tkp .
In a similar way to section 2.1 an overdetermined
system of linear equations in the unknowns z i ,
i  1..N and k p , j , j  1..M can be set up and
solved by linear least squares. The average of the
values of k p , j where the variation of is less than a
given tolerance is used to approximate the constant
term k p  k e  k h .
-0.016
-0.018
-0.02
-0.022
-0.024
-0.026
-0.028
-0.03
-0.032
However simulation has shown that k p and zi can
be most accurately found by allowing movement in
k p . That is a time varying k p is defined:
37.6
37.8
38
38.2
38.4
38.6
38.8
Figure 2. Comparison of Real and Estimated Displacements
Note how the drift error is corrected at 0.1s intervals.
This method of velocity and displacement estimation
makes the algorithm very robust in the presence of
noise.
Assuming that the bi-linear factor α (typically 0.050.1) is known for the Bouc-Wen element, k e and
k h can then be found:
k e  k p , k h  (1   )k p
(22)
Thus the permanent displacement parameter ( z  x) i
is defined:
( z  x ) i  Z i / k h
(23)
2.2.2 SDOF
2.2.3 2DOF
For a single DOF Equation (9) reduces to:
mx  cx  k e x (t )  k h z (t )  mxg
(16)
where the velocity and displacement are estimated
by adjusted integration using the procedure described in section 2.2.1, and the unknowns are k e , k h
and z.
A 2-DOF shear building model is defined:
m2 x2  c 2 x 2  k e 2 ( x 2  x1 )  k h 2 ( z 2  z1 )  m2 xg
(24)
Real Stiffness Variation
m1 x1   c1 x1  k e 2 ( x1  x 2 )  k e1 x1  k h 2 ( z1  z 2 )  k h1 z1 
 m1 xg
(25)
50
45
40
where the subscript 1 denotes the bottom floor.
Stiffness (N/m)
35
The unknowns in equation (25) are k e1 , k h1 , k e 2 ,
k h 2 , z1 and z2 , where z i is defined in Equation
(10).
30
25
20
15
Adding Equations (22) and (23) eliminates the unknown z2 which yields:
10
5
m1 x1  m2 x2  c1 x1  c2 x 2  (k e1  k h1 ) x1  k h1 ( z1  x1 ) 
 (m1  m2 ) xg
(26)
Equation (26) can be rewritten:
k p1d1  Z1 (t ) 
 (m1  m2 ) xg  m1 x1  m2 x2  c1v1  c2 v2
0
0
10
20
30
40
50
60
Time (s)
Figure 3. Real Stiffness Variation
Fitted Stiffness Variation
60
(27)
50
40
Stiffness (N/m)
Equation (27) is in the same form as Equation (17)
and thus the same procedure in section 2.2.2 can be
applied to find the parameters ( z1  x1 ) , ke1 and k h1 .
Rearranging Equation (24) and substituting the fitted
parameters ( z1  x1 ) ke1 and k h1 yields:
(k e 2  k h 2 ) x2  k h 2 ( z2  x2 ) 
(28)
 m2 xg  m2 x2  c2 x 2  k e 2 x1  k h 2 z1
30
20
10
\
0
0
10
20
30
40
50
60
Time (s)
Figure 4. 20% Fitted Stiffness Profile
where the only unknowns now are k e 2 , k h 2 and z2 .
The same procedure in section 2.2.2 can now be applied to find the parameters ( z2  x2 ) , k e 2 and k h 2 .
3.2 SDOF Model With Bouc-Wen Hysteresis
3.1 SDOF Bi-Linear Model
Simulations were conducted using the Kobe earthquake record and a 1kg mass with initial stiffness of
39.577N/m, bi-linear factor of 0.1, yield point of
45mm and a hysteresis shaping parameter n=2.
The algorithm was tested using a single degree of
freedom bi-linear elastic system subject to the El
Centro excitation with up to 10% randomly distributed noise applied to acceleration measurements.
Figure 5 shows the real variation of the permanent
displacement parameter z (t )  x(t ) over the course of
the excitation:
3 RESULTS
Real (Z-X)
0.15
The system had a mass of 1kg with a pre-yield stiffness of 39.577N/m and a bi-linear factor of 0.1. The
yield displacement was 45mm.
Permanent Displacement (m)
Figures 3 and 4 show the real and fitted stiffness variation, demonstrating that the algorithm accurately
identifies stiffness and the time and degree of yielding occurring, even in the presence of strong noise.
Stiffness is fitted at 0.2s intervals.
0.1
0.05
0
-0.05
-0.1
0
5
10
15
20
25
Time (s)
Figure 5. Real Variation of z (t )  x(t )
30
35
40
45
Parameters were fitted using data subject to 5% acceleration noise and 3% displacement noise. Stiffness was fitted at 3s intervals while permanent displacement was fitted at 0.6s intervals.
Figure 6 shows the variation of the fitted parameter
z (t )  x (t ) with time.
point of 45mm and shaping parameter n=2. The resulting structure had a fundamental natural frequency of 1Hz.
Figures 7 and 8 show the real value of the permanent
displacement z (t )  x (t ) for the top and bottom
floors:
Real (Z-X) - Top Floor
0.12
Fitted (Z-X)
0.1
Permanent Displacement (m)
0.15
Displacement (m)
0.1
0.05
0
0.08
0.06
0.04
0.02
-0.05
0
-0.1
-0.02
0
10
20
30
40
50
60
Time (s)
0
5
10
15
20
25
30
35
40
45
Time (s)
Figure 6. Fitted Variation of z (t )  x (t )
Figure 7. Top Floor Permanent Displacement (Real)
The algorithm reports a final residual displacement
as 15mm which compares well to the real value of
14.9mm.
x 10
14
12
Permanent Displacement (m)
The most accuracy in stiffness kp(t) was obtained
when z (t )  x (t ) was minimal. Thus only 3s intervals which had a maximum change of z (t )  x (t )
less than 1mm were used to calculate kp. As discussed in section 2.2.2 the average kp is used to approximate the linear stiffness and the results are
shown in Table 1.
Real (Z-X) - Bottom Floor
-3
16
10
8
6
4
2
0
Table
1. Fitted Linear Stiffness
__________________________
Period
Stiffness
___________________
N/m
__________________________
1
39.18
5
38.17
6
38.75
7
39.04
8
38.60
9
38.83
10
36.86
11
38.74
12
37.92
__________________________
Mean
38.45
Std Dev
0.72
3.3 2DOF Model with Bouc-Wen Hysteresis
The algorithm was tested using two identical
Bouc-Wen elements in a shear building arrangement,
subject to the El Centro record. The elements had
stiffness of 102.64N/m, bi-linear factor of 0.1, yield
-2
0
10
20
30
40
50
60
Time (s)
Figure 8. Bottom Floor Permanent Displacement (Real)
Stiffness and Displacement parameters were fitted
from acceleration data with 5% noise and displacement data with 3% noise. Stiffness was fitted at 4s
intervals while Displacement was fitted at 0.4s intervals.
Figures 9 and 10 show the fitted permanent displacements for the top and bottom floor respectively:
Fitted (Z-X) - Top Floor
identifies stiffness and permanent displacement.
Only the results of the fitted stiffness in time periods
where there is minimal change in z(t)-x(t) are used to
identify the overall linear stiffness kp. This avoids
potential tradeoff that can occur between kp and z(t)x(t) during periods of significant yielding.
0.1
0.09
Permanent Displacement (m)
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
60
Time (s)
Figure 9. Fitted Top Floor Permanent Displacement
The two degree of freedom example given shows
that the algorithm can accurately identify stiffness
and permanent stiffness in a multiple degree of freedom situation. Since the algorithm effectively decouples the fitting process into separate optimizations for each floor it is easily generalized to higher
degrees of freedom.
Fitted (Z-X) - Bottom Floor
0.02
Once the hysteresis parameters zi are identified, the
further hysteretic components of yield point Y and
shaping parameter n could be found from Equation
(10).
0.018
Permanent Displacement (m)
0.016
0.014
0.012
In summary this paper provides a highly efficient
and accurate method for identifying linear stiffness
and permanent displacement in multi-story buildings
under seismic loads as well as providing further information on Bouc-Wen hysteretic components.
0.01
0.008
0.006
0.004
0.002
0
0
10
20
30
40
50
60
Time (s)
5 REFERENCES
Figure 10. Fitted Bottom Floor Permanent Displacement
As with the single degree of freedom case, stiffness
values were only fitted where the change in bottom
floor permanent displacement ( z1  x1 ) was less
than a 2mm tolerance. Table 2 lists the values:
Table
2. Fitted Stiffness Values For Both Floors
______________________________________________
Top
Floor
Bottom
Floor
___________________
_____________________
Period
Stiffness (N/m)
Period Stiffness (N/m)
______________________________________________
2
104.43
2
98.27
3
103.47
3
98.67
4
104.35
4
99.16
5
104.53
5
99.12
6
103.54
6
99.02
7
104.48
7
100.19
8
105.12
8
99.28
______________________________________________
Mean 104.28
Std Dev 0.58
4
99.11
0.59
Bouc, R. 1967 “Forced Vibration of Mechanical Systems with
Hysteresis” Proceedings of the 4th Conference on Non-Linear
Oscillation. Prague, Czechoslovakia
Caicedo, J. M., Dyke, S. J. and Johnson, E. A. (2000) “Health
Monitoring Based on Component Transfer Functions” Proceedings of the 2000 International Conference on Advances in
Structural Dynamics, Hong Kong, December 13-15.
Doebling, S.W., Farrar, C.R., Prime, M.B., and Shevitz, D.W.
(1996a) “Damage Identification and Health Monitoring of
Structural and Mechanical Systems from Changes in Their Vibration Characteristics: a Literature Review” Los Alamos National Laboratory, Report LA-13070-MS.
Doherty, J. E. (1987) “Non-destructive Evaluation,” Chapter 12
in Handbook on Experimental Mechanics, A. S. Kobayashi
Edt., Society for Experimental Mechanics, Inc.
Loh, C.-H., Lin, C.-Y., and Huang, C.-C. (2000) “Time Domain Identification of Frames under Earthquake Loadings”
Journal of Engineering Mechanics, Vol.126, No.7, pp 693703.
DISCUSSION & CONCLUSIONS
In the Bi-Linear case the algorithm accurately
identifies changes in stiffness due to yielding using
only acceleration data with up to 20% normally distributed noise applied.
Using the Bouc-Wen Hysteresis model for a single
degree of freedom system the algorithm accurately
Lus, H. and Betti, R. (2000) “Damage Identification in Linear
Structural Systems” Proc. of the 14th ASCE Engineering Mechanics Conference, Austin, Texas, May 21–24.
Sato, T. and Qi, K. (1998) “Adaptive H Filter: Its Application
to Structural Identification” Journal of Engineering Mechanics,
Vol.124, No.11, pp 1233-1240.