Real Time Integral-Based
Structural Health Monitoring
I. Singh-Levett1, C. E. Hann1 J. G. Chase1, B.L. Deam2 , J.B. Mander2
1Department
of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
of Civil Engineering, University of Canterbury, Christchurch, New Zealand
2Department
Introduction
• Structural Health Monitoring (SHM) compares a structures condition relative to a baseline
state to identify damage
• Determining the existence, location and degree of damage is critical after a major event
to aid recovery/response
–
–
Visual and localized inspection methods can be complicated or time consuming
Just knowing its damaged and having to go check out how much is ok, but requires manpower
that may not be available.
• Main goal = autonomous ability to determine basic level and location of damage to make
a reliable safety/service estimate remotely
Current SHM Techniques:
• Modal parameter estimation
–
–
insensitive to localized damage and sensitive to noise
more suitable to linear structures
–
computationally intense, not suitable for real-time
• Flexibility based methods
• Time series methods
• Many variations on the above …
Benefits of Real-Time SHM
• Immediate assessment of instrumented structures
• Infer state of non-instrumented structures using fragility
relationships
• Optimization of response and recovery
• Reduction of social and economic impacts and costs
Integral SHM – Minimum
Requirements
• Measurements:
– Ground and structural accelerations
– Relatively very low frequency displacement (GPS, fibre-Optics or StoreyDrift Extensometers)
• Method uses simple Bouc-Wen non-linear model including permanent
displacements
• Integral-based fitting method
– Reformulated D.E.’s in terms of integration of measured/estimated motion
– Linear least squares, very fast  real-time capability
– Robust to noise and modelling error
Non-Linear Structural Model
• Bouc-Wen Hysteresis Model:
Equation of Relative Motion:
x , x , x
Mx(t)  Cx (t)  K e x(t)  K h z(t)  Mxg (t)
Permanent displacement is represented by
M = Mass Matrix
C = Damping Matrix
Ke = Elastic Stiffness Matrix
Kh = Hysteretic Stiffness Matrix
( x  z)
x(t) = Displacement
x (t) = Velocity
x(t) = Acceleration
x g (t) = Ground Acceleration
z(t) = Hysteretic Displacement
- for simulation, Bouc-Wen Hysteresis
- for fitting, constant piecewise function
 model independent
Displacement and Velocity via Integration
& Correction
• To use the model you need all the response quantities, but .. You didn’t
measure them
– The displacement is initially approximated by double integration of acceleration
– Displacement is measured at up to 1-10 Hz
• Drift error is corrected every 0.1-1.0 s with measured displacement
• However: this gives a discontinuity in displacement every 0.1-1.0 s
– Piecewise C(1) continuous cubic curves are fitted to the corrected displacements to
“smooth out” the joins at 0.1s intervals.
– The velocity and acceleration computed by numerical differentiation of corrected
displacements
– Important step as ensures displacement, velocity and acceleration are precisely
related by differentiation
– Otherwise solution is corrupted by the discontinuities
Parameter Identification
• Integrate of the Equations of Motion (2x)
• Discretise changes in stiffness and Bouc-Wen plastic terms
• Identify stiffness and permanent displacement from revised equation of motion:
k p d (t )  Z (t )  mxg (t )  ma(t )  cv(t )
k p  ke  k h
Z (t )  kh [ z(t )  x(t )]
k p = linear stiffness
Z (t ) = permanent displacement parameter
v(t ) = Estimated Velocity
d (t ) = Estimated Displacement
a (t ) = Estimated Acceleration
Result = Least Squares Solution
• Enabled by piecewise parameter variation
 kp 
 Z 


Ai  1   bi
Z 2 
Z3 
 d (ti1 ) 1 0 0
A i  d (ti 2 ) 0 1 0
 d (ti3 ) 0 0 1
  mxg (ti1 )  ma(ti1 )  cv(ti1 ) 


b i   mxg (ti 2 )  ma(ti 2 )  cv(ti 2 )
  mx (t )  ma(t )  cv(t ) 
g i3
i3
i3 

Algorithm Output
• Least squares procedure yields solution vector:
 kp 
 Z 
 1


Z 2 
Z 3 
• Can calculate estimated permanent displacement from results and BoucWen formulation:
D
 Z
 

k h 1 

 1 
Fitting Algorithm Overview
1.
Measure displacement at low sampling rate and acceleration at high rate
•
•
2.
Low = 1-10Hz
High = 1+ kHz
Estimate displacement via 2x-integration at high sampling rate
•
Correct high frequency displacement using low frequency displacement, and obtain
velocity and acceleration.
3.
Create system of linear equations by piecewise discretisation of unknowns over
time at useful rates/intervals of expected change
4.
Find unknowns using linear least squares
5.
Estimate permanent displacements from results
6.
Repeat for all time periods of interest
Results – Simulated Data
• Single DOF system with linear stiffness of 39.58N/m – 1s Period
• 10% uniformly distributed noise on all measurements
• Stiffness identified over 2s intervals, permanent displacement over 0.4s intervals
-3
45
2
Actual Stiffness
Fitted Stiffness
-2
Permanent Displacement (m)
35
30
Stiffness (N/m)
True
Identified
0
40
25
20
15
10
-4
-6
-8
-10
-12
-14
5
0
0
x 10
-16
10
20
30
Time (s)
40
50
60
-18
0
10
20
30
Time (s)
40
50
60
Generalises to MDOF
• 1-DOF algorithm is generalised for N-DOF situation
N
m1x1(t )   c1,b xb (t )  ke(2) [ x1(t )  x2 (t )]  ke(1) x1(t )  kh(2) z2 (t )  kh(1) z1(t )  m1xg (t )
b 1
Bottom Storey
N
mw xw (t )   cw,b xb (t )  ke( w) [ xw (t )  xw1 (t )]  ke( w1) [ xw (t )  xw1 (t )]
b 1
Intermediate Storeys
 kh( w) z w (t )  k h( w1) z w1 (t )  mw xg (t ) , w  2,.., N  1
N
mN xN (t )   c N ,b xb (t )  ke( N ) [ x N (t )  x N 1(t )]  kh( N ) z N (t )  mN xg (t )
b 1
Top Storey
• Taking a cumulative sum removes unknowns (by telescoping terms)
giving a generalised equation for any storey w :
Non-Linear Frame
•Four storey structure designed to demonstrate dynamic response
•Replaceable plastic hinges
•1/5th scale structure subjected to El Centro record
Steel Frame - Stiffness
•Identified stiffness compares well to pushover analysis results
•Similar behaviour observed for all storeys
•Discrepancy in final stiffness value attributed to strain hardening and strain rate effects
5
10
x 10
Actual Stiffness
Fitted Stiffness
9
8
Stiffness (N/m)
7
6
5
4
3
2
1
0
0
5
10
15
Time (s)
20
25
30
Steel Frame - Yielding
•Initial and final residual deformations are accurately captured
•Two major yields identified, both occur during peaks in displacement
•Final residual deformation identified to within 1.5% of true value for all storeys
Re-simulation – The ultimate check
Comparison of Measured and Resimulated Top Storey Displacements
Comparison of Measured and Resimulated Top Storey Displacements
0.03
Measured
Resimulated
Measured
Resimulated
0.015
0.02
0.01
0.01
Displacement (m)
Displacement (m)
0.005
0
-0.01
0
-0.005
-0.02
-0.01
-0.03
-0.015
-0.04
0
5
10
15
Time (s)
20
25
30
9
10
11
12
13
Time (s)
• Structural response is re-simulated using the fitted parameters
• Accurate re-simulation validates method and shows fitted parameters are realistic
14
Conclusions
• The algorithm is able to accurately identify stiffness and permanent
displacement in a non-linear steel frame structure and in many
simulated structures with significant random noise
• The algorithm has also been applied to recognize different regimes of
motion of a hybrid rocking structure
• Relatively minimal computations are required making the method very
suitable for real-time implementation
– Could be run on 10-20 MIPS
– i.e. your fancy cell phone does more work