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I
INTORDUCTION
Experimental Structural Damage
Detection in a biaxial bending torsion
coupled multi-span beam using
inverse static strain sensitivity
Vedhus Hoskere∗
BMS College of Engineering, Bangalore
[email protected]
I.
Intorduction
ull-scale structures ranging from highway bridges to dams are prone to safety hazards,
which translates to a critical need to monitor their performance to detect early damage to
the structure. This can affect its reliability and render it ineffective.
These problems have already been extensively explored.1 The authors developed a baseline
finite element model using nondestructive test data that assisted in planning, bridge inspections,
routine maintenance decisions, and rehabilitation throughout the life of the building.
The stiffness properties of a structure for each component can be quantified by parameter
estimation and model updating.w These parameters of the finite-element model can be any
appropriate geometric or material property, including cross-sectional properties of the structural
components. The parameter estimation results can be used to classify the damage as the major
differences between the estimated and expected parameters.3
The nondestructive test data is used to update the parameters of the previous model, the
changes in which are indicated in the updated model that may affect a structure’s load carrying
capacity. The input for parameter estimation is provided by the response of a structure to loads. In
cases that require only element stiffnesses for condition assessment, static load testing can prove
simpler and more cost effective than dynamic loading. Also, in many cases static loading is more
economical as well as imparts a deeper assessment of the damage.
The inverse problem is necessary to detect changes in the system properties and hence
considered to be one of the foremost tools in structural health monitoring of a system. Previously,
a method based on inverse sensitivity of the singular solutions of a system FRF matrix was
proposed which addressed the problem of structural damage detection based on the measured
frequency response functions of the structure in its damaged and undamaged states.4 In another
work, inverse problems associated with linear dynamical systems are studied where methods
based on inverse sensitivity analysis of eigensolutions, is considered.5
Displacement measurements from applied loads are used in the static finite element method,
which can be a demanding task, as a frame of reference must be defined. Strain gauges, however,
can be used for static strain measurements with higher accuracy than static displacement measurements. Therefore, in this study the strain measurements have been acquired as it is more sensitive
to local damage and capture element behavior more efficiently than displacement measurements.
F
∗ Vedhus
Hoskere, Senior Undergraduate, Civil Engineering, BMSCE
1
II
THEORETICAL BACKGROUND
The major objective of this paper is to use the static strain measurements for damage detection
of a multi-span bending-torsion coupled beam. The static response of the beam was measured,
after which an inverse problem was used to compute the damage vector under the application of
a series of specified loads.
II.
Theoretical Background
2.1 Finite element formulation
The section undergoes bending in both planes as well as torsion. A 2 noded general 3D beam
element with 6 degrees of freedom at each node has been used in the finite element formulation.
The elemental stiffness matrices due to twisting, axial deformation, bending in both planes are
computed separately and combined into a 12x12 elemental stiffness matrix. The bending in both
planes and torsion are uncoupled when the section is oriented such that the forces act along the
principal axes. Hence, this orientation is chosen as the local axis system.
Figure 1: Global Axis
Figure 2: Local Axis
We have, the element stiffness matrix
 AE
0
0
0
0
L
12EIz
 0
0
0
0
3

L

12EIy
6EIy
 0
0
0
− L2
L3

GJ
 0
0
0
0

L
6EIy
4EIy

0
− L2
0
 0
L

6EIz
 0
0
0
0
2
L
k=
− AE
0
0
0
0
 L
 0
12EIz
−
0
0
0

L3

12EI
6EI
y
y
 0
0
− L3
0

L2
 0
0
0
− GJ
0

L

6EIy
2EIy
 0
0
− L2
0
L
6EIz
0
0
0
0
L2
0
6EIz
L2
− AE
L
0
0
0
0
4EIz
L
0
z
− 6EI
L2
0
0
0
0
AE
L
0
0
0
2EIz
L
0
z
− 12EI
L3
0
0
0
z
− 6EI
L2
0
0
12EIz
L3
0
0
0
0
0
0
0
z
− 6EI
L2
0
0
12EI
− L3 y
0
0
0
0
0
0
− GJ
L
0
0
0
0
12EIy
L3
0
0
6EIy
L2
0
6EIy
L2
0
GJ
L
0
0
0
0
6EI
− L2 y
0
2EIy
L
0
0
0
6EIy
L2
0
4EIy
L
0
0
6EIz
L2




0 

0 


0 
2EIz 

L 
0 

6EIz 
− L2 

0 

0 


0 
4EIz
L
Having formulated the elemental stiffness matrix, it is now required to perform the transformation of the stiffness matrix such that the angle section can now be oriented according to the
global axis system as shown in Fig 1.
The basic formulation for transformation of stiffness matrix is:
K = [ T ]t [k][ T ]
Where, [k] is the element stiffness matrix,
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II

T3
0
T=
0
0
0
T3
0
0
0
0
T3
0

0
0

0
T3
THEORETICAL BACKGROUND

lu

T3 = lv
lw
mu
mv
mw

nu
nv 
nw
Direction cosines in T3 are as follows:
lu = cos(u, X ),
lv = cos(v, X ),
lw = cos(w, X ),
mu = cos(u, Y ),
mv = cos(v, Y ),
mw = cos(w, Y ),
nu = cos(u, Z )
nv = cos(v, Z )
nw = cos(w, Z )
The global stiffness matrix thus obtained by assembling all elemental stiffness matrices is used
to obtain the response.
2.2 Static loading
The response of a structure to static loading provides the input for parameter estimation. Using
a finite-element model (FEM) of the structure, measured and analytical responses are compared.
[K ] D = F
(1)
where D is the global nodal displacement vector (consisting of all displacement and rotation
values in three dimensions) and F is the global nodal force vector.
[ D ] = [ K ] −1 [ F ]
(2)
The following model was created using a MATLAB code. Responses were compared with
those of a similar model created in STAAD Pro 2007 and NISA 17.
Figure 3: FE model
2.2.1 Strain displacement relation
Sanayei and Salatnik3 derived a method to obtain strains in a general frame element. A
mapping matrix [ B] is constituted to transform the displacement vector to a strain vector. An
elemental mapping vector is obtained to find elemental strains from elemental displacements.
These strains are then transformed and assembled to obtain strains in the global axis system.
¯ n}
ε¯ n = { B¯ n }{U
(3)
¯ n } to global coordinates,
Transforming {U
{U¯ n } = {Un }[ T ]
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II
THEORETICAL BACKGROUND
Figure 4: Displacements from MATLAB Model, STAAD and NISA
where [ T ] is the transformation matrix. Thus,
{ B¯ n } = { Bn }[ T ]
For a system with n elements,the system mapping matrix [ B] is assembled by vertically augmenting
system strains and aligning system DOFs horizontally.
{ε} = [ B]{U }
(4)
Now, for bending in z-direction:
d2 v
(5)
dx2
where v is the bending displacement parallel to y-axis and y is the distnce from the neutral
axis to the strain measurement surface.
The Hermite shape functions are used to model the behaviour.
ε = −y
3x2
2x3
2x2
x3
+
;
H
(
x
)
=
x
−
+
;
2
l
l2
l3
l2
3x2
2x3
x2
x3
H3 ( x ) = 2 − 3 ; H4 ( x ) = − + 2 ;
l
l
l
l
Each function represents the bending deformation due to a unit displacement at a given DOF. The
entire bending deformation is represented by.
H1 ( x ) = 1 −
v( x ) = H1 vi + H2 θi + H3 vj + H4 θ j
Here, i and j are the node numbers of the element.Hence, equation 3 can be re-written in terms of
the shape functions as
 

 vi 


2
d {H} θvi 
ε n = −y
(6)
dx2 
 vj 



θvj
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II
THEORETICAL BACKGROUND
Figure 5: Strains from MATLAB Model and STAAD
{H} = H1
H2
H3
H4
Similarly, for bending in the z direction, it can be found that
 


 wi 
d2 {H}  θwi 
ε n = −z

 wj 
dx2 
 
θwj
(7)
ui
−1
uj
(8)
For axial deformation,
1
εn =
1
L
where ui and u j are axial displacements at the ends of each element.
From equations 6, 7 and 8, the elemental mapping matrix can be written as
{ B¯ n } = {
−1
L
1
L
6y
(2x − l )
l3
6y
− 3 (l − 2x )
l
−
6z
(2x − l )
l3
6z
− 3 (l − 2x )
l
−
0
0
2y
(3x − 2l )
l2
2y
− 2 (3x − l )
l
−
2z
(3x − 2l )
l2
2z
− 2 (3x − l )}
l
−
(9)
2.3 Inverse sensitivity analysis
Let (min=1 ) denote a set of system parameters that may be associated with mass or stiffness
characteristics of the system and Dk (m1 , m2 , .., mn ), k = 1, 2, ..Nk , represent set of response characteristics(static or dynamic) of the system . However let mu = (mui )in=1 represent the system
characteristics in reference state. Now, mdi = mui + ∆i represents the system characteristics in the
ith system due to occurrence of damage ∆i (change in system parameter).
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IV
EXPERIMENTAL INVESTIGATION
Based on these notations and using Taylor’s expansion the following expression can be written
:
n
n n
2
∂Dk
k
Dk (mu1 + ∆1 , . ., mun + ∆n ) = Dk (mu1 , .., mun ) + ∑ ∂m
∆i + 21 ∑ ∑ ∂m∂ D
∂m ∆i ∆ j
ui
i =1
i =1 j =1
ui
uj
The quantity ∆Dk = Dk (mu1 + ∆1 , mu2 + ∆2 , . ., mun + ∆n ) − Dk (mu1 , mu2 , .., mun ) represents
the change in response character due to occurrence of damage and this quantity is expected to be
measured based on experiments conducted on the structure in its damaged and reference states.
Change in response involving the second order expansion the Taylor series can be written as
∆Dk =
n
∂D
n
1
n
∂2 D
∑ ∂muik ∆i + 2 ∑ ∑ ∂mui ∂mk uj ∆i ∆ j , k = 1, 2.., n
(10)
i =1 j =1
i =1
This equation can be solved iteratively as follows
∆Dk =
n
1 n n
∂D
∂2 D
∑ ∂muik ∆i {q+1} + 2 ∑ ∑ ∂mui ∂mk uj ∆i ∆ j
i =1
i =1 j =1
{q}
(11)
The above equation is rewritten as
{ q +1}
{∆D } Nk×1 = [S] Nk×n { X } N ×1 + {∆D I I }{q}
(12)
{ X } N ×1 represents the damage vector. The matrix [S] Nk×n is the sensitivity matrix which is to be
evaluated for the structure. Thus,
{ X } N ×1 = [S]+ {∆D } − [S]+ {∆D I I }{q}
(13)
where + denotes the matrix pseudo-inverse. For a measured response in strain ε,
{∆Dk } = {∆ε}
the sensitivity matrix with respect to flexural rigidity of the ith zone, Ri is given by
{S} =
III.
∂ε
∂D
= [ B]
∂Ri
∂Ri
(14)
Numerical simulation
3.1 Numerical simulation
The MATLAB model was employed to compute the strain sensitivity matrices with respect to
the Young’s modulus of the beam. Strains were synthetically generated for two cases (i)Undamaged
state (ii)Damaged state. The difference in these strains was used to update the Young’s modulus
in both cases and ascertain damage. It was found that the properties converged to nearly exact
values in 8 global iterations. A plot of flexural rigidity values in the damaged beam against the
number of global iterations is shown in figure 6.
IV.
Experimental Investigation
4.1 Experimental setup
In this study, the proposed formulation is executed on a multi-span angle section as shown
in Fig 7. There are two over-hangs given at each end with values 0.6m and 0.4 m respectively.
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EXPERIMENTAL INVESTIGATION
Figure 6: Convergence of EI values with synthetic data
The length of remaining spans were 1.2m, 0.8m and 1.0m respectively. The finite element model
was divided into equal elements of 0.2m each, resulting in a total of 21 nodes. Static loads of
11.2 kg and 12.6 kg were applied at the last and first node respectively. A third quasi-static third
load of 36.58 kg was moved along the span, placing it at 8 points namely nodes 6,7 8,9,12,13,16
and 17. Strain gauges were installed at each of the 6 zones of the beam. Each strain gauge was
installed 18.9 mm from the lower inner edge of the angle section on which it was attached in order
to coincide with the centre of gravity of the beam.
Figure 7: Experimental setup
4.2 Measurements
The experiment was designed to calculate the damage vector of the elements of an angle-section
on application of loads along the length of the beam. First, the aluminium stiffeners were fastened
to the beam as shown in figure 4. This state of the beam is the undamaged state. Quasi-static
loads were applied and the resulting strains were measured at eight different points on the beam.
The stiffeners were then removed to "damage" the beam and the same exercise was carried out.
Below, plots of convergence of young’s modulus are shown for the measured strain values.
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EXPERIMENTAL INVESTIGATION
Figure 8: Convergence of Young’s modulus before introducing torsional springs
Figure 9: Young’s modulus and strain before introducing torsional springs
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EXPERIMENTAL INVESTIGATION
Figure 10: Experimental setup with stiffners in place
4.3 Boundary Conditions
The actual beam supports were not perfect hinge supports and had resistance to rotation. Thus,
torsional springs of unknown stiffness were included in the FE model to better represent the real
beam. The unknown stiffness values were identified through a parametric study on each of the
eight springs (2 at each support).
Figure 11: Torsional springs of unknown stiffness
The stiffness values were chosen to minimize the error in observed and predicted strains. To
make an optimal initial choice, a non-dimensional quantity, e is defined
n
e(K1y , K1z , K2y , ..K4z ) =
∑
i =1
ε i (K1y , ..K4z ) − ε mi (K1y , ..K4z )
ε i (K1y , ..K4z )2
2
(15)
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EXPERIMENTAL INVESTIGATION
Here, n is the number of strain readings available. The values of K1y ...K4z are initially chosen
to be equal such that e is minimized. Then each parameter is varied, one at a time, to further
minimize e.
Figure 12: Initial estimate of torsional spring stiffness
Figure 13: Variation of parameter e with increase in no. of iterations
The values of the spring stiffness estimated was found to be
Kyi × 107 Nmm/rad
Kzi × 107 Nmm/rad
1
0.5
15.9
2
5.45
0.15
3
7.65
0.0007
4
0.0001
426.34
A similar procedure was followed to update support displacements
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EXPERIMENTAL INVESTIGATION
Figure 14: Convergence of Young’s modulus after introducing torsional springs
Figure 15: Young’s modulus and strain after introducing torsional springs
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EXPERIMENTAL INVESTIGATION
Figure 16: Convergence of Young’s modulus after allowing support displacement
Figure 17: Young’s modulus and strain after allowing support displacement
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V
Dyi (mm)
Dzi (mm)
1
0.0154
-0.0129
2
-0.00473
-0.0123
3
-0.00175
0.00383
REFERENCES
4
-0.01307
0.00921
Tension Test
Figure 18: Tension test on steel coupon
The Young’s modulus was found to be 192 GPa
Figure 19: Flexural rigidity with stiffeners
V.
Figure 20: Calculated change in stiffness
References
1. Masoud Sanayei, John E. Phelps, Jesse D. Sipple, Erin S. Bell, and Brian R. Brenner. "Instrumentation, Nondestructive Testing, and Finite-Element Model Updating for Bridge Evaluation
Using Strain Measurements." Journal of Bridge engineering (2012).
2. Masoud Sanayei, Gregory R. Imbaro, Jennifer A. S. McClain, and Linfield C. Brown.
"Structural Model Updating Using Experimental Static Measurements." Journal of structural
engineering (1997). (CMC) (2009).
3. Masoud Sanayei, Michael J. Saletnik. "Parameter Estimation of Structures from Static Strain
Measurements. I: Formulation." Journal of structural engineering (1996).
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REFERENCES
Figure 21: Zones
4. S. Venkatesha, R. Rajender and C. S. Manohar. "Inverse Sensitivity Analysis of Singular
Solutions of Frf Matrix in Structural System Identification." Computer Modeling in Engineering
and Sciences (CMES) (2008).
5. R Sivaprasad, S. Venkatesha and C. S. Manohar. "Identiïnˇ Acation
˛
of Dynamical Systems with
Fractional Derivative Damping Models Using Inverse Sensitivity Analysis." Computers, Materials
and Continua (2008)
6. Luna Majumder and C. S. Manohar "A time-domain approachfor damage detection in beam
structures using vibration data witha moving oscillator as an excitation source." Journal of Sound
and Vibration (2003)
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