VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
INSTITUTE OF MECHANICS
o0o

TRẦN THANH HẢI
CRACK IDENTIFICATION FOR BEAM BY USING THE
VIBRATION METHOD Specialized in Mechanics of Solids
Code: 62 44 21 01
SUMMARY OF PhD THESIS


Hardcopy of the thesis can be found at
1. Vietnam National Library
2. Library of Institute of Mechanics
1

INTRODUCTION

Since early prognosis of cracks in structures is most important to
keep away from accident, a lot of efforts have been focused to study
cracked structures. This leads to many papers of the topic published in
journals on structural engineering; fracture mechanics; vibrations and
control etc.
Content of the intensive study consists mainly of two problems.
The first one is known as the forward problem that investigates
structural behaviour with given position and size of cracks. The
remaider acknowleged as the inverse problem relates to identify the
potential cracks from measured behaviour of structure under
consideration. The latter problem has recently received more interest
from practical point of view.
There are two approaches to the inverse problem, now could be
called generally structural damage detection problem or structural
health monitoring. The first one is based on “symptoms” of damage
occurrence that are directly extracted from the measured data by using
different methods of signal processing. This approach is termed by
direct technique. The other one makes use of not only the measured
data but also an artificial model of damaged structure for damage

Outline of thesis is as follows
Chapter 1. Overview on vibration-based method for structural
damage detection problem.
Chapter 2. Theoretical background for regularization method,
wavelet transform and model of cracked beam.
Chapter 3. Mode shape-based procedure for multiple crack
detection by using the Tikhonov’s regularization method.
Chapter 4. Reponse of cracked beam to moving load and crack
detection by the wavelet analysis of moving vehicle vibration.
Concluding remarks and Referenses.

Chapter 1. OVERVIEW
Structural damage is understood as a change in either physical
or geometrical properies of structure in comparison with a baseline
configuration of intact structure. Damage is often described by its
position and degree.
The structural damage detection problem was firstly
investigated by Adams etc [1] for a bar with single damage modeled
by an axial spring at a position in bar. The authors have obtained an
equation allowing one to determine the damage position from
3

measured pair of natural frequencies. Latter in [28] Liang and his
coworkers extended the result for the case of beam by establishing
general form of frequency equation of beam with single crack
represented by a torsional spring. Morassi [39] proposed a first order
approximate frequency equation for cracked beam with variable
stiffness. Narkis [40] has calculated analytical solution of the problem
for crack localization from measured two frequencies of simply
supported beam. Nguyen Tien Khiem và Dao Nhu Mai [41]

using frequencies is often non-unique. In such the case in order to
have unique solution one has to engage other features of structure that
could be axtracted from measurements. The mode shape of structure
was early used for structural damage detection by Rizos et al [51],
Yuen [60], West [65]…. At the first time, the mode shape has been
taken in use for calculating different damage indices such as Modal
Assurance Criteria (MAC) that is unable to be used for damage
localization in structure. Then, Kim and his coworkers [20] have
developed PMAC hay COMAC for the problem but they exposed to
be insensitive to damage. Despite that Ho and Ewins [16], Parloo et al
[46] proposed different damage indices calculated from given mode
shape or its derivatives, Pandey etc [44] demonstrated that the mode
shape is less sensitive to damage than mode shape curvature. Based on
the idea, Ratcliffe [49], Wahab and De Roeck [63] have developed
different procedures for damage localization from mode shape and
mode shape curvature.
Through studying vibration of multiple cracked beam, Li [27]
has derived a recurrent relationship between mode shape if the beam
in both sides of a crack. However, this is not a closed form solution
for the mode shape so that it cannot be used for crack detection from
measured mode shape. By using the step function Caddemi và Caliò
[4] obtained closed form solution for mode shape of beam with
arbitrary number of cracks that is an explicit expression of the mode
shape through crack parameters. The closed form solution of mode
shape has not been taken in use for multiple crack detection because it
contains Dirac delta function.
Although the mode shape of damaged structure could provide
more useful information on the damage circumstance of a structure,
measurement of mode shape is more difficult. The change in mode
shape due to damage is usually more difficult to be monitored than its

,bAx
=
(2.1)
where A is an arbitrary matrix (might be nonsquare or singular), b is a
vector that is determined only as an approximation to unknown exact
one
.b
Tikhonov và Arsenin [57], [58] suggested that solution of
equation (2.1) can be found from solution of the mean square problem

},{minarg
2
0
2
)xL(xbAxx −+−= α
x
RLS
(2.2)
with
0
x,, L
α
being regularization factor, regularization operator and a
prior information about solution respectively. Leaving outside the
equivalence between equations (2.1) and (2,2) one is going to consider
equation (2,2).
6

Theorema. For
0f

δα
=− bAx )( (2.4)
where
δ - noise level of vector b.
In turn, equation (2.3) can be solved by using the Singular
Value Decomposition (SVD) of matrix A

T
VΣUA = (2.5)
where
VU,
are square matrices of order m and n respectively and
n
T
m
T
IVV,IUU == , matrix ).,min(},, ,{)(
1
nmqdiagnm
q
=
=
×
σ
σ
Σ
Hence, solution of equation (2.3) with
0x =
0
is

of non-stationary processes when frequency is dependent upon time.
This gap can be fulfilled by using the newly developed wavelet
transform that is briefly described below.
2.2.1. Difinition of wavelet transform
The continuous wavelet transform is defined as

,)()(),(
,
∫
=
+∞
∞−
dtttfbaW
ba
ψ
(2.7)
where

),()/1()(
*
,
a
bt
at
ba
−
=
ψψ
(2.8)
a is a real number acknowledged as scale or dilation factor, b is

,
1
),()(
a
da
dbbaWCtf
bag
ψ
(2.9)
where

.
)(
ˆ
2
2
∞<
∫
=
∞+
∞−
ξ
ξ
ξψ
π
dC
g
(2.10)
From mathematical point of view, wavelet is convolution of the
signal and wavelet function, allowing compressing a signal.

π
=
= (2.12)
This idea constitutes the most simple and efficient model of crack in
beam member.
2.3.2. Model of beam with crack
The rotational spring model of crack allows one to represent a
crack at a section in beam in a form of compatibility conditions at
crack that should be satisfied by the displacements and forces of beam
8

in both sides of crack. The study of cracked beam with the crack
model can be carried out by dividing the beam into sub-beam
bordered by crack position and beam ends, Rizos và cộng sự [51].
This approach enables to use the governed equations without any
change for solving the problems of cracked structures.
2.3.3. Finite element model of beam with crack
Qian et al [48], have shown that stiffness matrix of a cracked
beam can be expressed as

.
Tee
c
TCTK
1
~
−
= (2.13)
Where
T

⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎣
⎡
=
EI
l
EI
l
EI
l
EI
l
ee
ee
C (2.14)
and

⎥
⎥
⎥
⎦

'
36
0
2
2
0
2
1
24
∫
=
∫
===
a
II
a
I
daaFRdaaFR
bhE
m
bhE
n
ππ
l
e
– the length of
element, a is crack depth.
Summary of Chapter 2
In this Chapter, the fundamental of the Tikhonov’s relarization
and Wavelet transform methods has been presented. The

(3.1)
where
,
2
k
ω
),(x
k
φ
,2,1=k denote the natural frequencies and
associated mode shapes. Suppose that the beam length is divided into
N segments
Njxx
jj
, ,1),,(
1
=
−
, each of them contains a crack ),(
jj
Ke
such that
Njxex
jjj
, ,1,
1
=
<<
−
, .,0

N
j
j
x
j
x
kj
N
j
kkN
j
x
j
x
jkjjkj
k
dxx
BLBxdxx
F
EI
1
1
2
1
1
1
22
2
)(
)0()()(")("

x
k
φ
′
′
)(
0
x
k
φ
′
′
′
everywhere along
the beam length (0, L) and linear functions

⎪
⎩
⎪
⎨
⎧
≤−
′′
≤
++=
−
−
10
1
),()(

sin.sin
3
)(2
sin41
sin21
1,
4
1
2
1
22
0
2
⎥
⎦
⎤
⎢
⎣
⎡
∑
+
∑
+
⎥
⎦
⎤
⎢
⎣
⎡
∑

,
jj
εη
γ
=
the asymptotic approximation of
equation (3,5) with regard to small parameter
,
ε
is

.sin21
1
22
0
2
⎥
⎦
⎤
⎢
⎣
⎡
∑
−=
=
N
j
jjkk
ek
πγωω

=
∑
=
(3.7)

[
]
.sin21
ˆ
22
0
2
jjkk
ek
πγωω
−= (3.8)
• For cantilever, with
,0
00
==
kk
DC the Rayleigh quotient is

⎥
⎦
⎤
⎢
⎣
⎡
∑

4
0
1
2
1
22
0
2
)()(
3
)(21
)(1
γγ
λ
γ
γω
ω
(3.9)
with the first order approximation

⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
∑
Φ−=

⎡
Φ−++
Φ+
==
jkjk
jkj
k
k
k
ee
e
γλγ
γ
ω
ω
ω
(3.11)
and

[
]
.)(1
~
2/1
2
jkjk
eΦ−=
γω
(3.12)
If beam has two cracks one has

kk
k
k
k
γγγγ
γγ
ω
ω
ω
(3.13)

[
]
2/1
2
2
21
2
1
)()(1
~
ee
kkk
Φ−Φ−=
γγω
. (3.14)
3.1.2. Explicit expression for mode shape
Consider the beam with n cracks at
nje
j


)sin)(sinh2/1()( xxxS
λ
λ
λ
+
=
(3.16)
One can express solution of equation (3.15) in segment
njee
jj
, ,1],,[
1
=
+
, satisfying the compatibility conditions at cracks, Li
[27] , as

njexSexx
jjjjjj
, ,1),()()()(
11
=
−
′
′
+
=
−−
φ

⎧
≤
=
0),(
00
)(
fxxS
x
xK
, (3.18)
One can express general solution of equation (3.15) in the form

,)()()(
1
0
∑
−+Φ=Φ
=
n
k
kk
exKxx
μ
(3.19)
where
)(
0
xΦ is general solution of equation (3.15) for uncracked beam
that can be expressed as


′′
+
′′
=
=
−
=
n
k
kkj
j
kj
j
jk
kjj
eeb
L
eeSxLDxLC
μλ
λ
γ
λμλλγμ
(3.21)
Using the matrix notations

{}
,, ,)(
1 n
diag
γ

= ,
equation (3.21) can be rewritten as

[]
.0)(),()(
0
=− μIBΓ
λλγ
Le
(3.23)
From equation (3.19) one can express modal slope and curvature in
the form

∑
=Φ=Φ
=
n
j
rjjr
l
r
l
r
l
exCxx
1
)()()(
),,()(),(
μλαλ
(3.24)

position on mode shape is also symmetrical. Two cracks at positions
symmetric through the middle of beam may annualize effect of each
other on the mode shape.
13

3.3. Multiple crack detection by frequency and mode shape
3.3.1. The problem and solution
Assume that m
f
natural frequencies
(
)
,, ,
**
1
mf
ωω
are available so
that corresponding eigenvalues
frr
mrFL ,,2,1,
4
2**
==
ρωλ
can be
calculated. Moreover, suppose that mode shapes have been measured
at positions (x
1
, …x

k
e if
0f
k
γ
. This is essence of the method of crack detection called crack
scanning technique that is described below.
Using equation (3.19) the mode shape can be calculated at the
mesh (x
1
, …x
m
) and as result one gets the system of linear algebraic
equations

rr
bμA =
(3.25)
where
{
}
;, ,
**
1 rmrr
φφ
=b ];, ,1;, ,1),([)(
**
nkmja
rjkrr
====

by using the Tikhonov’s regularization method.
Using SVD method solution of equations (3.25) can be
ontained as

∑
+
=
=
R
k
k
k
r
T
kk
r
1
2
)(
v
bu
μ
σα
σ
(3.27)
14

with the regularization factor
α
is chosen from equation

r
T
k
rrr
bu
bu
bμA . (3.28)
Afterward, the vector of crack magnitudes can be calculated as

., ,1,
),(
)(
ˆ
,
*
*
0
nj
μb
μL
kr
rkrjk
r
rjr
j
=
∑
∑
=
e

}
1, 0
21
≤
≤
n
eee pp for scanning cracks in
beam of the unknown magnitudes
{
}
.,
1
T
n
γγ
=γ

Step 2:
Based on the crack mesh
), ,(
1 n
ee
, the equations (3.25), (3.26)
and expression (3.27) are constructuted for every mode.
Step 3:
Solving equation (3.28) with a given noise level leads first the
factor
α
ˆ
to be found and then solution )

=
γ versus crack
position mesh
{}
ci
ni , ,1, =e in the ),(
γ
e - plane obvious peaks give
thedetected crack positions.
Step 6:
Handling the steps 1-4 with the obtained above detected crack
mesh one will have got finnaly desird crack magnitudes.
3.3.3. Numerical results
The noisy left hand side of equation (3.25) is simulated by

mj,
m
E
j
p
jj
, ,1
*
=+= N
σ
φφ
, (3.30)
15

with

Table 3.4. Result of crack detection extracted from Figure 3.2
(number of measurement points 40) Table 3.5. Result of crack detection extracted from Figure 3.3 (40
measurement points and noise level 2%)

16 Figure 3.1. Crack magnitude versus crack scanning mesh with noise
level 3%, crack depth a/h = 0,5; number of points in the scanning
mesh 10,…, 60.

Figure 3.2. Crack magnitude versus crack scanning mesh with crack
depth 50%, number of measurement points 40, noise level from 0% to
10%.

Figure 3.3. Crack magnitude versus crack scanning mesh for noise
level 2%, 40 measurement points, crack depth from 10% to 60%.

Observing the numerical results leads to the conclusions
Cracks in beam might be exactly localized by spairse
measurements at 10 sites if scanning mesh is chosen properly and
increasing the number of measurement points of course improves the
result of crack localization. However, some false cracks can be
observed near the free end of beam even in the case of small noise
17

level. This miscalculation disappears for suitablely chosen


(
)
(
)
,0
010
1
1
=−+−+ wykwycym
&&&&
(4.1)
Finite element model of cracked beam under moving load

,
0
f
T
NfKddCdM ==++
&&&
(4.2)
18

f
0
- the load from moving vehicle. Figure 4.1. Simply supported beam under moving load of ¼ vehicle.


⎢
⎢
⎢
⎢
⎣
⎡
+−
+−−
−−+−
−−+
+
⎪
⎪
⎭
⎪
⎪
⎬
⎫
⎪
⎪
⎩
⎪
⎪
⎨
⎧
⎥
⎥
⎥
⎥
⎥

0
000
000
000
000
d
d
d
d
ccccb
ccccb
cccccbcb
cbcbcbcbcbcb
d
d
d
d
m
m
m
I
&
&
&
&
&&
&&
&&
&&


⎪
⎨
⎧
+
+
=
⎪
⎪
⎭
⎪
⎪
⎬
⎫
⎪
⎪
⎩
⎪
⎪
⎨
⎧
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢

&&&
(4.4)
19

4.2. Response of beam to the moving load
• The model of ¼ vehicle : (Figure 4.1)
Consider beam with L = 50m, ρ = 7860 kg/m
3
, E =
2,1×10
11
N/m
2
, cross section b×h = 0,5×1,0m. Assume that two cracks
appear at positions L/3 and 2L/3 of the beam. The parameters of
vehicle are taken those of Reference Majumder [37].

a) v = 1m/s b) v = 4m/s
Figure 4.3. Vibration of vehicle for the case of uncracked ( ) and
cracked ( ) beam in different speeds of moving.

a) vận tốc v = 1m/s b) vận tốc v = 5m/s
Figure 4.4. Vibration of vehicle in the case of different crack depth.

• Model of ½ vehicle: (Figure 4.2)
Beam parameters are ρ = 7855kg/m
3
, E = 2,1×10
11
N/m

y – calculated vertical displacement of vehicle, E
p
noise level, N
oise
a
random vector of normal distribution with mean 0 and standard
deviation 1,
σ
(y) standard deviation of y, y
oise
noisy displacement of
vehicle.
4.4. Crack detection by wavelet
• Wavelet transform of beam response to ¼ vehicle

Figure 4.6. a/h = 0 , v =1 m/s.

Figure 4.7. a/h=0,1 v = 1 m/s.
21 Figure 4.8. a/h=0,3 v=1 m/s.

Figure 4.9. a/h=0,5 v=1 m/s.

Figure 4.10. a/h=0,5; v = 0,5 m/s.
noise level E
p
= 0%, noise level E
p

passed across the crack position L/3 and peacks at t = 13,7s và t =
16,7s correspond to passing of the legs across second crack at 2L/3.
Consequently, peacks at the wavelet diagrams are really
indicators of presence of cracks and the peacks are apparent for the
measurement noise up to 6%.
23

Summary of Chapter 4
In this Chapter the models of ¼ and ½ vehicle moving on the
cracked beam have been proposed and analyzed in time domain with
the wavelet transform. It has demonstrated influence of crack
parameters, speed of moving vehicle and measurement noise on the
wavelet diagram of the vehicle response. This analysis shows that

GENERAL CONCLUSIONS

The major results obtained in this thesis are as follow:
1. Explicit expressions of natural frequencies and mode shapes have
been derived for multiple cracked beams in term of crack
parameters that can be usefully employed for crack detection by
using the vibration-based method. Particularly, the expressions
are useful tool for modal analysis of multiple cracked beam.
2. Model of cracked beam subject to moving vehicle has been
established and analyzed by using the Finte Element Method and
Newmark procedure.
3. A new procedure has been developed for multiple crack
identification for beam from noisy measured mode shape with
application of the Tikhonov’s regularization method that allows
for obtaining consistent solution of the inverse problem for the
noise level up to 10%.