Crack is a damage usually happened in structural members
but dangerous for safety of structure if it is not early detected.
However, cracks are often difficult to identify by visual
inspection as they occurred at the unfeasible locations.
Therefore, cracks could be indirectly detected from measured
total dynamical characteristics of structures such as natural
frequencies, mode shapes and frequency or time history
response. In order to identify location and size of a crack in a
structure the problem of analysis of the crack’s effect on the
dynamic properties is of great importance. It could give also
useful tool for crack localization and size evaluation. On the
other hand, beams are frequently used as structural member in
the practice of structural engineering. So, crack detection for
beam-like structures gets to be an important problem. Crack
detection problem of beam with uniform cross section is
thoroughly studied, but vibration of cracked beam with varying
cross section presents a difficult problem. It is because vibration
of such the structure is described by differential equations with
varying coefficients that are nowadays not generally solved.
The beam with piecewise uniform beam, acknowledged as
stepped beam is the simplest model of beam with varying cross
section. Although, vibration analysis and crack detection for
stepped beam have been studied in some publications,
developing more efficient methods for solving the problems of
various types of stepped beams is really demanded

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MINISTRY OF EDUTATION
AND TRAINING
VIETNAM ACADEMY OF SCIENCE
AND TECHNOLOGY
GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY
-----------------------------
VU THI AN NINH
MODAL ANALYSIS AND CRACK DETECTION
IN STEPPED BEAMS
Specialization: Engineering Mechanics
Code: 62 52 01 01
ABSTRACT OF DOCTOR THESIS
IN MECHANICAL ENGINEERING AND ENGINEERING
MECHANICS
HANOI - 2018
The thesis has been completed at: Graduate University of Science and
Technology - Vietnam Academy of Science and Technology
Supervisions:
1: Prof.DrSc. Nguyen Tien Khiem
2: Dr. Tran Thanh Hai
Reviewer 1: Prof.DrSc. Nguyen Van Khang
Reviewer 2: Prof.Dr. Nguyen Manh Yen
Reviewer 3: Assoc.Prof.Dr. Nguyen Đang To
Thesis is defended at Graduate University of Science and Technology
- Vietnam Academy of Science and Technology at, on
datemonth201
Hardcopy of the thesis be found at:
- Library of Graduate University of Science and Technology
- Vietnam National Library
1
INTRODUCTION
1. Necessarily of the thesis
Crack is a damage usually happened in structural members
but dangerous for safety of structure if it is not early detected.
However, cracks are often difficult to identify by visual
inspection as they occurred at the unfeasible locations.
Therefore, cracks could be indirectly detected from measured
total dynamical characteristics of structures such as natural
frequencies, mode shapes and frequency or time history
response. In order to identify location and size of a crack in a
structure the problem of analysis of the crack’s effect on the
dynamic properties is of great importance. It could give also
useful tool for crack localization and size evaluation. On the
other hand, beams are frequently used as structural member in
the practice of structural engineering. So, crack detection for
beam-like structures gets to be an important problem. Crack
detection problem of beam with uniform cross section is
thoroughly studied, but vibration of cracked beam with varying
cross section presents a difficult problem. It is because vibration
of such the structure is described by differential equations with
varying coefficients that are nowadays not generally solved.
The beam with piecewise uniform beam, acknowledged as
stepped beam is the simplest model of beam with varying cross
section. Although, vibration analysis and crack detection for
stepped beam have been studied in some publications,
developing more efficient methods for solving the problems of
various types of stepped beams is really demanded.
2
2. Objective of the thesis
The aim of this thesis is to study crack-induced change in
natural frequencies and to develop a procedure for detecting
cracks in stepped beams by measurement of natural frequencies.
3. The research contents of the thesis
(1) Developing the Transfer Matrix Method (TMM) for
modal analysis of stepped Euler – Bernoulli, Timosheko and
FGM beams with arbitrary number of cracks.
(2) Expanding the Rayleigh formula for computing natural
frequencies of stepped beams with multiple cracks.
(3) Employing the extended Rayleigh formula for
developing an algorithm to detect unknown number of cracks in
stepped beams by natural frequencies.
(4) Experimental study of cracked stepped beams to
validate the developed theory.
Thesis composes of Introduction, 4 Chapters and
Conclusion. Chapter 1 describes an overview on the subject
literature; Chapter 2 – development of TMM; Chapter 3 – the
Rayleigh method and Chapter 4 presents the experimental
study.
CHAPTER 1. OVERVIEW ON THE MODELS,
METHODS AND PUBLISHED RESULTS
1.1. Model of cracked beams
1.1.1. On the beam theories
Consider a homogeneous beam with axial and flexural
displacements ( , , )u x z t , ( , , )w x z t at cross section x. Based on
some assumptions the displacements can be represented as:
3
0 0 0( , , ) ( , ) ( , ) ( ) ( , ); ( , , ) ( , ),u x z t u x t zw x t z x t w x z t w x t
where u0(x, t), w0(x, t) are the displacements at the neutral axis,
(x,t) – shear slop, z is heigh from the neutral axis. Function
(z), representing shear distribution can be chosen as follow:
(a) ( ) 0z - for Euler-Bernoulli beam theory (the classical
beam theory).
(b) ( )z z - for Timoshenko beam theory or the first order
shear beam theory.
(c) 2 2( ) 1 4 / 3z z z h - the second shear beaa theory.
(d)
2{ 2( / ) }( ) e z hz z - the exponent shear beam theory.
Recently, one of the composites is produced and called
Functionally Graded Material (FGM), mechanical properties of
which are varying continuously along corrdinate z or x.
Denoting elasticity modulus E, shear modulus G and material
density , a model of the FGM is represented as
( ) ( ) ( )b t bz g z
where ,b t stand for the characteristics (E, , G) at the
bottom and top beam surfaces and function g(z) could be chosen
in the following forms:
a) P-FGM: ( ) ( 2) /
n
g z z h h - the power law material.
b) E-FGM: (1 2 / )( ) , 0.5ln( / )z ht t bE z E e E E
- the exponent
law material.
c) S-FGM: 1( ) 1 0.5 1 2 /
n
g z z h , 0 / 2z h .
d) 2 ( ) 1 2 / / 2
n
g z z h , / 2 0h z - Sigmoid law material.
In this thesis only the FGM of power law is investigated.
4
1.1.2. Crack model in homogeneous beams
Fig 1.2. edge crack model.
Consider a homogeneous beam as shown in Fig. 1.2 that
contains a crack with depth a at position e. Based on the
fracture mechanics, Chondros, Dimagrogonas and Yao have
proved that the crack can be represented by a rotational spring
of stiffness
where EI is bending stiffness, h is heigh of beam and function
Hence, compatibility conditions at the cracked sections are
For Timoshenko beam the conditions take the form
( 0, ) ( 0, );w e t w e t ( 0, ) ( 0, ) ( , );x x xe t e t e t
( 0, ) ( 0, ) ( )c xe t e t e ; w ( 0, ) w ( 0, ) ( , )x x c xe t e t e t .
M
a
e
' M e-0
' e+0
2
.
6 (1 ) ( / )
c
c
EI
hI a h
K
2 2 3 4
5 6 7 8
( ) (0.6272 0.17248 5.92134 10.7054 31.5685
67.47 139.123 146.682 92.3552 ),
cI z z z z z z
z z z z
2
2
0 0
2 2 3 3
2 2 3 3
w(x,t) w(x,t) ( ) w( , )
w( 0, ) w( 0, ),
w( 0, ) w( 0, ) w( 0, ) w( 0, )
, .
c
x e x e c
M e x t
x x K x
e t e t
e t e t e t e t
x x x x
5
1.1.3. Modeling crack in FGM beam
Crack in FGM beam can be modeled by a spring of
stiffness calculated as
1.2. Vibration of cracked beams
1.2.1. Homogeneous beams
Consider an Euler-Bernoulli beam with n cracks at
positions 1 20 ... ne e e L and depth , 1,2,..., .ja j n Free
vibration of the beam is described by equation
4 4 4 4 2( ) / ( ) 0, / .d x dx x F EI
in every beam segment 1 0 1( , ), 1,..., 1, 0,j j ne e i n e e L ,
general solution of which is
1( ) cosh sinh cos sin , ( , ).j j j j j j jx A x B x C x D x x e e
Substituting the solution into conditions at the crack positions
1 1 1( ) ( ), ( ) ( , ( ) ( ),j j j j j j j j j j j je e e e e e
1( ) ( ) ( ), 1,2,..., .j j j j j j je e e j n
one obtains 4n equations for 4(n+1) unknowns
1 1 1 1 1 1 1 1{ , , , ,..., , , , }
T
n n n nA B C D A B C D C .
Therefore, combining the equations with 4 boundary conditions
allows one to get closed system of equations
1 1[ ( , ,..., , ,..., )]. 0n ne e D C .
2 2/
2
0
72 (1 ) ( )
1/ ; ,
( )
a h F
K C C d
h E h
2 3 4
5 6 7
2 1
( ) 1.910 2.752 4.782 146.776 770.75
1947.83 2409.17 1177.98 , / 0.2;
F
E E
2 3 4
5 6 7
2 1
( ) 1.150 1.662 21.667 192.451 909.375
2124.31 2395.83 1031.75 , / 1.0;
F
E E
2 3 4
5 6 7
2 1
( ) 0.650 0.859 12.511 72.627 267.91
535.236 545.139 211.706 , / 5.0.
F
E E
6
for determining the unknown constants. Hence, frequency
equation can be obtained as
1 1det[ ( , ,..., , ,..., )] 0,n ne e D
that could be solved to give roots , 1,2,3,...k k from which
natural frequencies are calculated as
2 / , 1,2,3,....k k EI F k
For Timoshenko beam, equations of free vibration are
2 W( ) (W ) 0x G ; 2 ( ) ( ) (W ) 0I x EI x GA ,
that would be solved together with conditions at cracks
W( 0) W( 0) W( )j j je e e ; ( 0) ( 0) ( );j j je e e
( 0) ( 0) ( )j j j je e e ; ( 0) ( 0) ( ).j j jW e W e e
Similarly, putting general solution
1 1 2 2W ( ) cosh sinh cos sin ;j j j j jx A k x B k x C k x D k x
1 1 1 1 2 2 2 2( ) sinh cosh sin cos ,j j j j jx r A k x r B k x r C k x r D k x
2 2 2 2
1 1 1 2 2 2( ) / ; ( ) / ;r Gk Gk r Gk Gk
2 2
1 2( 4 ) / 2, ( 4 ) / 2k b c b k b c b
2(1 ); ( ); / ; / ; /b c E E G F I .
in beam segment 1( , )j je e into condittions at cracks and
boundaries, frequency equation is obtained also in the form
1 1det ( , ,..., , ,..., ) 0n ne e D
for determining natural frequencies , 1,2,3,...k k
1.2.2. Vibration of FGM beams
Based on the model of Timoshenko FGM beam and taking
account for actual position of neutral plane equations of motion
of the beam can be established in the form
7
11 11 12 0I u A u I ; 11 33( ) 0I w A w ;
12 22 22 33( ) 0;I u I A A w
with coefficients 11 22 33 11 12 22, , , , ,A A A I I I calculated from the
material constants , , , , , ,...b t b tE E n Beside, from condition of
neutral plane, actual position of the axis measured from the
midplane is determined as
0 [ ] / / .( 1) [2( 2)( )],e e e t bh n r h n n r r E E
Seeking solution of the equations of motion given above in the
form
( , ) ( ) ; ( ,t)=W( ) ; ( ,t)= ( )i t i t i tu x t U x e w x x e x x e ,
one has got the equations
2 2
11 11 12( ) 0I U A U I ; 2 11 33( ) 0I W A W ;
2 2
22 22 12 33( ) ( ) 0,I A I U A W
that in turn give rise general solution
0 0( , ) ( , )x x z G C ,
where
0
,( , ) { ( , ), ( , ), ( , )}Tx U x x W x z 1 6,...,C{C }
TC =
and
2 2 2 2 2
12 11 11 33 11 33/ ( ); / ( ), 1,2,3.j j j j jI I k A k A I k A j
In case, if the beam is cracked at position e the solution gets to
be
( ) ( ). ,c cx xz Φ C 0 0( ) ( , ) ( ) ( , )c x x x e e Φ G K G .
3 31 2 1 2
3 31 2 1 2
3 31 2 1 2
1 2 3 1 2 3
0
1 2 3 1 2 3
( , ) ;
k x k xk x k x k x k x
k x k xk x k x k x k x
k x k xk x k x k x k x
e e e e e e
x e e e e e e
e e e e e e
G
( ) : 0; ( ) : 0;
( ) ( )
0 : 0 0 : 0
c cx x x x
x x
x x
G G
K K
8
1.2.3. Conventional formulation of TMM
In this section, an Euler-Bernoulli homogeneous beam
composed of uniform beam elements with the material and
geometry constants:
{ , , , , }, 1,2,...,j j j j jE A I L j n ,
It is well known that general solution of free vibration problem
in every beam segment is expressed in the form
( ) cosh sinh cos sin , (0, ),j j j j j j j j j jx A x B x C x D x x L
with 2 1/4( ) ( / )j j j j j jA E I . Introducing the state vector
{ ( ), ( ),M ( ),Q ( )}j j j j jx x x x V ,
( ) ( ); ( ) ( )j j j j j j j jM x E I x Q x E I x
we would have got the expression
( ) ( )j j jx xV H C ; { , , , }
T
j j j j jA B C DC
and ( )j xH is a matrix function acknowledged as shape function
matrix. From the continuity conditions at joints of the beam
segments 1( ) (0)j j jL V V one gets
1
1 , 1( 1) (0). ( ). ( ) . ( )j j j j jj L j j
V H H V T V
or
, 1 1, 1 21( ) . ... (1) . (1)n n n nn V T T T .V T V ,
with T being called transfer matrix of the beam. Applying
boundary conditions for the latter connection allows one to get
0 1 1{ (0)} 0; { (1)} 0n B V B V .
or ( ). (1)=0.B V Consequently, frequency equation is obtained as
det ( ) =0. B
This is content of the so-called Transfer Matrix Method that is
appropriate for modal analysis of stepped beams.
9
1.2.4. Rayleigh method
For a standard beam flexural deflection in in vibration of
frequency is ( , ) ( )sinv x t x t with function ( )x called
mode shape of vibration. In that case, potential and kinetic
energies are
2 2
0
(1/ 2)(sin ) ( ) ;
L
xxt EI x dx
2 2 2
0
(1/ 2)(cos ) ( ) .
L
T t A x dx .
Obviously, one of the energies reaches maximum when the
other gets to be minimum equal to 0. So that, due to the energy
conservation one gets 2 2 2
0 0
( ) ( )
L L
xxEI x dx A x dx , from that
frequency can be calculated as
2 2 2
0 0
( ) / ( )
L L
xxEI x dx A x dx .
This is classical form of Rayleigh formula or ratio that
expresses relationship between exact mode shape and frequency
of undamped free vibration. The Rayleigh formula in the exact
form has no meaning for application to calculate frequency
from mode shape because both the modal parameters, the
frequency and mode shape, are usually found together.
However, if we might select approximately a function for mode
shape, then natural frequency could be easily calculated by
using the Rayleigh formula. Off course, this is an
approximation of the natural frequency and it converges to the
exact frequency if the chosen mode shape gets to be approached
to the exact one. Such calculating natural frequency from
appropriately chosen shape function is acknowledged as
Rayleigh method. The Rayleigh formula was expanded for
multiple cracked Euler-Bernoulli beam by N.T. Khiem and T.T.
10
Hai and applied for calculating natural frequencies of just
uniform Euler-Bernoulli beams.
1.3. Crack detection problem for beams
Contents of the crack detection problem is to localize and
evaluate severity of crack based on the measured data gathered
from testing on the structure of interest. There are two
appoaches to solve the problem: first approach is based only on
measured data that are often response of the structure to a given
load; the second one involves additionally a model of the
structure with assumed cracks of unknown loacation and depth.
The crucial tool for the first approach is the method used for
signal processing such as, for example, the Fourier or wavelet
transform. The second approach finds the way to connect the
measured data with the structure model in form of diagnostic
equations of unknown crack parameters. The advantage of the
model-based approach to crack detection is that enables to
apply the latest achievements in both theoretical and numerical
development of the system identification theory. In this thesis,
the model-based approach is applied and the crack parameters
are determined from the equations connecting the measured and
calculated natural frequencies.
1.4. Overview on vibration of stepped beams
1.4.1. Spepped beams without cracks
Free vibration of stepped beams was studied by numerious
authors such as Jang and Bert; Jaworski and Dowell, Cunha et
al.; Kukla et al. and Yang, ... The most important obtained
results demonstrate that natural frequencies of stepped beam are
11
significantly affected by abrupt change in cross section area of
stepped beams and the natural frequency variation is dependent
also on the boundary conditions. Sato studied an interesting
problem that proposed to calculate natural frequency of beam
with a groove in dependence on size of the groove. Using a
model of stepped beam and the Transfer Matrix Method
combined with Finite Element Method the author demonstrated
that (a) fundamental frequency of the structure increases with
growing thickness and reducing length of the mid-step; (b) the
mid-step could be modeled by a beam element, therefore, the
TMM is reliably applicable for the stepped beam if ratio of its
length to the beam thickness (r=L2/h) is equals or greater 4.0.
Comparing with experimental results the author concluded that
error of the TMM may be up to 20% if the ratio is less than 0.2.
1.4.2. Cracked stepped beams
Kukla studied a cracked onestep column with a crack at the
step under compression loading. Zheng et al calculted
fundamental frequency of cracked Euler-Bernoulli stepped
beam by using the Rayleigh method. Li solved the problem of
free vibration of stepped beam with multiple cracks and
concentrated masses by using recurent connection between
vibration mode of beam steps. The crack detection problem for
stepped beams was first solved by Tsai and Wang, then, it was
studied by Nandwana and Maiti based on the so-called contour
method for identification of single crack in three-step beam.
Zhang vet al. solved the problem for multistep beam using
wavelet analysis and TMM. Besides, Maghsoodi et al have
12
proposed an explicit expression of natural frequencies of
stepped beam through crack magnitudes based on the energy
method and solved the problem of detecting cracks by
measurements of natural frequencies. The classical TMM was
completely developed by Attar for both the forward and inverse
problem of multistep beam with arbitrary number of cracks.
Neverthenless, the frequency equation used for solving the
inverse problem is still very complicated so that cannot be
usefully employed for the case of nember of cracks larger than
2.
1.5. Formulation of problem for the thesis
Based on the overview there will be formulated subjects for
the thesis as follow:
(1) Further developing the TMM for modal analysis of
stepped Euler – Bernoulli; Timoshenko and FGM beams;
(2) Extending the Rayleigh formula for calculating natural
frequencies of stepped beam with multiple cracks;
(3) Using the established Rayleigh formula to propose an
algorithm for multi-crack detection in stepped beam from
natural frequencies;
(4) Overall, carrying out an experimental study on cracked
stepped beam to validate the developed theories.
13
CHAPTER 2. THE TRANSFER MATRIX METHOD FOR
VIBRATION ANALYSIS OF STEPPED BEAMS WITH
MULTIPLE CRACKS
2.1. Stepped Euler-Bernoulli beam with multiple cracks
2.1.1. General solution for uniform homogeneous Euler-
Bernoulli beam element is
1 1 2 2 3 3 4 4( ) ( ) ( ) ( ) ( )x C L x C L x C L x C L x ,
where
0
1
( ) ( ) ( ), 1,2,3,4
n
k k kj j
j
L x L x K x e k
;
01 02( ) (cos os ) / 2; ( ) (sin sin ) / 2;L x h x c x L x h x x
03 04( ) (cos os ) / 2; ( ) (sin sin ) / 2;L x h x c x L x h x x
1
0
1
( ) ( ) , 1,2,3,4
j
kj j k j ki j i
i
L e S e e k
.
2.1.2. The transfer matrix
Using the solution for mode shape, transfer matrix for the
beam with cracks is conducted in the form
, 1 1, 1 21( ) . ...n n n n T T T T ;
1(j) = ( ). (0)j j jL
T H H ;
2.1.3. Numerical results
For illustration, two types of stepped beam as shown in Fig.
2.1 are numerical examined herein. The first is denoted by B1S
and the second – B2S. Three lowest natural frequencies of the
beams with single crack are computed versus crack location
(Fig. 2.2).
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) .
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
j j j j
j j j j
j
j j j j j j j j j j j j
j j j j j j j j