30120130405012

International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –
6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME
103
BILINEAR RESPONSE ANALYSIS OF BREATHING CRACK IN
CANTILEVER BEAM
Dr.Animesh Chatterjee1
,Ashwini Baghele2
, Nagnath Kakde3
,Vishal Deshbhratar4
1
Mechanical Department, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra India
2
Mechanical Department, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra India
3
Mechanical Department, Dr. Babasaheb Ambedkar college of Engineering and Research,
Nagpur, Maharashtra India
4
Mechanical Department, Vidarbha Institute of Technology,Umrer Road, Nagpur
ABSTRACT
Vibration measurements offer an effective, inexpensive and fast means of non- destructive
testing of structures and various engineering components. There are mainly two approaches to crack
detection through vibration testing; open crack model with emphasis on changes in modal parameters
and secondly, the breathing crack model focusing on nonlinear response characteristics. The open
crack model based on linear response characteristics can identify the crack only at an advanced stage.
Researchers have shown that a structure with a breathing crack behaves more like a nonlinear
system, the nonlinear response characteristics can very well be investigated to identify the presence
of the crack. In the present study, the switching behavior of crack are analyzed and converted the
time-domain data into frequency domain. The effect of crack severity on the response harmonic
amplitudes are investigated and a new procedure is suggested whereby the crack severity can be
estimated through measurement of the first and second harmonic amplitudes.
Keywords:- Vibration, Breathing Crack, Cantilever Beam
1. INTRODUCTION
Fatigue crack often exist in structural member that are subjected to repeated loading ,which
compromises the structural integrity. Many studies have been carried out on the dynamic response
of fatigue cracks, in an attempt to find viable vibration methods for non destructive inspection and
health monitoring. The crack models used in these analyses fall largely into two categories: open
crack models and breathing crack models .Most researchers have used open crack models in their
studies and have claimed that the change in natural frequency might be a parameter used to detect the
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
AND TECHNOLOGY (IJMET)
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 4, Issue 5, September - October (2013), pp. 103-110
© IAEME: www.iaeme.com/ijmet.asp
Journal Impact Factor (2013): 5.7731 (Calculated by GISI)
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IJMET
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104
presense of cracks However the assumption that cracks are always open in vibration is not realistic
because compressive loads may close the cracks .Recently, efforts have focused on vibration analysis
using opening and closing models to simulate a fatigue crack .Their fatigue crack model considers
the bilinear behavior of an elastic crack. In this model ,the structure has only two characteristic
stiffness values: larger value corresponding to the state of crack closing and a smaller value for
crack opening. This fatigue crack model, however, only represents an idealized situation in which
the crack has two perfectly flat surfaces and can only exist in fully open or fully closed states. Due to
the lack of a systematic theory regarding the breathing crack, it is difficult to interpret the
experimental results. The effect of the breathing crack in the vibration response of cracked structural
members have been recognized long ago.
2. LITERATURE REVIEW
Kirmsher in 1944 [1]reported that if a crack in a concrete beam is filled with dirt or
crystallized material, or is narrow enough so that interference occurs, the effect on the natural
frequency is the same as that of a crack of lesser depth. This observation was the basis for
systematic investigation of the effects of opening and closing of cracks .Actis and Dimarogonas [2],
used the finite element method to study the simply supported cracked beam. The crack was assumed
to be a breathing crack. They assumed that when the bending moment changes sign, the crack
changes from open to closed, or from closed to open.
Ibrahim et al. [3] presented a bond graph technique that models the crack as a torsional spring
with two spring constants, one when it is open, and the other when it is closed. A numerical
simulation procedure is used for the prediction of the non-linear behaviour of a cantilever beam with
a fatigue crack located near its root. Qian et al. [4] investigated the effects of an opening and closing
crack on the dynamic behaviour of a cantilever beam using a finite element model for the cracked
member. A numerical method and Hermitian interpolation was introduced for the solution of the
resulting non-linear equations of motion.
Friswell and Penny [5]have simulated the non-linear behavior of a beam with a closing crack
vibrating in its first mode of vibration through a simple 1 d.o.f .model with bilinear stiffness .Shen
and Chu[6] simulated, with a bilinear equation of motion for each mode of vibration, the dynamic
response of simply supported beams with a closing crack ,analyzing the response spectra in order to
detect changes which are potentially useful for damage assessment . Krawczuk and Ostachowicz [7]
studied the transverse vibrations of a beam with a closing crack simulated through springs with
periodically time variant stiffness. In a recent paper these author presented an analysis of the forced
vibrations of cantilever beam with a closing crack, in which the equation of motion were solved
using harmonic balance method[8].
Abraham and Brandon [9]applied a piece-wise linear approach to analyse vibrations of a
cantilever beam with a breathing crack. Their formulation was a hybrid frequency-domain/time-
domain method. For the majority of the vibration ,the crack section is unambiguously either open or
closed. During this time a mode superposition is used to develop the response of the system. They
proposed fourier series to simulate the continuous change of stiffness in crack breathing.
In this paper,a simple non-linear fatigue crack model of cantilever beam is developed.
Equations of motion have solved by newmark beta method[12] and analyse the response in time
domain and in frequency domain by fast fourier transform. A procedure is then suggested for
estimating the structural damage through measurement of the first and second harmonic amplitudes.
The method is illustrated with numerical simulation for two different damage levels and it is found
that the procedure provides an accurate estimation of the damage, even when the crack size is very
small.

105
3. CRACK MODEL DESCRIPTION
The presence of crack in a structural member alters the local compliance that would affect the
vibration response under external loads. This effect is related to the crack tip stress intensity factors.
Consider the cantilever beam as shown in Fig.1.The length of beam is L and it has a crack at a
distance l1 from the fixed end. The beam is divided into elements. The beam stiffness matrix is
derived using energy method. The strain energy for the uncracked beam element, subjected to a total
bending moment Mb is given by
U0=1/2 ∫
l
0
Mb
2
dx/EI (1)
Where E is the Young’s modulus ,I is the geometric moment of inertia and l is the length of
the element. The total bending moment maybe written as
Mb=Tl+M, (2)
Where T is the shear force and M is the bending moment on the element.
Fig 1. Schematic model of beam
The additional strain energy due to the presence of crack is given by Tada et al.[10] as
U1= ∫A
1/AE’
[KI
2
+KII
2
+1/(1-ν)KIII
2
]dA, (3)
Where A is the crack cross-section ,E’
=E for plane stress and E’=E/1-v2
for plane strain, v is
the poisson’s ratio and KI,KII,KIII are stress intensity factors for opening, sliding and tearing type
cracks ,respectively. To apply the linear fracture mechanics theory, it is necessary to consider a plain
strain state. Neglecting the effect of axial force, for a beam with cross sectional area bxh and a crack
with depth a,Eq.(3) may be written as
U1=b ∫
a
0
{[(KlM+KlP)2
+KIIP
2
]/E’}da. (4)
The stress intensity factors are then given as [10]
KlM = 6M/bh2
aπ FI(s),
KlP = 3TL/bh2
aπ FI(s),
KlIP=T/bh aπ FII(s), (5)
l1
a
T
L
M
b
h

106
Where s=a/h and FI(s) and FII(s) for a rectangular cross-section are expressed as follows:
FI(s) = )2/tan()/2( ss ππ 0.923+0.199[1-sin(π s/2)]4
/ cos(π s/2)),
FII(s)=(3s-2s2
)1.122-0.561+0.085s2
+0.18s3
/( s−1 ), (6)
From the definition of the compliance, the flexibility coefficient for an element without crack is
obtained as
Cij
(0)
=∂ 2
U0/∂ Ti ∂ Tj
T1=T, T2=M, i,j=1,2. (7)
The additional flexibility coefficient introduced due to crack is
Cij
(1)
=∂ 2
U1/∂ Ti ∂ Tj ,
T1=T, T2=M, i,j=1,2. (8)
3.1 Cracked beam finite element
In order to model the effect that the crack introduces to a structure, the stiffness matrix of a
cracked beam element is derived. The element is assumed to have a transverse crack under bending
and shear forces.Applying the principle of work,the stiffness matrix of cracked beam element is
defined as
[Kc]=[T]T
[C]-1
[T]. (9)
Where [T] is the transformation matrix given as
[T] = -1 -l 1 0
0 -1 0 1
From eqs. (1),(2) and (4)-(8) the elements of flexibility matrix [C] may be obtained as
C11=l3
/3EI + 2B1[9l2
B2 + h2
B3],
C12=l2
/2EI + 36lB1B2 ,
C22=l/EI+72B1B2 (10)
Where, B1 = π (1-v2
)/Ebh2
, B2 =
∫
s.
0
sFI
2
(s)ds , , B3=
∫
s.
0
sFII
2
(s)ds ,
Using Eq.(9) ,the cracked element stiffness matrix becomes
(11)

107
The integrals B2 and B3 were evaluated using the MATLAB program. Due to negligible
effect of crack on mass of the beam element ,mass matrix of cracked element is assumed to be the
same as that of uncracked element.
3.2 Cracked element behavior
To show the vibration behavior of cracked beam, consider a cantilever beam with ten
elements as a model and MATLAB[11] program ,the natural frequencies of the cracked beam are
obtained for aedge crack located at normalized distance x/L from the fixed end and with a
normalized depth a/h in each location. The normalized natural frequencies are defined as the ratio of
cracked beam natural frequency to the uncracked beam natural frequency.
Variations of the first four normalized natural frequencies of the beam model in terms of
crack depth ratio and cracked element location are constructed shown in Fig. 2. Variations of the
normalized natural frequencies of the beam model in terms of the cracked element number, for
different crack depth ratios, are illustrated in Fig. 3. As can be seen from Fig. 2, natural frequencies
of the cracked beam decrease as crack grows deeper. The fundamental frequency is mostly affected
when crack is located on the first element which is near the fixed end. The reason is that the presence
of a crack near the fixed end reduces the stiffness near the support significantly. The second, third
and fourth natural frequencies change rapidly as the crack is located on the sixth element.
From the results it can be seen that the decrease in the frequencies is greatest for a crack
located at the point of maximum bending moment. The trends of changes of the second, third and
fourth natural frequencies, also, are not monotonic as in the first natural frequency (see Fig. 3). It can
be concluded that both crack location and crack depth have influence on the frequencies of the
cracked beam. Also, a certain frequency may correspond to different crack depths and locations.
Based on this, the contour line which has the same normalized frequency change can be plotted
having crack location and crack depth as its axis. The location and depth corresponding to any point
on this curve becomes a possible crack location and depth.
Fig.2 Variation of normalized natural frequencies verses crack depth ratio for different crack element
locations:

108
1 2 3 4 5 6 7 8 9 10
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
1
Element Number
Frequencyofcrackedbeam/Frequencyofuncrackedbeam
1 2 3 4 5 6 7 8 9 10
0.7
0.75
0.8
0.85
0.9
0.95
1
Element number
1 2 3 4 5 6 7 8 9 10
0.8
0.85
0.9
0.95
1
1.05
Element Number
1 2 3 4 5 6 7 8 9 10
0.8
0.85
0.9
0.95
1
1.05
Element Number
Fig.3 Variation of normalized natural frequencies verses cracked element number for different crack
depth ratio

109
4. TIME DOMAIN/FREQUENCY DOMAIN ANALYSIS
In this paper have solved the numerical integration at each time step with switching behavior
of crack by the newmark beta method[9] and find out the displacement per unit time. Frequency-
domain data are obtained by converting time-domain data using a mathematical technique referred to
as a fast Fourier transform (FFT).This FFT gives two frequency spectrum first for the uncracked
beam and second peak gives the assessment of crack in the beam.
Consider a cantilever beam with uniform cross-section having
L=200mm,b=20mm,d=20mm,E=208e9,µ=0.3,ρ=7850kg/m3
,dt=0.2sec.,number of elements=10
Fig 4 the response amplitude for the first and second harmonic
Figure 4 shows that FFT results for computing number of DFT at consistent frequency and
gives the two peaks.First peak for the uncracked and second peak for the structural damage or crack
of the beam.As the crack depth varies the second peak varies continuously and peak ratio decreases
and when move fixed end to free end second peak decreases and first peak remain unchanged
5. CONCLUSION
The Bilinear response of a cracked beam is analysed using fast fourier transform
representation. The effect of crack severity on the response harmonic amplitudes are investigated and
a new technique is suggested whereby the crack severity can be estimated through measurement of
the first and second harmonic amplitudes. The estimate is found to be more accurate for smaller
crack size. The procedure also discusses proper location and depth of structural damage.

110
6. REFERENCES
[1] P.G.Kirmsher, “The effect of discontinuities on the natural frequencies of beam” Proceedings
of the American Society of testing and materials 44,897-904.
[2] R.I.Actis and A.D.Dimargonas,” Non-linear effect due to closing crack in vibrating beams”
12th
ASME Conference on Mechanical Engineering, Vibration and Noise, Montreal Canada,17-
20 september.
[3] A.Ibrahim,F.Ismail and H.R.Martin, “Modelling of the dynamics of a continuous beam
including non linear fatigue crack” International journal of Analytic and Experimental Model
Analysis 2, 76-82.
[4] G.L.Qian,S.N.Gu and J.S.Jiang,”The dynamic behavior and crack detection of beam with a
crack” Journal of sound and vibration 138, 233-243.
[5] M.I.Friswell and J.E.T.Penny, “A simple nonlinear modal of a cracked beam”. In: 10th
Int.
Modal Analysis Conf.,San Diego, Calif.,pp.516-521 (1992).
[6] M.H.Shen and Y.C.Chu, “Vibration of beams with a fatigue crack”. Comput. Struct. 45(1), 79-
93 (1992).
[7] W.Ostachowicz and M.Krawczuk, “Vibration analysis of cracked beam”. Comput.
Struct.36,245-250 (1990)
[8] M.Krawczuk and W.Ostachowicz, “Forced vibration of a cantilever Timoshenko beam with a
closing crack” 1994 proceedings of the 19th
International Seminar on Modal Analysis, K.U.
Leuven, Belgium, Vol. 3, 1067-1078.
[9] O.N..L.Abraham and J.A.Brandon, “The modelling of opening and closing of a crack”
Transaction of the American Society of Mechanical Engineers:Journal of Vibration and
Acoustics 117,370-377.
[10] Tada H,Paris P,Irwin G. “The stress analysis of cracks” handbook. Hellertown ,PA: Del
Rsearch Corp; 1973.
[11] Jabbari M. “Crack detection in beams using vibration test data and finite element method”.
M.Sc. Thesis, Isfanan University of Technology,Isfanan,Iran,2004.
[12] Klaus Jurgan Bathe and Edward L.Wilson. “Numerical methods in finite elements analysis”
Prentice Hall of India 1987.
[13] Akash.D.A, Anand.A, G.V.Gnanendra Reddy And Sudev.L.J, “Determination Of Stress
Intensity Factor For A Crack Emanating From A Hole In A Pressurized Cylinder Using
Displacement Extrapolation Method” International Journal of Civil Engineering & Technology
(IJCIET), Volume 4, Issue 2, 2013, pp. 373 - 382, ISSN Print: 0976 – 6308, ISSN Online:
0976 – 6316, Published by IAEME.
[14] Mohammed S. Al-Ansar, “Flexural Safety Cost Of Optimized Reinforced Concrete Beams”
International Journal of Civil Engineering & Technology (IJCIET), Volume 4, Issue 2, 2013,
pp. 373 - 382, ISSN Print: 0976 – 6308, ISSN Online: 0976 – 6316, Published by IAEME.
[15] Sharad V. Kshirsagar, Dr. Lalit B. Bhuyar, “Signature Analysis Of Cracked Cantilever Beam”
International Journal of Advanced Research in Engineering & Technology (IJARET), Volume
1, Issue 1, 2010, pp. 105 - 117, ISSN Print: 0976-6480, ISSN Online: 0976-6499, Published
by IAEME.

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