a ctip
t K
f Eng
m 29
e 19 D
dyna
graphite bre-reinforced polyamide. The nite element and the componentmode synthesismethods are used tomodel the problem. The
defects such as cracks which, as time progresses, lead to
quencies and mode shapes of the structure contain in-
formation about the location and dimensions of thedamage. Vibration analysis, which can be used to detect
structural defects such as cracks, of any structure oers
an eective, inexpensive and fast means of non-
the past [17] and reviewed by many researchers such as
of a prismatic beam with a central crack. Nikpour [12]
studied the buckling of cracked composite columns andshowed that the instability increases with the column
slenderness and the crack depth. Oral [13] developed a
shear exible nite element for non-uniform laminated
composite beams. He tested the performance of the ele-
ment with isotropic and composite materials, constant
ology*the catastrophic failure or breakdown of the structure.
Thus, the importance of inspection in the quality as-
surance of manufactured products is well understood.
Several methods, such as non-destructive tests, can beused to monitor the condition of a structure. It is clear
that new reliable and inexpensive methods to monitor
structural defects such as cracks should be explored.
Cracks or other defects in a structural element inuence
its dynamical behaviour and change its stiness and
damping properties. Consequently, the natural fre-
Wauer [8] and Dimarogonas [9].
Over the past decade, several techniques have been
explored for detecting and monitoring of the defects in
the composite materials. Adams et al. [10] showed thatany defect in bre-reinforced plastics could be detected
by reduction in natural frequencies and increase in
damping. Nikpour and Dimarogonas [11] studied the
variation of the mixed term in the energy release rate for
various angles of inclination of the material axes of
symmetry and they derived the local compliance matrixcantilever composite beam divided into several components from the crack sections. Stiness decreases due to cracks are derived from
the fracturemechanics theory as the inverse of the compliancematrix calculatedwith theproper stress intensity factors and strain energy
release rate expressions. The eects of the location and depth of the cracks, and the volume fraction and orientation of the bre on the
natural frequencies and mode shapes of the beam with transverse non-propagating open cracks, are explored. The results of the study
lead to conclusions that, presented method is adequate for the vibration analysis of cracked cantilever composite beams, and by using
the drop in the natural frequencies and the change in the mode shapes, the presence and nature of cracks in a structure can be detected.
2003 Elsevier Ltd. All rights reserved.
Keywords: B. Defects; B. Vibration; C. Finite element analysis; Non-destructive testing; Component mode synthesis
1. Introduction
During operation, all structures are subjected to de-
generative eects that may cause initiation of structural
destructive testing. What types of changes occur in thevibration characteristics, how these changes can be de-
tected and how the condition of the structure is inter-
preted has been the topic of several research studies inFree vibration analysis ofwith mul
Mura
Mechanical Engineering Department, Faculty o
Received 8 July 2003; received in revised for
Available onlin
Abstract
This study is an investigation of the eects of cracks on the
Composites Science and TechnTel.: +90-533-4524865; fax: +90-414-3440031.
E-mail address: [email protected] (M. Kisa).
0266-3538/$ - see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.compscitech.2003.11.002antilever composite beamle cracks
isa *
ineering, Harran University, Sanliurfa, Turkey
September 2003; accepted 3 November 2003
ecember 2003
mical characteristics of a cantilever composite beam, made of
64 (2004) 13911402
www.elsevier.com/locate/compscitech
COMPOSITESSCIENCE ANDTECHNOLOGYand variable cross-sections, and straight and curved
geometries. Krawczuk [14] developed a new nite element
for the static and dynamic analysis of cracked composite
beams. He assumed that the crack changes only the
stiness of the element whereas the mass of the element is
unchanged. Krawczuk and Ostachowicz [15] investigatedthe eigenfrequencies of a cracked cantilever composite
beam. They presented twomodels of the beam. In the rst
model, the crack was modelled by a massless spring and
in the second model the cracked part of the beam re-
placed by a cracked element. Krawczuk et al. [16] pro-
posed an algorithm to nd the characteristic matrices of a
composite beam with a single transverse fatigue crack. In
the literature, exception of a few papers [1721], wherethe vibration analysis of beams with two or multiple
cracks was explored, there is a small number of research
works addressing particularly to the problem of free vi-
bration of composite beams having multiple open-edge
cracks. Recently, Song et al. [22] investigated the dy-
namics of anisotropic composite cantilevers. They pre-
is replaced by several smaller ones. In many respects, the
original rationale for such substructuring techniques has
been rendered obsolete by the widespread availability of
high performance computers. However, there are appli-
cations where alternative justications are valid, for ex-ample where the results of independent analysis of
individual structural modules are to be used to predict
the dynamics of an assembled structure. In the present
context, the author wishes to examine the response pre-
diction of an assembled structure under a variety of as-
sumptions concerning nonlinearity at the interface of
substructures which are otherwise linear.
2. Mathematical model
The model chosen is a cantilever composite beam of
uniform cross-section A, having multiple open-edge
transverse cracks of various depths ai at variable posi-tions Li (i 1; . . . ; n). The width, length and height ofthe beam are B, L and H, respectively, Fig. 1. The anglebetween the bres and the axis of the beam is a. The
H1a
x
y
z
Matrix
B
Fibre
2a ia na
1L2L
nLiL
L
Le
A
m m m
1392 M. Kisa / Composites Science and Technology 64 (2004) 13911402sented an exact solution methodology utilising Laplace
transform technique to study the bending free vibrationof cantilever composite beams with multiple open cracks.
The full eigensolution of a structure containing sub-
structures each having large numbers of degrees of
freedom can be cumbersome and costly in computing
time. A method proposed by Hurty [23] enabled the
problem to be broken up into separate elements and thus
considerably reduced its complexity. His method con-
sisted of considering the structure in terms of substruc-tures and was called as substructuring. Essentially, themethod required the derivation of the dynamic equations
for each component and these equations were then
connected mathematically by matrices which represent
the physical displacements of interface connection points
on each component. In this way, one large eigenproblem
1F
1Q1M
u1
v1
1
1
1 2
A1
1
A2
1m1 m2Fig. 2. Components of the composite beam and div2F
2Q2M
u2
v2
2
2
1I 1 N N+1Fig. 1. Geometry of the cantilever composite beamwithmultiple cracks.
I AN AN+1iding them into the nite number of elements.
M. Kisa / Composites Science and Technology 64 (2004) 13911402 1393cantilever beam is partitioned into several components
from the crack sections enabling a substructure ap-
proach, Fig. 2. By separating the whole beam into parts,
global non-linear system can be separated into linear
subsystems joined by local stiness discontinuities. Inthe current study, each component is also divided into
nite elements with two nodes and three degrees of
freedom at each node as shown in Fig. 2.
2.1. Stiness and mass matrices for composite beam
element
The stiness and mass matrices are developed fromthe procedure given by Krawczuk [15] and modied to
three degrees of freedom for each node, d fu; v; hg. InFig. 2, a general nite element, the applied system forces
F fF1;Q1;M1; F2;Q2;M2g and the corresponding dis-placements d fu1; v1; h1; u2; v2; h2g are shown. Thestiness matrix for a two-noded composite beam ele-
ment with three degrees of freedom d fu; v; hg at eachnode, for the case of bending in the xy plane, are givenas follows [15]:
Kel kij66; 1where kij i; j 1; . . . ; 6 are given ask11 k55 7BHS33=3Le;k12 k21 k56 k65 BHS33=2;k13 k31 k35 k53 8BHS33=3Le;k14 k41 k36 k63 k23 k32
k45 k54 2BHS33=3;k15 k51 BHS33=3Le;k16 k61 k25 k52 BHS33=6; 2k22 k66 BH7H 2S11=36Le LeS33=9;k24 k42 k46 k64 BH2H 2S11=9Le LeS33=9;k26 k62 BHH 2S11=36Le LeS33=18;k33 16BHS33=3Le;k44 BH4H 2S11=9Le 4LeS33=9;k34 k43 0;where B, H and Le are the dimensions of the compositebeam element. S11 and S33 are the stressstrain constantsand given as [24]
S11 S11m4 2S12 2S33m2n2 S22n4; 3
S33 S11 2S12 S22 2S33m2n2 S33m4 n4; 4where m cos a, n sin a and Sij terms are determinedfrom the relations [24]
S11 E111 m212E22=E11; S22 S11E22=E11; 5S12 m12S22; S33 G12;where E11, E22, G12 and m12 are the mechanical propertiesof the composite and can be determined as shown in
Appendix A.
The mass matrix of the composite beam element can
be given as [15]
Mel mij66; 6where mij i; j 1; . . . ; 6 arem11 m55 2qBHLe=15;m12 m21 m56 m65 qBHL2e=180;m13 m31 m35 m53 qBHLe=15;m14 m41 m45 m54 qBHL2e=90;m36 m63 m23 m32 m34 m43 0;m15 m51 qBHLe=30;m16 m61 m25 m52 qBHL2e=180;m22 m66 qBHLeL2e=1890 H 2=360;m24 m42 m46 m64
qBHLeL2e=945 H 2=180;m26 m62 qBHLeL2e=1890 H 2=360;m33 8qBHLe=15;m44 qBHLe2L2e=945 2H 2=45;
7
where q is the mass density of the element.
2.2. The stiness matrix for the crack
According to the St. Venants principle, the stresseld is inuenced only in the region near to the crack.
The additional strain energy due to crack leads to ex-ibility coecients expressed by stress intensity factors
derived by means of Castiglianos theorem in the linearelastic range. In this study, the bending-stretching eect
due to mid-plane asymmetry induced by the cracks is
neglected. The compliance coecients Cij induced bycrack are derived from the strain energy release rate, J,
developed in GrithIrwin theory [25]. J can be given as
J oUPi;AoA
; 8where A is the area of the crack section, Pi are the cor-responding loads, U is the strain energy of the beam due
to crack and can be expressed as [11]
U ZA
D1XiNi1
K2Ii
D12
XiNi1
KIiXjNj1
KIIjD2XiNi1
K2IIi
!dA;
9where KI and KII are the stress intensity factors forfracture modes of I and II. D1, D12 and D2 are the co-ecients depending on the materials parameters [11]
D 0:5b Im s1 s2
; 101 22 s1s2
Kji ripa
pYjnFjia=H; 13
nd Technology 64 (2004) 13911402where ri is the stress for the corresponding fracturemode, Fjia=H is the correction factor for the nitespecimen size, Yj(n) is the correction factor for the an-isotropic material [11], a is the crack depth and H is the
element height. Castiglianos theorem [27] implies thatthe additional displacement due to crack, according to
the direction of the Pi, is
ui oUPi;AoPi : 14
Substituting the strain energy release rate J into Eq.
(14), the relation between displacement and strain en-
ergy release rate J can be written as follows:
ui ooPi
ZAJPi;AdA: 15
The exibility coecients, which are the functions of the
crack shape and the stress intensity factors, can be in-
troduced as follows [25]:
cij ouioPj o2
oPioPj
ZAJPi;AdA o
2UoPioPj
: 16
The compliance coecients matrix, after being derivedfrom above equation, can be given according to the
displacement vector d fu; v; hg asC cij33; 17where cij (i; j 1; 2; 3) are derived by using Eqs. (8)(16).
The inverse of the compliance coecients matrix,
C1, is the stiness matrix due to crack. Considering thecracked node as a cracked element of zero length and
zero mass [5], the crack stiness matrix can be repre-
sented by equivalent compliance coecients. Finally,
resulting stiness matrix for the crack can be given as
Kc C1 C1
C1 C1
66: 18
3. Component mode analysis
The equation of motion of a mid-plane symmetrical
composite beam is [24]
IS11o4yx; t=ox4 qAo2yx; t=ot2 f t; 19where I, q, A and yx; t are the geometrical moment ofD12 b11 Ims1s2; 11
D2 0:5b11 Ims1 s2: 12The coecients s1, s2 and bij are given in Appendix A.The mode I and II stress intensity factors, KI and KII, fora composite beam with a crack are expressed as [26]
1394 M. Kisa / Composites Science ainertia of the beam cross-section, material density, cross-sectional area of the beam and transverse deection of
the beam, respectively. Now, consider the component
A1, Fig. 2, for undamped vibration analysis, Eq. (19), in
matrix notation, can be given as
MA1qA1 KA1qA1 fA1t; 20where MA1 and KA1 are the mass and stiness matrices ofthe component A1, respectively, qA1 and fA1t are thegeneralised displacement and external force vectors, re-
spectively. Assuming that
fqA1g f/A1g sinxA1 t b;fqA1g x2A1f/A1g sinxA1 t b
21
and substituting them into Eq. (20), one ends up with
the standard free vibration equation for the component
A1 as,
x2A1MA1/A1 KA1/A1 ; 22which gives eigenvalues x2A11; . . . ;x
2A1n
and modal ma-
trix /A1 for the component A1. Making the transfor-mation
qA1 /A1pA1 ; 23where pA1 is the principal coordinate vector. Bypremultiplying /TA1 and substituting Eq. (23), Eq. (20)becomes
/TA1MA1/A1pA1 /TA1KA1/A1pA1 /TA1fA1t; 24where
/TA1MA1/A1 mm;/TA1KA1/A1 km;
25
where [mm] and [km] are modal mass and stiness matri-ces, respectively. Mass normalising the modal matrix by
wij /ijm
pjj
; 26
where wij is mass normalised mode vector. By using thetransformation
qA1 wA1sA1 27by premultiplying wTA1 and substituting Eq. (27), Eq. (20)becomes
IsA1 x2A1sA1 wTA1fA1t; 28
where x2A1 is a diagonal matrix comprising the eigen-values of A1.
3.1. Coupling of the components
Consider components, A1;A2; . . . ;AN , joined to-gether by means of springs capable of carrying axial,
shearing and bending eects, Fig. 3. The kinetic andstrain energy of the components, in terms of principalmodal coordinates, can be given as
Fig. 3. Composite beam components connected by springs.
nd TeT 12_sTM _s;
U 12sTKs;
29
where T and U are kinetic and strain energy, respec-tively. M and K in Eq. (29) are
M
I 0 00 I 0 0 0 I
26664
37775;
K
x2A1 0 00 x2A2 0 0 0 x2AN
266664
377775:
30
The strain energy of the connectors, in terms of princi-
pal modal coordinates, is
UC 12sTwTKCws; 31
where KC is the stiness matrix of the cracked nodalelement and can be calculated by using Eq. (18). w in Eq.(31) can be written as
w wA1 0 00 wA2 0 0 0 wAN
2664
3775: 32
The total strain energy of the system is, therefore,
UT 12sTK wTKCws; 33
where K has been given by Eq. (30). The equation of
motion of the complete structure is
s K wTKCws wTf t; 34wherew has been given by Eq. (32), f t is the global forcevector for the system. From Eq. (34), the eigenvalues and
mode shapes of the cracked system can be determined.
After solving these equations, the displacements for eachAIAI-1 AI+1k I-1, I k I, I+1
M. Kisa / Composites Science acomponent are calculated by using Eq. (27).
4. Results and discussion
4.1. Validation of the current approach
In order to check the accuracy of the present method,
the case considered in [16] is adopted here. The beamassumed to be made of unidirectional graphite bre-re-
inforced polyamide. The geometrical characteristics and
material properties of the beam were chosen as the same
of those used in [16]. The material properties of the
graphite bre-reinforced polyamide composite, in termsof bres and matrix, identied by the indices f and m,
respectively, are
The geometrical characteristics, the length (L), height
(H) and width (B) of the composite beam, as consistentwith [16], were chosen as 0.6 m, 0.025 m and 0.05 m,
respectively.
Firstly, the presented method has been applied for the
free vibration analysis of a non-cracked composite
cantilever beam. The three lowest eigenfrequencies for
various values of the angle of the bre (a) and the vol-ume fraction of bres (V) are determined. As shown in
Fig. 4, the results found by using a four elements modelare compared with the analytical and numerical solu-
tions found in the literature [16,24]. The non-dimen-
sional natural frequencies are normalised according to
the following relation [16]:
-i LxiH=
S11=12q
qr; 35
where L and H show the length and height of the beam,
respectively. xi is the ith dimensional natural frequency.As can be seen from the gures, an excellent agreement
has been found between the results.
Secondly, the natural frequencies and mode shapes of
the cantilever composite beam having single open-edge
crack are analysed. The calculations have been carried
out for various volume fractions of the bres (V), thebre angles (a) and the crack ratios (a=H ). The naturalfrequencies of the cracked cantilever composite beam
are lower than those of the corresponding intact beam,
as expected. In Fig. 5, the changes in the rst natural
frequency of the cracked beam are given as a function of
the dierent crack ratios (a=H ) and the bre orientations(a) for several volume fractions (V). First non-dimen-sional natural frequencies are normalised according tothe following equation:
x xaxnca ; 36
where xa and xnca denote the natural frequency ofthe cracked and non-cracked cantilever composite beam
as a function of the angle of the bre a, respectively.As seen in Fig. 5, when the crack is perpendicular to the
Modulus of elasticity Em 2:756 GPa,Ef 275:6 GPa
Modulus of rigidity Gm 1:036 GPa,Gf 114:8 GPa
Poissons ratio mm 0.33, mf 0:2Mass density qm 1600 kg/m3,
qf 1900 kg/m3
chnology 64 (2004) 13911402 1395bre direction, the decrease in the rst natural frequency
nd Te4
6
8
en
sio
nal f
requ
en
cie
sV = 0.30
0
2
4
6
8
0 15 30 45 60 75 90
No
n-d
imen
sio
nal f
requ
en
cie
s
Present solutionKrawczukVinson & Sierakowski
Angle of fibre (deg)
1396 M. Kisa / Composites Science ais highest. As the angle of the bre increases, the changesin the rst frequency reduce. For the value of the angle
of bre is greater than 45 these changes are very lowand thus, the fundamental frequency of the cantilever
composite beam with cracks does not dier too much
from that of the corresponding non-cracked beam.
Fig. 5 shows that the reduction in the rst natural fre-
quency is higher for the volume fraction of the bres is
between 0.3 and 0.5.In Fig. 6, the variation of the rst natural frequencies
of the cracked cantilever composite beam is presented as
a function of relative crack positions (L1=L) and depths(a=H ). In the analysis, the volume and angle of brewere assumed to be 0.1 and 0, respectively. Non-di-mensional natural frequencies are normalised according
to Eq. (36). Due to the bending moment along the beam,
which is concentrated at the xed end, a crack near thefree end will have a smaller eect on the fundamental
frequency than a crack closer to the xed end, and as
seen from Fig. 6, it can be concluded that the frequen-
cies are almost unchanged when the crack is located
away from the xed end. These conclusions are in per-
fect agreement to those outlined by Krawczuk and
Ostachowicz [16].
V = 0.700
2
0 15 30
No
n-d
im
Angle of
Fig. 4. Non-dimensional natural frequencies of the intactPresent solutionKrawczukVinson & Sierakowski
V = 0.500
2
4
6
8
0 15 30 45 60 75 90
No
n-d
imen
sio
nal f
requ
encie
s
Present solutionKrawczukVinson & Sierakowski
Angle of fibre (deg)
chnology 64 (2004) 139114024.2. Vibration of composite beam with multiple cracks
After verication of the present method, the approach
is applied to a composite beam with multiple cracks as
shown in Fig. 1. The numerical illustrations were carried
out for a composite cantilever beam having the same
geometrical characteristics and material properties as the
ones supplied in [22]. The material properties of the
composite beam were the same as those used in the pre-vious case. The geometrical beam characteristics, consis-
tent with [22], were L 1 m, H 0:025 m, B 0:025 m.In Fig. 7, the variation of the rst three lowest natural
frequencies of the composite beam with multiple cracks
is shown as a function of bre orientation (a) for thedierent crack locations (Li=L). In this gure, three cases,labelled as E, F and G, were considered. In the model,
the number of the cracks assumed to be three. The cracklocations (L1=L, L2=L, L3=L) for the cases E, F and G,were chosen as, 0:05; 0:15; 0:25, 0:45; 0:55; 0:65,0:75; 0:85; 0:95, respectively. For numerical calcula-tions, the volume of the bre (V) and the crack ratio
(a=H ) were assumed to be 0.5 and 0.2, respectively. Thenon-dimensional natural frequencies, for the present and
subsequent cases, are normalised according to Eq. (36).
45 60 75 90 fibre (deg)
composite beam as a function of the bre angle a.
nd TeV = 0.101,05
M. Kisa / Composites Science aIt can be clearly seen from the Fig. 7 that, when the
cracks are placed near the xed end the decreases in
the rst natural frequency are highest, whereas, when
0,75
0,8
0,85
0,9
0,95
1
0 15 30 45 60 75 90
Angle of fibre (deg)
Rel
ativ
e fi
rst
freq
uenc
y
a/H=0.0a/H=0.2a/H=0.4a/H=0.6
V = 0.50
0,75
0,8
0,85
0,9
0,95
1
1,05
0 15 30 45 60 75 90
Angle of fibre (deg)
Rel
ativ
e fi
rst
freq
uenc
y
a/H=0.0a/H=0.2a/H=0.4a/H=0.6
Fig. 5. Changes in the rst natural frequency of the cracked composite
Fig. 6. Changes in the rst natural frequency for various relative crack
depth and location.V = 0.30
0,75
0,8
0,85
0,9
0,95
1
1,05
0 15 30 45 60 75 90
Angle of fibre (deg)
Rel
ativ
e fi
rst
freq
uenc
y
a/H=0.0a/H=0.2a/H=0.4a/H=0.6
V = 0.701,05
chnology 64 (2004) 13911402 1397the cracks are located near the free end, the rst natu-
ral frequencies are almost unaected. This observationgoes to the conclusion that, the rst, second and third
natural frequencies are most aected when the cracks
located at the near of the xed end, the middle of the
beam and the free end, respectively, Fig. 7. This con-
clusion is clearly seen from Fig. 8, which illustrates the
rst three natural bending mode shapes of the cracked
composite beam.
Fig. 9 shows the rst three natural frequencies as afunction of the bre orientation (a) for dierent crackratios (a=H ). In the model, the composite beam has fourcracks which were located as L1=L 0:05, L2=L 0:35,L3=L 0:65, L4=L 0:95, and the volume of bre (V)was 0.5. It is noticeable that decreases in the natural
frequencies become more intensive with the growth of
the crack depth. The most dierence in frequency occurs
when the angle of the bre (a) is 0. When the value ofthe angle of bre is greater than 45, the eects ofthe cracks on the frequencies decrease. This can be
0,75
0,8
0,85
0,9
0,95
1
0 15 30 45 60 75 90
Angle of fibre (deg)
Rel
ativ
e fi
rst
freq
uenc
y
a/H=0.0a/H=0.2a/H=0.4a/H=0.6
beam as a function of the angle of bre for various crack ratios.
00.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90
Angle of fibre (deg)
1st n
on-di
men
sio
nal
fre
quen
cies
E=0.05; 0.15; 0.25F=0.45; 0.55; 0.65G=0.75; 0.85; 0.95
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90
Angle of fibre (deg)
2nd
no
n-di
men
sio
nal
fre
quen
cies
E=0.05; 015; 0.25F=0.45; 0.55; 0.65G=0.75; 0.85; 0.95
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90
Angle of fibre (deg)
3rd
no
n-d
imen
sion
al fre
quen
cies
E=0.05; 0.15; 0.25F=0.45; 0.55; 0.65G=0.75; 0.85; 0.95
Fig. 7. Variation of the rst three non-dimensional natural frequencies as a function of bre orientation for the case of three cracks located dif-
ferently, as indicated a=H 0:2 and V 0:5.
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
x (m)
1st n
atu
ral b
endi
ng
mo
de
IntactE=0.05; 0.15; 0.25F=0.45; 0.55; 0.65G=0.75; 0.85; 0.95
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
x (m)
2nd
natu
ral b
endi
ng
mo
de Intact
E=0.05; 0.15; 0.25F=0.45; 0.55; 0.65G=0.75; 0.85; 0.95
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
x (m)
3rd
natu
ralb
en
din
g m
ode
IntactE=0.05; 0.15; 0.25F=0.45; 0.55; 0.65G=0.75; 0.85; 0.95
Fig. 8. Normalised rst three mode shapes that correspond to the cases in Fig. 7.
1398 M. Kisa / Composites Science and Technology 64 (2004) 13911402
00.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90Angle of fibre (deg)
1st n
on-di
men
sio
nal
frequ
en
cie
s
a/H=0.2a/H=0.4a/H=0.6
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90
Angle of fibre (deg)
2nd
non-
dim
en
sio
nal
frequ
en
cie
s
a/H=0.2a/H=0.4a/H=0.6
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90
Angle of fibre (deg)
3rd
non
-dim
ensi
on
al fre
quen
cies
a/H=0.2a/H=0.4a/H=0.6
Fig. 9. First three non-dimensional natural frequencies as a function of bre orientation for dierent crack ratios a=H 0:2, 0.4, 0.6, volume of thebre V is 0.5.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
1st n
on-di
men
sio
nal
fre
quen
cy
a/H=0.2a/H=0.4a/H=0.6
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Volume of fibre (V)Volume of fibre (V)
Volume of fibre (V)
2nd
non-
dim
ensi
on
al fre
quen
cy
a/H=0.2a/H=0.4a/H=0.6
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
3rd
non-
dim
ensi
on
al fre
quen
cy
a/H=0.2a/H=0.4a/H=0.6
Fig. 10. First three non-dimensional natural frequencies as a function of the bre volume fraction for dierent crack ratios a=H 0:2, 0.4, 0.6, angleof the bre a is 0.
M. Kisa / Composites Science and Technology 64 (2004) 13911402 1399
00.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90
Angle of fibre (deg)
1st n
on-d
imen
sio
nal
frequ
ency
E1F1G1
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90
Angle of fibre (deg)
2nd
no
n-di
men
sio
nal
fre
quen
cy
E1F1G1
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50 60 70 80 90
Angle of fibre (deg)
3rd
non-
dim
en
sio
nal
frequ
en
cy
E1F1G1
Fig. 11. Variation of the rst three non-dimensional natural frequencies as a function of bre orientation for the case of three cracks featuring various
depths. For the cases E1, F1 and G1, the crack locations and ratios Li=L ai=H are; 0:15 0:2; 0:35 0:4; 0:55 0:6, 0:15 0:2; 0:35 0:6;0:55 0:4, 0:15 0:6; 0:35 0:4; 0:55 0:2, respectively, as indicated V 0:5.
-1
-0.8
-0.6
-0.4
-0.2
0
0 0.2 0.4 0.6 0.8 1
x (m)
1st n
atu
ral b
endi
ng
mo
de
IntactE1F1G1
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8
x (m)
2nd
natu
ral b
endi
ng
mo
de
IntactE1F1G1
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
x (m)
3rd
natu
ral b
endi
ng
mo
de IntactE1F1G1
1
Fig. 12. Normalised rst three mode shapes that correspond to the cases in Fig. 11, angle of the bre a is 0.
1400 M. Kisa / Composites Science and Technology 64 (2004) 13911402
of the bre is between 0.2 and 0.8 and the crack ratio is
getting higher, the frequency reductions are relatively
E22 Em Ef Em Ef EmV
;
nd Tehigh, Fig. 10.
In all previous cases, a unique crack depth was con-
sidered for the all cracks. Figs. 11 and 12 show the eect
of the dierent crack ratios and the values of the angle of
the bre on the natural frequencies and mode shapes of acomposite beam having three cracks. In the analysis
three cases, shown as E1, F1 and G1, were considered.
The crack locations and ratios (L1=L a1=H;L2=L a2=H; L3=L a3=H) for the cases E1, F1 and G1,were chosen as, 0:15 0:2; 0:35 0:4; 0:55 0:6, 0:150:2; 0:35 0:6; 0:55 0:4, 0:15 0:6; 0:35 0:4; 0:550:2, respectively. In the natural frequency analysis,the volume of the bre (V) was assumed to be 0.5 whilein the natural mode shape calculations the volume (V)
and angle (a) of the bre were 0.5 and 0, respectively.From these gures, it is apparent that the maximum
fundamental frequency drop occurs when the large
cracks are located closer to the xed end, while the
second and third natural frequencies and mode shapes
exhibit higher changes when the large cracks are situ-
ated closer to the middle of the beam and at a distanceequal to 33% of the beam length from the xed end,
respectively.
5. Concluding remarks
This paper has presented a new method for the nu-
merical modelling of the free vibration of a cantilevercomposite beam having multiple open and non-propa-
gating cracks. The method integrates the fracture me-
chanics and the joint interface mechanics to couple
substructures. In the methods observed in the literature
compromises have been made either in the representa-
tion of the physics of the nonlinearities in defective
structures or in the complexity of the structure which
can be analysed. For example, nite element studiesusually use a simplistic representation of the interface
mechanics whereas analytical studies require simpleexplained as the exibility due to crack is negligible
when the angle of the bre is greater then 45 [16], es-pecially when the crack ratio is relatively low.
Fig. 10 presents the rst three natural frequencies as a
function of the volume of the bre (V) for the severalvalues of the crack ratios (a=H ). As the previous case,the composite beam has four cracks which were located
as L1=L 0:05, L2=L 0:35, L3=L 0:65, L4=L 0:95and the angle of the bre (a) was 0. As can be seen fromthe gure, the natural frequencies are aected by the
values of the volume of the bre (V) and the crack ratios
(a=H ), as expected. The exibility due to cracks is highwhen the volume of the bre is between 0.2 and 0.8, andmaximum when V 0:45 [16]. Therefore, if the volume
M. Kisa / Composites Science aboundary conditions. Neither is satisfactory in practicalEf Em Ef EmV
m12 mfV mm1 V ;
m23 mfV mm1 V 1 mm m12Em=E111 m2m mmm12Em=E11
;
G12 Gm Gf Gm Gf GmVGf Gm Gf GmV
; G23 E2221 m23 ;
where indices m and f denote matrix and bre, respec-design studies. It is believed that the current approach
addresses both of these issues leading to the develop-
ment of design tools. Consequently, the present ap-
proach can be used for the analysis of non-linear
interface eects such as contact that occurs when thecracks close. To illustrate the eectiveness of the ap-
proach, results have been compared with previous
studies published in the literature leading to condence
in the validity of this approach.
Appendix A
The roots of following characteristic equation give
the complex constants s1 and s2 [24]:
b11s4 2b16s3 2b12 b66s2 2b26s b22 0;where bij constants are
b11 b11m4 2b12 b66m2n2 b22n4;b22 b11n4 2b12 b66m2n2 b22m4;b12 b11 b22 b66m2n2 b12m4 n4;b16 2b11 2b12 b66m3n 2b22 2b12 b66mn3;b26 2b11 2b12 b66n3m b22 2b12 b66nm3;b66 22b11 4b12 2b22 b66m2n2 b66m4 n4;where m cos a, n sin a and bij are compliance con-stants of the composite along the principal axes. bij canbe related to the mechanical constants of the material by
b11 1E11 1 m212
E22E11
; b22 1E22 1 m
223;
b12 m12E11 1 m23; b66 1=G12; b44 1=G23;b55 b66;where E11, E22, G12, G23, m12, m23 and q are the mechanicalproperties of the composite and calculated using the
following formulae [24]:
q qfV qm1 V ; E11 EfV Em1 V ;
chnology 64 (2004) 13911402 1401tively. E, G, m and q are the modulus of elasticity, the
modulus of rigidity, the Poissons ratio and the massdensity, respectively.
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Free vibration analysis of a cantilever composite beam with multiple cracksIntroductionMathematical modelStiffness and mass matrices for composite beam elementThe stiffness matrix for the crack
Component mode analysisCoupling of the components
Results and discussionValidation of the current approachVibration of composite beam with multiple cracks
Concluding remarksAppendix AReferences
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