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Exact free vibration study of rectangular Mindlin plates with all-over
part-through open cracks
Sh. Hosseini-Hashemi a,*, Heydar Roohi Gh. a, Hossein Rokni D.T. b
a Impact Research Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iranb School of Engineering, University of British Columbia Okanagan, Kelowna, BC, Canada V1V 1V7
a r t i c l e i n f o
Article history:
Received 21 January 2010
Accepted 11 June 2010
Available online 13 July 2010
Keywords:
Free vibration
Exact solution
Rectangular plate
Part-through crack
Mindlin theory
a b s t r a c t
Based on Mindlin plate theory (MPT), a set of exact closed-form characteristic equations incorporating
shear deformation and rotary inertia are proposed for the first time to analyze free vibration problem
of moderately thick rectangular plates with an arbitrary number of all-over part-through cracks. The pro-
posed rectangular plates have two opposite edges simply supported while six possible combinations of
free, simply supported and clamped boundary conditions are taken into account for two other edges.
The crack is assumed to be open, non-propagating and perpendicular to two opposite simply supported
edges. A continuously distributed line-spring model is used to describe the elastic behavior of an all-over
part-through crack. The accuracy of the current approach is investigated through comparing the present
exact natural frequencies with those of 3D finite element method obtained by ABAQUS software package.
A parametric study is undertaken to show the effect of crack depth, crack location, number of cracks and
thickness-to-length ratio on natural frequencies of rectangular moderately thick plates with different
boundary conditions in tabular and graphical forms. Finally, the effect of shearing and tearing modes
on the modeling of cracks located at the nodal line is shown.
2010 Elsevier Ltd. All rights reserved.
1. Introduction
Structures consisting of plates with different shapes, sizes,
thickness variations and boundary conditions are widely observed
in aerospace, civil, optical, electronic, automotive, mechanical, and
shipbuilding industries. Owing to their wide practical applications,
it is very likely that cracks, which are one of the most common
damages, appear in the vibrating plates. It is well known that the
crack can lead to changes in dynamic and stability characteristics
of plates. Therefore, examining the effect of crack damage on trans-
verse free vibration of thin and moderately thick plates is crucial
for engineers and designers.
Depending on the value of plate thickness, two main theoriesmay be considered for modeling a rectangular plate containing
an open crack. The classical plate theory, referred to as Kirchhoffs
theory [1], must be employed for thin plates due to ignoring the ef-
fect of shear deformation through the plate thickness. A consider-
able research work pertaining to the free vibration of thin cracked
plates has been performed. An extensive literature survey of the
vibration of cracked structures, reported until 1996, can be found
in the work of Dimarogonas[2].
Early research effort for free vibration analysis of thin rectangu-
lar plates with cracks dates back to the work of Lynn and Kumbasar
[3], who used Greens functions to represent the transverse dis-
placements of simply supported rectangular plates, resulting in
Fredholmintegral equations of the first kind. Stahl and Keer [4] for-
mulated freely vibrating thin simply supported rectangular plates
as dual series equations, reducing homogeneous Fredholm integral
equations from the first kind to the second one. Hirano and
Okazaki [5] utilized Levys form of solution and further matched
the boundary conditions using a weighted residual method to
investigate free vibrations of cracked rectangular plates with two
opposite edges simply supported and the remaining sides free or
clamped. Aggarwala and Ariel[6]determined the natural frequen-cies of simply supported cracked plates by using the homogeneous
Fredholm integral of the second kind by taking the stress singular-
ity at the crack tips into account. Neku [7]analyzed free vibration
of a simply supported rectangular plate with straight through cen-
tral or side cracks by means of finite Fourier transformation. The
flexural vibration of a simply supported rectangular plate with
arbitrarily located rectilinear cracks were studied by Solecki [8]
using the finite Fourier transformation of discontinuous functions.
At the beginning of 21st century, Khadem and Rezaee [9] devel-
oped an analytical approach for crack detection in rectangular thin
plates with an all-over part-through crack and subjected to uni-
form external loads using vibration analysis.
0045-7949/$ - see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.compstruc.2010.06.004
* Corresponding author. Tel.: +98 2177 240 190; fax: +98 2177 24 0488.
E-mail addresses: [email protected] (Sh. Hosseini-Hashemi), [email protected]
(Hossein Rokni D.T.).
Computers and Structures 88 (2010) 10151032
Contents lists available at ScienceDirect
Computers and Structures
j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m p s t r u c
http://dx.doi.org/10.1016/j.compstruc.2010.06.004mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruc.2010.06.004http://www.sciencedirect.com/science/journal/00457949http://www.elsevier.com/locate/compstruchttp://www.elsevier.com/locate/compstruchttp://www.sciencedirect.com/science/journal/00457949http://dx.doi.org/10.1016/j.compstruc.2010.06.004mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.compstruc.2010.06.004 -
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The dynamic characteristics of centrally located cracked plates
in tension were analyzed by Petyt[10]by the use of the finite ele-
ment (FE) displacement method. A finite element method (FEM) of
cracked plates was employed by Qian et al. [11]to investigate the
vibration analysis of simply supported and cantilevered square
thin plates having a through-thickness crack, parallel to one side
of the plate, using the integral of stress intensity factor. Prabhakara
and Datta [12] examined the static stability and vibrations of dam-
aged rectangular thin plates. Krawczuk[13]presented a FE model
based on the stiffness matrix to investigate the influence of the
crack location and its length on the changes of the natural frequen-
cies of the simply supported and cantilever rectangular thin plates.
Later, Krawczuk and Ostachowicz[14]used the same approach as
Krawczuk [13]did in order to investigate the influence of the crack
length and its location upon the amplitudes of the transverse
forced vibration for aluminum cantilevered cracked plates. Han
and Ren[15]analyzed transverse vibration of cracked rectangular
plates by means of a model of zero dimension elements with crack.
Su et al.[16]extended the two-level FEM to analyze the free vibra-
tion of a thin cracked plate with arbitrary boundary conditions.
Krawczuk et al.[17]studied elasto-plastic FE model of simply sup-
ported and cantilever square plates with a through crack. Fujimoto
et al.[18]reported a vibration analysis of rectangular plates sub-
jected to a tensile load and containing a centrally located crack
along the line of symmetry perpendicular to the direction of the
tensile load using the FEM. Very recently, Saito et al. [19]investi-
gated the linear and nonlinear vibration response of a cantilevered
rectangular plate with a transverse crack using a FE model.
By decomposing a rectangular plate into various domains and
introducing artificial springs at the interconnecting boundaries be-
tween the domains, Yuan and Dickinson [20] used the Rayleigh
Ritz method with regular admissible functions to find the solutions
for the flexural vibration of rectangular plates. Lee[21]employed
the Rayleigh method to obtain the fundamental frequency of annu-
lar plates with internal concentric cracks. Lee and Lim [22] used
the RayleighRitz approach for the study of the free vibration of
rectangular plates with a centrally located crack by consideringtransverse shear deformation and rotary inertia. The decomposi-
tion method was used by Liewet al. [23] to determine the vibration
frequencies of cracked plates with any combination of boundary
conditions. They assumed the cracked plate domain to be an
assemblage of small subdomains with the appropriate functions
formed and led to a governing eigenvalue equation. Ramamurti
and Neogy [24] applied the generalized RayleighRitz method to
determine the natural frequency of cracked cantilevered square
plates. Khadem and Rezaee[25]employed the modified compari-
son functions to obtain the natural frequencies of a simply sup-
ported rectangular cracked plate using the RayleighRitz method.
Very recently, free vibrations of simply supported and completely
free rectangular thin plates with a side crack were studied by
Huang and Leissa[26]using the famous Ritz method with special
displacement functions.
By applying the Galerkins method to von Karmans plate the-
ory, Wu and Shih [27] determined natural frequencies of simply
supported square and rectangular cracked plates subjected to the
pulsating in-plane forces by using incremental harmonic balance
method. Hu and Fu[28]introduced linear free vibration of visco-
elastic simply supported rectangular plates with a crack using
the Galerkins method. Transverse free vibration of viscoelastic
rectangular thin plate with linearly varying thickness and multiple
all-over part-through cracks was studied by Wang and Wang[29]
using the differential quadrature method. Very recently, a new
analytical model was presented by Israr et al.[30]for the vibration
analysis of cracked rectangular plates subjected to transverse load-
ing at some specified position with different sets of boundary con-
ditions using Galerkins method.
Adams and Cawley[31]developed an experimental technique
to estimate the location and depth of a crack from changes in nat-
ural frequencies of a rectangular plate. Maruyama and Ichinomiya
[32]obtained experimentally the natural frequencies for rectangu-
lar plates with straight narrow slits, which the effects of lengths,
positions and inclination angles of slits on natural frequency and
mode shape are discussed. Variations of natural frequencies and
associated mode shapes with respect to changes in crack length
for a square plate with an edge crack were reported by Ma and
Huang[33]using both experiments and FEM. A baseline finite ele-
ment model along with modal test data were utilized by Wu and
Law [34]to obtain natural frequencies of a completely free rectan-
gular Aluminum thin plate with an inclined through crack.In order to eliminate the deficiency of the classical plate theory
for moderately thick plates, the first-order shear deformation the-
ory was proposed by Reissner [35], and developed further for the
deformable plates in statics and dynamics by Mindlin et al. [36].
Although considerable papers exist for vibration analysisof Mindlin
Fig. 1. Strip with edge crack under bending moment.
Table 1
Values for coefficient Cnbb
.
n Cnbb
0 1.9710
1 4.42772 34.4952
3 165.73214 626.3926
5 2144.46516 7043.4169
7 19003.21998 37853.3028
9 52595.468110 48079.2948
11
25980.1559
12 6334.2425Fig. 3. Geometry and dimension of a rectangular cracked plate.
Fig. 2. A Mindlin plate with coordinate convention.
1016 Sh. Hosseini-Hashemi et al. / Computers and Structures 88 (2010) 10151032
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rectangular plates [3740], according to the authors knowledge,
only one publication has been devoted to the free vibration analy-
sis of their cracked counterparts. Bachene et al. [41]adopted the
extended finite element method (X-FEM) to analyze free vibrations
of rectangular plates containing through edge and central cracks
based on the MPT. The formulation was first implemented numer-
ically using two types of quadrilateral elements. Then, they devel-
oped a FORTRAN computer code on the basis of the X-FEM
formulation and consequently calculated natural frequencies of
the cracked plates.
The line-spring model, adopted in this study to solve part-
through crack plates, has been widely used in the fracture mechan-
ics analysis of plate components containing surface or internal
cracks. The line-spring model was first proposed by Rice and Levy
[42] to give an approximate treatment for three-dimensional prob-
lem of a surface crack penetrating partly through the thickness of
an elastic plate. The problem was simplified to a two-dimensional
through crack problem in plate theory, in which the constraint ef-
fects of net ligaments from the three-dimensional problem were
incorporated in the form of a membrane load and bending moment
imposed on the through crack. Delale and Edrogan[43]reformu-
lated the line-spring model in the context of the Reissner plate the-
ory to include transverse shear effects.
The aim of this paper is to extract exact closed-form character-
istic equations for rectangular plats with all-over part-through
cracks based on the Mindlin plate theory. The plate has a pair of
opposite simply supported edges, called the levy plate [44], while
the other two edges may be taken any combinations of free, simply
supported andclamped boundary conditions. It is assumed that the
crack is open, non-propagating and perpendicular to two opposite
simply supported edges. The local flexibility, induced by an all-over
part-through crack in the plate, is modeledby the mode (I) fracture,
which is generated by the bending moment. Using the well-known
commercial software ABAQUS [45], 3D finite element method is
Fig. 4. A meshed rectangular plate with an all-over part-through crack whena= 0.5,f = 0.5,g = 2/3 andd = 0.1. Total number of elements: 32306; total number of singularelement: 1472; total number of nodes 141347.
Table 2
First seven frequency parameter (b) for SFSF cracked rectangular thin plate when g = 2/3.
f a Methods Frequency parametersmn
d= 0.05
0 0.5 Present 9.646311 12.841612 22.535513 38.395521 39.356814 41.761522 52.693823
3D FEM 9.646411 12.840612 22.533313 38.400121 39.356614 41.763722 52.694523
% Diff 0.00 0.01 0.01 0.01 0.00 0.01 0.000.5 0.25 Present 9.626511 12.797012 22.430013 37.381314 38.288521 41.630622 52.665723
3D FEM 9.626811 12.760412 22.304413 37.389114 38.302921 41.605722 52.460723
% Diff 0.00 0.29 0.56 0.02 0.04 0.06 0.390.5 0.5 Present 9.619911 12.841612 21.665513 38.286921 39.356814 41.761522 51.829023
3D FEM 9.621511 12.791712 21.708413 38.386921 39.127814 41.701222 51.928923
% Diff 0.02 0.39 0.20 0.26 0.58 0.14 0.19
d= 0.1
0 0.5 Present 9.515711 12.571112 21.749013 36.594121 37.301914 39.588622 49.291223
3D FEM 9.516511 12.569012 21.746313 36.609521 37.307614 39.597022 49.303623
% Diff 0.01 0.02 0.01 0.04 0.02 0.02 0.030.5 0.25 Present 9.4849 12.506412 21.556913 34.332814 36.446921 39.420722 49.242423
3D FEM 9.4860 12.438912 21.331613 34.239914 36.481921 39.384622 48.899423
% Diff 0.01 0.54 1.05 0.27 0.10 0.09 0.700.5 0.5 Present 9.4740 12.571112 20.442313 36.443221 37.301914 39.588622 48.147023
3D FEM 9.4786 12.477612 20.530413 36.483621 37.017914 39.495822 48.330623
% Diff 0.05 0.74 0.43 0.11 0.76 0.23 0.38
d= 0.2
0 0.5 Present 9.066611 11.750512 19.556113 31.578221 31.942714 33.788222 40.887023
3D FEM 9.069011 11.747412 19.552013 31.601521 31.957414 33.810222 40.917923
% Diff 0.03 0.03 0.02 0.07 0.05 0.07 0.080.5 0.25 Present 9.028711 11.681912 19.213213 28.391214 31.433021 33.641022 40.802423
3D FEM 9.032011 11.561812 18.832413 28.187614 31.496521 33.596122 40.314523
% Diff 0.04 1.03 1.98 0.72 0.20 0.13 1.200.5 0.5 Present 9.012111 11.750512 17.957613 31.419021 31.942714 33.788222 39.758623
3D FEM 9.020511 11.589712 18.030613 31.494021 31.591214 33.652622 39.962923
% Diff 0.09 1.37 0.41 0.24 1.10 0.40 0.51
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adopted to show the merit of the present analytical approach. The
influence of crack depth, crack location, and thickness-to-length ra-
tio on frequency parameters of cracked rectangular plates is inves-
tigated for different boundary conditions. In addition, the effect of
second and third modes of the crack on natural frequencies of the
plate is discussed. Finally, fundamental frequency parameters of
rectangular plates with an arbitrary number of cracks are given
for six possible combinations of boundary conditions.
2. Governing equations of cracked Mindlin plate
2.1. Modeling of crack in vibrating plates
Consider an edge-cracked elastic strip subjected to a bending
momentMb, as shown inFig. 1. The stress intensity factor of sucha loaded cracked strip is given by[42]:
KI H12rbgb; 1
where His the strip thickness andrbis the nominal stress at a pointfar from the crack tip which is imposed by the bending moment
(rb= 6Mb/H2). Functiongb can be expressed in dimensionless form
as[46]:
where f is the ratio of crack depth to strip thickness (f= h/H). Eq. (2)
is valid when the parameter f changes within the range of
0 < f< 0.8. Also, the relationship between stress intensity factor
(KI) and the strain energy release rate (G), in a state of plane strain,
is expressed as
G 1 m2
E K2I; 3
where m and E are Poissons ratio and Youngs modulus, respec-
tively. By substituting Eq.(1)into Eq.(3), the strain energy releaserate is derived as
Table 3
First seven frequency parameter (b) for SSSF cracked rectangular thin plate when g = 2/3.
f a Methods Frequency parametersmn
d= 0.05
0 0.5 Present 10.602811 18.079412 33.064613 39.376221 47.294222 56.005914 62.727823
3D FEM 10.602811 18.079612 33.065613 39.379721 47.299122 56.013714 62.737823
% Diff 0.00 0.00 0.00 0.01 0.01 0.01 0.020.5 0.25 Present 10.589411 17.616112 31.577513 39.342721 46.770022 55.131014 61.481523
3D FEM 10.575011 17.620312 31.715513 39.335221 46.810122 55.057114 61.588623
% Diff 0.14 0.02 0.44 0.02 0.09 0.13 0.170.5 0.5 Present 10.559811 17.672712 32.644313 39.256421 46.896722 53.371714 62.247523
3D FEM 10.551911
17.668012
32.566713
39.264421
46.897422
53.456814
62.118823
% Diff 0.07 0.03 0.24 0.02 0.00 0.16 0.210.5 0.75 Present 10.549511 18.078912 31.888713 39.171121 47.277122 52.575714 62.105723
3D FEM 10.543211 17.969312 31.835713 39.188721 47.141022 52.633614 61.978823
% Diff 0.06 0.61 0.17 0.04 0.29 0.11 0.20
d= 0.1
0 0.5 Present 10.440411 17.603312 31.634513 37.480621 44.603322 52.323114 58.165223
3D FEM 10.440911 17.603812 31.643013 37.496221 44.625322 52.357814 58.202723
% Diff 0.00 0.00 0.03 0.04 0.05 0.07 0.060.5 0.25 Present 10.418911 16.878312 29.532013 37.433721 43.883322 51.173914 56.574123
3D FEM 10.392411 16.898312 29.761213 37.431121 43.974122 50.952614 56.757123
% Diff 0.25 0.12 0.78 0.01 0.21 0.43 0.320.5 0.5 Present 10.372511 16.968712 31.029413 37.314321 44.052922 48.806514 57.558723
3D FEM 10.360311 16.969912 30.881613 37.339721 44.075422 48.777214 57.355623
% Diff 0.12 0.01 0.48 0.07 0.05 0.06 0.350.5 0.75 Present 10.359311 17.600412 29.745213 37.204121 44.587022 47.799414 57.255723
3D FEM 10.360311 17.418112 29.616313 37.243421 44.364322 47.783914 57.012523
% Diff 0.01 1.04 0.43 0.11 0.50 0.03 0.42
d= 0.2
0 0.5 Present 9.897211 16.154612 27.647613 32.248521 37.575822 43.181314 47.283923
3D FEM 9.899311 16.161112 27.681313 32.294221 37.641022 43.270914 47.388823
% Diff 0.02 0.04 0.12 0.14 0.17 0.21 0.220.5 0.25 Present 9.867811 15.223912 25.371713 32.198621 36.833422 42.089214 45.806423
3D FEM 9.822711 15.243612 25.639013 32.219321 36.955322 41.741514 45.960523
% Diff 0.46 0.13 1.05 0.06 0.33 0.83 0.340.5 0.5 Present 9.808311 15.332012 26.980313 31.936721 36.998122 39.829514 46.734023
3D FEM 9.788911 15.318912 26.719713 31.971221 37.043022 39.955514 46.429823
% Diff 0.20 0.09 0.97 0.11 0.12 0.32 0.650.5 0.75 Present 9.800311 16.135812 25.117013 31.983521 37.571022 38.739914 46.202023
3D FEM 9.781111 15.870812 24.869813 32.042721 37.273522 38.701214 45.725923
% Diff 0.20 1.64 0.98 0.19 0.79 0.10 1.03
gbf ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffipf1:1202 1:8872f 18:0143f2 87:3851f3 241:9124f4 319:9402f5 168:0105f6
q ; 2
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G H1 m2
E r2bg
2b : 4
One may consider two identical elastic strips, one of which has
an edge crack, as shown in Fig. 1. If both strips are subjected to
equal moment, the presence of the crack will cause the rotation
of one end relative to the other to increase more than the
rotation of an uncracked strip. Accordingly, one can write G as
the sum of potential energy release rates due to bending momentas[42]:
G 12 rb
o
oh
H2h
6
!" #: 5
By comparing Eqs.(4) and (5), one can obtain the added slope
(h) as[42]:
h 121 m2
E rbabb; 6
whereabb is the non-dimensional bending compliance factor givenby[46]:
abb
f2 X12
n0C
nbbf
n;
7
and the values ofC
nbb are listed inTable 1.
As an approximation, one may assume that the stress intensity
factor at a point along the crack front of a plate with an all-over
part-through constant depth crack, is identical to the stress inten-
sity factor for an edge-cracked strip in-plane strain state with the
same loading conditions and crack depth[42].
2.2. Governing equations of Mindlin plate
Consider a moderately thick rectangular plate of width a, length
b, and uniform thickness H, oriented so that its undeformed middle
surface contains thex1 andx2 axes of a Cartesian coordinate system
(x1,x2, x3), as shown inFig. 2.
The displacements along the x1 and x2 axes are denoted by U1andU2, respectively, while the displacement in the direction per-
pendicular to the undeformed middle surface is denoted by U3.
In the Mindlin plate theory, the displacement components are as-
sumed to be given by
U1 x3w1 x1;x2; t ;U2 x3w2 x1;x2; t ;U3 w3 x1;x2; t ;
8ac
wheretis the time variable, w1 andw2 are the rotational displace-ments about the x2 and x1 axes at the middle surface of the plate,
Table 4
First seven frequency parameter (b) for SFSC cracked rectangular thin plate when g = 2/3.
f a Methods Frequency parametersmn
d= 0.05
0 0.5 Present 10.895111 20.025712 37.060313 39.501821 48.484122 61.992414 65.519523
3D FEM 10.896011 20.034012 37.086213 39.506521 48.497722 62.048014 65.551623
% Diff 0.01 0.04 0.07 0.01 0.03 0.09 0.050.5 0.25 Present 10.834511 20.002212 35.295913 39.287321 48.478822 58.500814 64.591223
3D FEM 10.824911 19.877512 35.303113 39.304021 48.319922 58.578914 64.535723
% Diff 0.09 0.62 0.02 0.04 0.33 0.13 0.090.5 0.5 Present 10.860811 19.365912 36.985813 39.387821 47.929722 58.330814 65.286323
3D FEM 10.847411
19.403412
36.870513
39.392921
47.969222
58.546014
65.067523
% Diff 0.12 0.19 0.31 0.01 0.08 0.37 0.340.5 0.75 Present 10.895011 19.830012 35.399113 39.483421 48.070522 59.537114 63.977423
3D FEM 10.847411 19.815012 35.568213 39.471921 48.082822 59.688614 64.190023
% Diff 0.44 0.08 0.48 0.03 0.03 0.25 0.33
d= 0.1
0 0.5 Present 10.709911 19.349812 35.019213 37.579321 46.530222 56.975414 60.229323
3D FEM 10.711811 19.363812 35.063513 37.595821 46.551622 57.074914 60.297823
% Diff 0.02 0.07 0.13 0.04 0.05 0.17 0.110.5 0.25 Present 10.618711 19.303612 32.373813 37.292421 45.525922 52.549714 58.974923
3D FEM 10.602311 19.062612 32.318913 37.330721 45.268222 52.200614 58.882723
% Diff 0.15 1.25 0.17 0.10 0.57 0.66 0.160.5 0.5 Present 10.655911 18.368612 34.889413 37.420321 44.800622 52.326414 59.900923
3D FEM 10.633611 18.446012 34.675813 37.440221 44.879622 52.454114 59.567623
% Diff 0.21 0.42 0.61 0.05 0.18 0.24 0.560.5 0.75 Present 10.709811 19.040212 32.701413 37.552621 44.952122 54.169614 58.327123
3D FEM 10.683011 19.031312 33.017813 37.545621 45.010522 54.030614 58.681023
% Diff 0.25 0.05 0.97 0.02 0.13 0.26 0.61
d= 0.2
0 0.5 Present 10.106011 17.388812 29.670713 32.300321 38.044222 45.411014 48.207023
3D FEM 10.110011 17.412012 29.745913 32.346521 38.121022 45.581914 48.343823
% Diff 0.04 0.13 0.25 0.14 0.20 0.38 0.280.5 0.25 Present 10.000511 17.300212 26.560613 32.028921 38.043722 41.229114 46.910023
3D FEM 9.972111 16.969112 26.382213 32.086321 37.710622 40.737614 46.709723
% Diff 0.28 1.91 0.67 0.18 0.88 1.19 0.430.5 0.5 Present 10.036011 16.248812 29.428213 32.131821 37.349722 41.368314 47.829523
3D FEM 10.003411 16.322012 29.023513 32.181821 37.439822 41.493714 47.430023
% Diff 0.32 0.45 1.38 0.16 0.24 0.30 0.840.5 0.75 Present 10.106011 16.984712 27.377613 32.270121 37.444822 43.518414 46.618723
3D FEM 10.063111 16.980612 27.785013 32.288421 37.560422 43.252114 47.014323
% Diff 0.42 0.02 1.49 0.06 0.31 0.61 0.85
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respectively, andw3is the transverse displacement. For a harmonicsolution, the rotational and transverse displacements are assumed
to be:
w1 x1;x2; t ~w1 X1;X2 eixt;w2 x1;x2; t ~w2 X1;X2 eixt;
w3 x1;
x2;
t 1
a~
w3 X1;
X2 eixt
;
9ac
wherex denotes the natural frequency of vibration in radians andi
ffiffiffiffiffiffiffi1
p . It should be noted that each parameter having the over-
tilde is non-dimensional. Introducing the dimensionless
parameters:
X1x1a
; X2x2b
; d Ha
; g ab
; b xa2ffiffiffiffiffiffiffiqHD
r ; 10ad
governing equations of motion in dimensionless form can be ex-
pressed as[37]:
~w1;11 g2~w1;22 m2m1~w1;11 g~w2;12
12K2
d2
~w1 ~w3;1
b2d2
12m1~w1;
~w2;11 g2~w2;22 m2m1 g ~w1;12 g~w2;22
12K2
d2~w2 g~w3;2
b
2d2
12m1~w2;
~w3;11 g2~w3;22 ~w1;1 g~w2;2
b2d2
12K2m1~w3:
11ac
where comma-subscript convention represents the partial differen-
tiation with respect to the normalized coordinates, b is the fre-
quency parameter, m1= (1 m)/2, m2= ( 1 + m)/2, K2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:86667
p is
the shear correction factor to account for the fact that the
transverse shear strains are not truly independent of the thickness
coordinate. Finally, The general solutions to Eq. (11), in terms of
the three dimensionless potentials W1, W2 and W3, may be ex-
pressed as
~w1 C1W1;1 C2W2;1 gW3;2;~w2 C1gW1;2 C2gW2;2 W3;1;~w3 W1 W2;
12ac
where
C1 1 a22
m1a23; C2 1 a
21
m1a23;
a21;a22
b2
2
d2
12
1
K2m1 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2
12
!21
K2m1 1
2 4b2
vuut264
375;
a23 12K2
b2d
2a21a
22 12K
2
d2
b2
d4
144K2m1 1 !:
13
Based on these dimensionless potentials, the governing equa-
tions of motion may now be given by
W1;11 g2W1;22 a21W1;W2;11 g2W2;22 a22W2;W3;11 g2W3;22 a23W3:
14ac
Table 5
First seven frequency parameter (b) for SSSS cracked rectangular thin plate when g = 2/3.
f a Methods Frequency parametersmn
d= 0.05
0 0.5 Present 14.166211 27.086612 43.033821 48.300613 55.634122 76.336023 77.361614
3D FEM 14.167011 27.089012 43.041421 48.310013 55.648222 76.360523 77.387914
% Diff 0.01 0.01 0.02 0.02 0.03 0.03 0.030.5 0.25 Present 13.991611 25.821212 42.802121 46.969813 54.567522 75.261123 77.361614
3D FEM 13.969111 25.883812 42.797721 46.955013 54.694922 75.203123 77.185414
% Diff 0.16 0.24 0.01 0.03 0.23 0.08 0.230.5 0.5 Present 13.837311 27.086612 42.593921 45.484513 55.634122 74.082823 77.361614
3D FEM 13.860111 26.957312 42.652821 45.649413 55.406222 74.351823 77.187014
% Diff 0.16 0.48 0.14 0.36 0.41 0.36 0.23
d= 0.1
0 0.5 Present 13.908511 26.180312 40.846721 45.584613 52.100122 70.021923 70.892914
3D FEM 13.911611 26.191712 40.874721 45.619713 52.146222 70.104823 70.978314
% Diff 0.02 0.04 0.07 0.08 0.09 0.12 0.120.5 0.25 Present 13.624611 24.315712 40.516621 43.822413 50.688822 68.704423 70.892914
3D FEM 13.598111 24.340712 40.530621 43.749413 50.844622 68.609923 70.550214
% Diff 0.19 0.10 0.03 0.17 0.31 0.14 0.480.5 0.5 Present 13.393811 26.180312 40.236821 41.736513 52.100122 67.195423 70.892914
3D FEM 13.437511 25.938912 40.362621 41.954713 51.733822 67.560323 70.560914
% Diff 0.33 0.92 0.31 0.52 0.70 0.54 0.47
d= 0.2
0 0.5 Present 13.025011 23.400512 34.855821 38.384713 43.123622 55.586023 56.173114
3D FEM 13.036611 23.437212 34.936821 38.482813 43.248022 55.785723 56.378414
% Diff 0.09 0.16 0.23 0.26 0.29 0.36 0.370.5 0.25 Present 12.634811 21.246912 34.496621 36.678513 41.746722 54.436023 56.173114
3D FEM 12.574711 21.349512 34.545321 36.514113 41.939122 54.254523 55.654014
% Diff 0.48 0.48 0.14 0.45 0.46 0.33 0.920.5 0.5 Present 12.351911 23.400512 34.212921 34.523613 43.123622 53.057223 56.173114
3D FEM 12.408911 23.100512 34.379421 34.728513 42.671822 53.464223 55.456414
% Diff 0.46 1.28 0.49 0.59 1.05 0.77 1.28
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One set of solutions toEq. (14)are:
W1 A1 sink1X2 A2 cosk1X2 sinl1X1 B1 sink1X2 B2 cosk1X2 cosl1X1;
W2 A3 sinhk2X2 A4 coshk2X2 sinl2X1 B3 sinhk2X2 B4 coshk2X2 cosl2X1;
W3
A5
sinhk
3X
2 A
6cosh
k
3X
2cos
l3X
1 B
5sinh
k
3X
2 B6 coshk3X2 sinl3X1;15
provided thata21> 0;a22< 0;a
23< 0. These conditions must hold in
order to avoid imaginary roots in the characteristic equations of
the plate.Ai andBi are the arbitrary constants.ki andli are also re-lated to theai by
a21 l21 g2k21;a22 l22 g2k22;a23 l23 g2k23:
16ac
On the assumption of a simply supported edge at both X1= 0
and,X1= 1 Eq.(15)may be written as[37]:
W1 A1 sin k1X2 A2 cos k1X2 sin mpX1 ;W2 A3 sinh k2X2 A4 cosh k2X2 sin mpX1 ;W3 A5 sinh k3X2 A6 cosh k3X2 cos mpX1 :
17ac
In this theory, the bending moments, twisting moments, and
the transverse shear forces in terms ofw1,w2 andw3 are obtainedby integrating the stresses and moment of the stresses through the
thickness of the plate. These are given by
M11Z H
2
H2
r11x3dx3 D w1;1 mw2;2
;
M22Z H
2
H2
r22x3dx3 D w2;2 mw1;1
;
M12Z H
2
H2
r12x3dx3 D2
1 m w1;2 w2;1
;
Q1Z H
2
H2
r13dx3 K2GH w1 w3;1
;
Q2 Z H
2
H
2
r23dx3 K2GH w2 w3;2 ;
18ae
Table 6
First seven frequency parameter (b) for SSSC cracked rectangular thin plate when g = 2/3.
f a Methods Frequency parametersmn
d= 0.05
0 0.5 Present 15.449511 30.561812 43.674921 53.856813 57.825522 80.383423 84.733614
3D FEM 15.456011 30.584412 43.686821 53.907913 57.856122 80.447023 84.826514
% Diff 0.04 0.07 0.03 0.09 0.05 0.08 0.110.5 0.25 Present 15.162711 29.049512 43.383521 52.819213 56.613122 79.462623 84.562714
3D FEM 15.150411 29.135512 43.387121 52.801513 56.764522 79.360323 84.431414
% Diff 0.08 0.30 0.01 0.03 0.27 0.13 0.160.5 0.5 Present 15.040411 30.419912 43.183721 50.935113 57.774922 77.978823 84.102614
3D FEM 15.074411
30.316012
43.252121
51.148513
57.546622
78.272023
84.021414
% Diff 0.23 0.34 0.16 0.42 0.40 0.38 0.100.5 0.75 Present 15.414011 29.471312 43.520321 51.251213 56.725922 78.415323 83.617914
3D FEM 15.377111 29.583312 43.495821 51.466313 56.879822 78.601323 83.598314
% Diff 0.24 0.38 0.06 0.42 0.27 0.24 0.02
d= 0.1
0 0.5 Present 15.088411 29.199612 41.359221 50.023813 53.771422 72.894423 76.203414
3D FEM 15.100411 29.240912 41.394121 50.116213 53.842122 73.027323 76.377814
% Diff 0.08 0.14 0.08 0.18 0.13 0.18 0.230.5 0.25 Present 14.637711 27.049712 40.956421 48.636213 52.210222 71.745223 76.046314
3D FEM 14.621311 27.205012 40.984221 48.559513 52.478022 71.594123 75.771914
% Diff 0.11 0.57 0.07 0.16 0.51 0.21 0.360.5 0.5 Present 14.460111 29.017812 40.688321 46.090213 53.720922 69.920923 75.624914
3D FEM 14.538911 28.816912 40.826121 46.387813 53.354722 70.396323 75.346314
% Diff 0.55 0.69 0.34 0.65 0.68 0.68 0.370.5 0.75 Present 15.028511 27.586212 41.132321 46.850613 52.322722 70.718123 75.241014
3D FEM 14.967811 27.835612 41.119221 47.106513 52.623722 70.943723 75.003014
% Diff 0.40 0.90 0.03 0.55 0.58 0.32 0.32
d= 0.2
0 0.5 Present 13.911311 25.323512 35.127321 40.658313 43.918322 56.749523 58.307414
3D FEM 13.934711 25.395612 35.216221 40.818313 44.067122 57.000023 58.601114
% Diff 0.17 0.28 0.25 0.39 0.34 0.44 0.500.5 0.25 Present 13.343511 22.992712 34.716221 39.257713 42.463722 55.702323 58.259414
3D FEM 13.307411 23.101712 34.776221 39.010513 42.758222 55.522323 57.979614
% Diff 0.27 0.47 0.17 0.63 0.69 0.32 0.480.5 0.5 Present 13.133311 25.200912 34.443421 36.772813 43.897222 54.150623 58.116914
3D FEM 13.262511 24.797912 34.642521 37.122813 43.437422 54.505623 57.263414
% Diff 0.98 1.60 0.58 0.95 1.05 0.66 1.470.5 0.75 Present 13.823411 23.546412 34.871721 38.221013 42.578822 55.250923 58.069314
3D FEM 13.735911 23.841812 34.908421 38.585713 42.943422 55.300023 57.605014
% Diff 0.63 1.25 0.11 0.95 0.86 0.09 0.80
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whereG = E/2(1 + m) is the shear modulus, D = EH3/12(1 m2) is theflexural rigidity andrij(i,j = 1, 2 and 3) represents normal stresseswheni = j and the shear stresses when i j.
2.3. Geometrical configuration of cracked plate
The problem considered in this study is an elastic flat rectangu-
lar moderately thick plate with an arbitrary number of all-over
part-through cracks. They are perpendicular to one pair of opposite
simply supported edges, and their location is defined by the
parameter a , as shown in Fig. 3. Since an exact solution may beachieved for rectangular plates having at least two opposite simply
supported edges, central and side cracks, either oblique or parallel
to one edge of the plate, cannot taken into account in this study. In
addition, an oblique all-over part-through crack divides the rectan-
gular plate into two trapezoidal plates which, to the best knowl-
edge of the authors, no exact solution has been proposed for
these types of plates. For the sake of convenience, following formu-
lations are given for the plate with only one crack. As shown in
Fig. 3, the plate has thickness H, widtha and length b, while the
crack has a uniform depthh and constant lengtha. The plate is di-
vided by the crack line into two regions (I) and (II). The cracked
section is represented as continuously distributed line-spring
model so that the plate vibrates as an entire system. In order to ap-
ply dynamic and geometric boundary conditions, two Cartesian
coordinate systems (x1, x2, x3) and x01;x
02;x
03
are needed to define
for regions (I) and (II), respectively, which are related to each other
by the following form:
x01 x1; x022x2 1
2 ; x03x3; 19
x2= 0 andx02 0 are located in the middle of the plate. It should be
pointed out that considering the Cartesian coordinate systems like
those shown inFig. 3nullifies ill-conditioning phenomenon in the
mass and the stiffness matrices of the cracked plate.
2.4. Compatibility conditions for connection of two regions
At the crack location (X2a2or,X02
a
1
2 ), continuity conditionscan be written along the crack line as
w1I w1II;w3I w3II;M22I M22II;M12I M12II;Q2I Q2II;
20ae
where Eqs. 20(a)(e) express the equality of the slope of the plate in
X1 direction (w1), transverse displacement (w3), bending moment(M22), twisting moment (M12) and shearing force (Q2) at the two
sides of the crack location, respectively. The slope compatibility
condition of the crack in the X2 direction at the crack location
X2a
2
is satisfied by taking the following form:w2I w2II h; 21
wheretheaddedslope angle (h) induced bythe crackis givenby Eq.(6).
2.5. Classical boundary conditions
In this study, plates may take any classical boundary conditions
at non-simply supported edges, including free, simply supported
Table 7
First seven frequency parameter (b) for SCSC cracked rectangular thin plate when g = 2/3.
f a Methods Frequency parametersmn
d= 0.05
0 0.5 Present 17.178011 34.574812 44.459021 59.884113 60.353522 84.836423 90.433031
3D FEM 17.193811 34.624812 44.478721 59.986613 60.408122 84.946723 90.477131
% Diff 0.09 0.14 0.04 0.17 0.09 0.13 0.050.5 0.25 Present 17.091411 33.044812 44.255221 57.298913 59.023222 82.881723 90.105731
3D FEM 17.062111 33.230212 44.236721 57.530113 59.230922 83.055023 90.142931
% Diff 0.17 0.56 0.04 0.40 0.35 0.21 0.040.5 0.5 Present 16.611011 34.574812 43.891021 56.179613 60.353522 81.956423 89.696631
3D FEM 16.671911 34.460112 43.974921 56.527413 60.095322 82.360223 89.843331
% Diff 0.37 0.33 0.19 0.62 0.43 0.49 0.16
d= 0.1
0 0.5 Present 16.645511 32.587612 41.970021 54.672813 55.645022 75.955523 81.572614
3D FEM 16.672711 32.670412 42.014021 54.833213 55.746222 76.147323 81.844814
% Diff 0.16 0.25 0.10 0.29 0.18 0.25 0.330.5 0.25 Present 16.509211 30.451112 41.682421 51.672413 53.969522 73.859123 81.119614
3D FEM 16.465311 30.812412 41.680621 51.933813 54.337722 74.049723 80.626514
% Diff 0.27 1.19 0.00 0.51 0.68 0.26 0.610.5 0.5 Present 15.805211 32.587612 41.213021 49.963213 55.645022 72.573923 80.843431
3D FEM 15.935111 32.365812 41.375221 50.485813 55.213622 73.169723 81.119531
% Diff 0.82 0.68 0.39 1.05 0.78 0.82 0.34
d= 0.2
0 0.5 Present 15.014711 27.340012 35.434621 42.893613 44.762422 57.930323 60.357614
3D FEM 15.056311 27.454512 35.531913 43.120413 44.939722 58.234823 60.745614
% Diff 0.28 0.42 0.27 0.53 0.40 0.53 0.640.5 0.25 Present 14.845211 25.235112 35.134821 40.719313 43.319322 56.520423 60.277014
3D FEM 14.780611 25.620812 35.182921 40.986613 43.721422 56.810623 59.675414
% Diff 0.44 1.53 0.14 0.66 0.93 0.51 1.000.5 0.5 Present 14.061211 27.340012 34.697321 38.660213 44.762422 55.184223 60.357614
3D FEM 14.254511 26.937912 34.914121 38.990713 44.277022 55.604623 59.584914
% Diff 1.37 1.47 0.62 0.85 1.08 0.76 1.28
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and clamped. The boundary conditions along the edges X2 a2andX02 1a2 are satisfied by the following relations:
(a) for a free edge
M11 M12 Q1 0: 22(b) for a simply supported edge
M11 ~w2 ~w3 0: 23(c) for a clamped edge
~w1 ~w2 ~w3 0: 24
By application of the aforementioned boundary conditions and
the crack continuity conditions, a set of homogenous equation are
obtained. The coefficient matrix of rectangular moderately thick
plates with an all-over part-through crack under six possible com-
binations of boundary conditions has been given in Appendix A.
Vanishing the determinant of the coefficient matrix results in the
frequency equation of the cracked plate.
3. Results and discussion
By solving the closed-form characteristic equations for six pos-
sible combinations of boundary conditions, frequency parameters
of rectangular moderately thick plates with an all-over part-
through crack are obtained as a function of crack location a,relative crack depth f, aspect ratio g and wave number m. For
the brevity, SCSF denotes that the edges X1= 0,X2= 0,X1= 1, andX2 = 1 are simply supported, clamped, simply supported and free,
respectively. All frequencies are expressed in terms of the dimen-
sionless parameterb xa2 ffiffiffiffiffiffiffiffiffiffiffiffiffiqH=D
p .
For the sake of comparison, ABAQUS software package was used
to demonstrate the merit and high accuracy of the present exact
solution procedure. All the specimens were modeled and analyzed
with quadratic hexahedral 3D stress elements of type C3D20in or-
der to consider the transverse shear deformation effect. Quarter-
point singular elements of type C3D20 were used at the first ring
of the elements around the crack tip to bring models closer to real-
ity. A mesh sensitivity analysis was carried out to ensure indepen-
dency of FEM results from the number of elements. For example, a
total number of 32306 3D elements and 1472 singular elements
produce sufficiently converged results for a rectangular Mindlin
Fig. 5. The variation ofbC/bUCwith respect to the crack locationa for an SFSF rectangular plate (g= 0.5, d = 0.1).
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plate (d= 0.1,g = 2/3) with an all-over part-through crack (a= 0.5,f= 0.5), as shown inFig. 4.
3.1. Effect of crack depth
The first seven natural frequency of cracked rectangular plates
(g= 2/3), obtained by the present analytical solution and 3D FEmodel, are listed in Tables 27 for SFSF, SSSF, SFSC, SSSS, SSSC
and SCSC boundary conditions, respectively. These results are gi-
ven for two relative crack depths (f= 0, 0.5), three relative thick-nesses (d = 0.05, 0.1 and 0.2) and three crack locations (a= 0.25,0.5 and 0.75). Needless to say that f= 0 indicates an uncracked
plate. It should be noted that when the plate adopts symmetrical
boundary conditions abouta= b/2 (i.e., SFSF, SSSS and SCSC cases),natural frequencies fora= 0.25anda= 0.75arethe same.In Tables27, the corresponding mode shapes m and n denote the number of
half-sine waves in the X1 andX2 direction, respectively. The per-
centage difference given inTables 27is defined as follows:
%DiffAnalytical method 3D FEM Analytical method 100:
It is evident fromTables 27that the discrepancy between the
results of the present methodand 3D FEM is small and does not ex-
ceed 2% for the worst case. It is worth noting that the reduction ofnatural frequencies in the vibrating modes (n,m) = (1, 4) and (3, 1)
for SFSF and SCSC cracked plates, respectively, is so dominant that
the phenomenon of mode switching occurs. The vibrating mode (n,
m) = (1, 4) switches from the fifth frequency to fourth one and the
vibrating mode (n, m) = (3, 1) from the seventh frequency to sixth
one. InTables 2 and 7, these types of modes are labeled in bold.
3.2. Effect of crack location
The variation ofbC/bUC(subscriptsCand UC refer to cracked and
uncracked, respectively) against the crack location a for the firstfour mode shapes of rectangular moderately thick plates (g= 0.5)is shown inFigs. 510for six possible combinations of boundary
conditions when f= 0.2, 0.4 and 0.6. To gain further insight into
the effect of the crack location on frequency parameters of cracked
plates, 3D mode shapes of their uncracked counterparts are also
plotted.
It can be observed from Figs. 510that when the crack location
is close to the nodal line (i.e., the line of zero displacement), the ef-
fect of crack on the natural frequencies decreases and frequency
parameters of cracked plates approach to those of the plate with-
out crack (i.e., bC/bUC? 1). In contrast, if the crack is located at
the middle of two successive nodal lines, this effect can be maxi-
mized. It is also evident fromFigs. 7, 9 and 10that when the crack
gets much closer to the clamped edge of the plate, a considerablereduction in the natural frequencies of the cracked plate occurs.
Fig. 6. The variation ofbC/bUCwith respect to the crack location a for an SSSF rectangular plate (g= 0.5, d = 0.1).
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This may be attributed to the fact that the additional rotation in-
duced by the crack violates the clamped boundary condition,
where the slope is assumed to be zero. It is worthy to note from
Figs. 510that the natural frequencies decrease with the increase
of the relative crack depth f. This is due to the fact that the pres-
ence of the deeper crack causes a more reduction in the overall
stiffness of the plate.
3.3. Effect of stress intensity factor
In Fig. 11, the discrepancy between the present analytical meth-
od and 3D FEM for the first seven natural frequency of an SSSS
cracked rectangular plate (g= 2/3) is plotted in terms of the modenumber when a= 0.5,f= 0.5 andd= 0.05, 0.1 and 0.2. It can be ob-served fromFig. 11 that the most discrepancy takes place in the
modes (1, 2), (2, 2) and (1, 4), where the crack is located at the no-
dal line. This is due to the fact that only the first stress intensity
factor (KI) is considered in the present analytical solution for mod-
eling the crack. However, it seems that the second and third stress
intensity factors (KIIandKIII) should play a key role when the crack
lies on the nodal line.
A crack in a body may be subjected to three types of loading,which involve displacement of the crack surfaces (see Fig. 12).
The mechanical behavior of a solid containing a crack of a specific
geometry and size can be predicted by evaluating the stress inten-
sity factors (SIFs) KI, KIIand KIIIwhich are related to deformation
modesI,IIandIII. Hence, calculating SIFs is the first step for eval-
uating various crack deformations.
A lot of methods have been developed for determining the SIFs.
However, a very simple method for extracting these parameters is
called displacement extrapolation. In this approach, SIFs are com-
puted from the nodal displacements near the crack tip. In fact,
the nodal displacements in the vicinity of the crack are fitted to
the displacement equations and consequently the SIFs can be com-
puted from simple equations. By the use of collapsed-node isopara-
metric quadrilateral elements in the first ring around the crack tip,
the values of the SIFs are directly computed from nodal-point dis-
placements on opposite sides of the crack plane through the fol-
lowing relations[47]:
KIffiffiffiffiffiffiffi
2pp
2G1k jDVj / jDVj
KIIffiffiffiffiffiffiffi
2pp
2G1k jDUj / jDUj
KIIIffiffiffiffiffiffiffi
2pp
2GjDWj / jDWj;
k 3
4v For plane strain or axisymmetric;
3v1v For plane stress;
( 25ac
Fig. 7. The variation ofbC/bUCwith respect to the crack location a for an SFSC rectangular plate (g= 0.5,d = 0.1).
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whereG = E/2(1 m) is the shear modulus and DV, DUand DWarethe difference of the nodal displacements in opposite sides of the
crack inX1,X2 andX3 directions, respectively. For instance, the no-
dal displacement of opposite sides of the crack inX2 direction (i.e.,
DV) is shown inFig. 13. According toEq. (25), the magnitude ofKI,
KIIandKIIIis directly proportional to the values ofDV, DUand DW,
respectively. Therefore, the displacement differences DV, DU and
DW are properly utilized to compare the values of KI, KII and KIII
with each other. For example, for deformation mode I, the valueofDV is significantly greater than the other displacement differ-
ences DUand DW.
The aforementioned method is used to compare the deforma-
tion modesI, IIandIIIat the crack tip. In order to show the effect
of KII and KIII on natural frequencies of plates, first two shape
modes of an SSSS cracked rectangular plate (g= 2/3) are illustratedinFig. 14fora = 0.5 andf = 0.5. The values ofDV, DUand DWforthree locationsX1= 0,X1= 0.25andX1= 0.5 are listed in Table 8 for
the first two modes of the SSSS cracked plate. Note that these val-
ues have been calculated for the maximum deflection of the vibrat-
ing plate at the crack location.
It can be seen from Table 8 that the stress intensity factor (KI) of
mode Iis zero when the plate with a crack located at the nodal line
is vibrating in the second mode (1, 2). Under this circumstance, thepresent analytical method gives incorrect results and the natural
frequencies of cracked and uncracked plates become identical. It
can be attributed to the fact that the deformation modes II and
IIIare dominant for the cracks located at the nodal line. It is note-
worthy that in fracture mechanics, modeling of deformation modes
II and III are much more complicated than that of deformation
modeI. That is why the function gb, given by Eq.(2)for the defor-
mation mode I [46], has not been reported in literature for the
deformation modes IIand III. If this function was available, it could
be used in the present solution to model other two fracture modes.
3.4. Effect of multiple cracks
The present analytical approach can be extended to the plates
with several all-over part-through cracks. There are six unknown
coefficients in Eq. (17) which can be determined by applying
boundary conditions at two non-simply supported edges. As earlier
mentioned, one crack divides the plate into two segments resulting
in 12 unknown coefficients. They can be determined by applying
six boundary conditions at two non-simply supported edges and
other six boundary conditions defined by the continuity equations
(i.e.,Eqs. (20) and (21)) at the crack line. Similarly, a 12 12 coef-ficient matrix for one crack, for example, turns into an 18 18coefficient matrix for two cracks. Table 9shows the fundamentalfrequency parameters of a cracked rectangular plate (g= 2/3,
Fig. 8. The variation ofbC/bUCwith respect to the crack locationa for an SSSS rectangular plate (g= 0.5, d = 0.1).
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d= 0.1,f = 0.5) with six possible combinations of boundary condi-
tions, while there exist either one, two or three cracks in the plate.
It is seen fromTable 9that the more the cracks are introduced in
the plate, the more the natural frequencies decrease, as expected.
4. Conclusion
Based on the Mindlin plate theory, free vibration analysis of
rectangular moderately thick plates with an all-over part-throughcrack was investigated for different classical boundary conditions
of free, simply supported and clamped. The present solution was
given for the case where the cracks were perpendicular to two
opposite edges simply supported and remained open during the
entire cycle of motion. They were also treated as the mode (I) frac-
ture. ABAQUS software package was used to demonstrate the high
accuracy of the present analytical method. The influence of the
crack parameters, number of cracks, thickness-to-length ratios
and boundary conditions on natural frequencies of cracked rectan-
gular plates was investigated in the tabular and graphical form. It
was shown that the effect of cracks on the reduction of natural fre-
quencies is so tangible for some vibrating modes and boundary
conditions that the mode switching occurs. It was also found that
the natural frequencies decrease with the increase of the relativecrack depth and the number of cracks. The crack which is very
close to the clamped edge, results in a considerable reduction in
the natural frequencies of the cracked plate. Finally, it was ob-
served that the closer the crack is to the nodal line, the more dom-
inant the effect of the second and third stress intensity factors (KIIandKIII) on the frequency parameter will be. The significant merit
of the present paper lies in the fact that the exact closed-form char-
acteristic equations for six cases have been given in an explicit
form which is easy to compute natural frequencies of cracked rect-
angular plates.
Appendix A
There exist closed-form exact solutions to the characteristic
equations of cracked rectangular Mindlin plates under six combi-
nations of free, simply supported and clamped boundary condi-
tions. For the sake of clarity, the characteristic equations are
represented in a matrix form and the determinant will not be ex-
panded. For each case, an exact solution can be obtained by setting
the determinant of the matrices equal to zero. Roots of the deter-
minant are the natural frequencies of cracked rectangular Mindlin
plates for given wave number and dimensions of the plate and
crack. The characteristic determinants for the six boundary condi-
tions (i.e., SFSF, SSSF, SFSC, SSSS, SSSC and SCSC) are expressed asfollows:
Fig. 9. The variation ofbC/bUCwith respect to the crack locationa for an SSSC rectangular plate (g= 0.5,d = 0.1).
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Case 1. SFSF
Case 2. SSSF
Case 3. SFSC
C11Z4 S11Z4 C12Z5 S12Z5 S13Z6Z0 C13Z6
Z00 0 0 0 0 0
S11Z1 C11Z1 S12Z2 C12Z2 C13Z3Z0S13Z3
Z00 0 0 0 0 0
C11Z7
S11Z7 C12Z8 S12Z8
S13lZ0
C13lZ0
0 0 0 0 0 0
S21c1l C21c1l S22c2l C22c2l C23gk3Z0 S23gk3
Z0S31c1l C31c1l S32c2l C32c2l C33gk3Z0 S33g
k3Z0
S21 C21 S22 C22 0 0 S31 C31 S32 C32 0 0S21Z1 C21Z1 S22Z2 C22Z2 C23Z3Z0
S23Z3Z0
S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3
Z0
C21Z7 S21Z7 A10Z8 A9Z8 A11lZ0 A12l
Z0C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0
C33lZ0
C21Z4 S21Z4 A10Z5 A9Z5 A11Z6Z0 A12Z6
Z0C31Z7 S31Z7 C32Z8 S32Z8 S33Z6Z0
C33Z6Z0
Z9 Z10 Z11 Z12 Z13 Z14 C31c1gk1 S31c1gk1 C32c2gk2 S32c2gk2 S33lZ0C33l
Z0
0 0 0 0 0 0 C41Z4 S41Z4 C42Z5 S42Z5 S43Z6Z0 C43Z6
Z0
0 0 0 0 0 0 S41Z1 C41Z1 S42Z2 C42Z2 C43Z3Z0S43Z3
Z0
0 0 0 0 0 0 C41Z7 S41Z7 C42Z8 S42Z8 S43lZ0 C43l
Z0
0: A:1
S11c1l C11c1l S12c2l C12c2l C13gk3Z0 S13gk3
Z00 0 0 0 0 0
S11 C11 S12 C12 0 0 0 0 0 0 0 0
S11Z1 C11Z1 S12Z2 C12Z2 C13Z3Z0S13Z3
Z00 0 0 0 0 0
S21c1l C21c1l S22c2l C22c2l C23gk3Z0 S23gk3
Z0S31c1l C31c1l S32c2l C32c2l C33gk3Z0
S33gk3Z0
S21 C21 S22 C22 0 0 S31 C31 S32 C32 0 0S21Z1 C21Z1 S22Z2 C22Z2 C23Z3Z0
S23Z3Z0
S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3
Z0
C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0
C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33l
Z0
C21Z4
S21Z4 C22Z5 S22Z5
S23Z6
Z0
C23Z6
Z0
C31Z7 S31Z7
C32Z8
S32Z8
S33Z6Z0
C33Z6Z0
Z9 Z10 Z11 Z12 Z13 Z14 C31c1gk1 S31c1gk1 C32c2gk2 S32c2gk2 S33lZ0C33l
Z0
0 0 0 0 0 0 C41Z4 S41Z4 C42Z5 S42Z5 S43Z6Z0 C43Z6
Z0
0 0 0 0 0 0 S41Z1 C41Z1 S42Z2 C42Z2 C43Z3Z0S43Z3
Z0
0 0 0 0 0 0 C41Z7 S41Z7 C42Z8 S42Z8 S43lZ0 C43l
Z0
0: A:2
C11Z4 S11Z4 C12Z5 S12Z5 S13Z6Z0 C13Z6
Z00 0 0 0 0 0
S11
Z1
C11
Z1
S12
Z2
C12
Z2
C13Z3
Z0
S13Z3
Z00 0 0 0 0 0
C11Z7 S11Z7 C12Z8 S12Z8 S13lZ0 C13lZ0
0 0 0 0 0 0
S21c1l C21c1l S22c2l C22c2l C23gk3Z0 S23gk3
Z0S31c1l C31c1l S32c2l C32c2l C33gk3Z0
S33gk3Z0
S21 C21 S22 C22 0 0 S31 C31 S32 C32 0 0S21Z1 C21Z1 S22Z2 C22Z2 C23Z3Z0
S23Z3Z0
S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3
Z0
C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0
C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33lZ0
C21Z4 S21Z4 C22Z5 S22Z5 S23Z6Z0 C23Z6
Z0C31Z7 S31Z7 C32Z8 S32Z8 S33Z6Z0
C33Z6Z0
Z9 Z10 Z11 Z12 Z13 Z14 C31c1gk1 S31c1gk1 C32c2gk2 S32c2gk2 S33lZ0C33lZ0
0 0 0 0 0 0 S41c1l C41c1l S42c2l C42c2l C43gk3Z0 S43gk3
Z0
0 0 0 0 0 0 C41c1gk1 S41c1gk1 C42c2gk2 S42c2gk2 S43lZ0C43lZ0
0 0 0 0 0 0 S41 C41 S42 C42 0 0
0: A:3
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Case 4. SSSS
Case 5. SSSC
Case 6. SCSC
S11c1l C11c1l S12c2l C12c2l C13gk3Z0 S13gk3
Z00 0 0 0 0 0
S11 C11 S12 C12 0 0 0 0 0 0 0 0
S11Z1 C11Z1 S12Z2 C12Z2 C13Z3Z0S13Z3
Z00 0 0 0 0 0
S21c1l C21c1l S22c2l C22c2l
C23gk3
Z0 S23gk3
Z0 S31c1l
C31c1l
S32c2l
C32c2l
C33gk3
Z0
S33gk3
Z0
S21 C21 S22 C22 0 0 S31 C31 S32 C32 0 0S21Z1 C21Z1 S22Z2 C22Z2 C23Z3Z0
S23Z3Z0
S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3
Z0
C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0
C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33lZ0
C21Z4 S21Z4 C22Z5 S22Z5 S23Z6Z0 C23Z6
Z0C31Z7 S31Z7 C32Z8 S32Z8 S33Z6Z0
C33Z6Z0
Z9 Z10 Z11 Z12 Z13 Z14 C31c1gk1 S31c1gk1 C32c2gk2 S32c2gk2 S33lZ0C33lZ0
0 0 0 0 0 0 S41c1l C41c1l S42c2l C42c2l C43gk3A0 S43gk3
A0
0 0 0 0 0 0 S41 C41 S42 C42 0 0
0 0 0 0 0 0 S41Z1 C41Z1 S42Z2 C42Z2 C43Z3A0S43Z3
A0
0: A:4
S11c1l C11c1l S12c2l C12c2l C13gk3Z0 S13gk3
Z00 0 0 0 0 0
S11 C11 S12 C12 0 0 0 0 0 0 0 0
S11Z1 C11Z1 S12Z2 C12Z2 C13Z3Z0S13Z3
Z00 0 0 0 0 0
S21c1l C21c1l S22c2l C22c2l C23gk3Z0 S23gk3
Z0S31c1l C31c1l S32c2l C32c2l C33gk3Z0
S33gk3Z0
S21 C21 S22 C22 0 0 S31 C31 S32 C32 0 0S21Z1 C21Z1 S22Z2 C22Z2 C23Z3Z0
S23Z3Z0
S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3
Z0
C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0
C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33lZ0
C21
Z4
S21
Z4
C22
Z5
S22
Z5
S23Z6
Z0 C23Z6
Z0 C
31Z
7 S
31Z
7 C
32Z
8 S
32Z
8
S33Z6
Z0
C33Z6
Z0
Z9 Z10 Z11 Z12 Z13 Z14 C31c1gk1 S31c1gk1 C32c2gk2 S32c2gk2 S33lZ0C33lZ0
0 0 0 0 0 0 S41c1l C41c1l S42c2l C42c2l C43gk3Z0 S43gk3
Z0
0 0 0 0 0 0 S41 C41 S42 C42 0 0
0 0 0 0 0 0 C41c1gk1 S41c1gk1 C42c2gk2 S42c2gk2 S43lZ0 C43lZ0
0: A:5
S11c1l C11c1l S12c2l C12c2l C13gk3Z0 S13gk3
Z00 0 0 0 0 0
S11 C11 S12 C12 0 0 0 0 0 0 0 0
C11c1gk1 S11c1gk1 C12c2gk2 S12c2gk2 S13lZ0 C13lZ0 0 0 0 0 0 0S21c1l C21c1l S22c2l C22c2l C23gk3Z0
S23gk3Z0
S31c1l C31c1l S32c2l C32c2l C33gk3Z0S33gk3
Z0
S21 C21 S22 C22 0 0 S31 C31 S32 C32 0 0S21Z1 C21Z1 S22Z2 C22Z2 C23Z3Z0
S23Z3Z0
S31Z1 C31Z1 S32Z2 C32Z2 C33Z3Z0 S33Z3
Z0
C21Z7 S21Z7 C22Z8 S22Z8 S23lZ0 C23lZ0
C31Z7 S31Z7 C32Z8 S32Z8 S33lZ0C33l
Z0
C21Z4 S21Z4 C22Z5 S22Z5 S23Z6Z0 C23Z6
Z0C31Z7 S31Z7 C32Z8 S32Z8 S33Z6Z0
C33Z6Z0
Z9 Z10 Z11 Z12 Z13 Z14 C31c1gk1 S31c1gk1 C32c2gk2 S32c2gk2 S33lZ0C33l
Z0
0 0 0 0 0 0 S41c1l C41c1l S42c2l C42c2l C43gk3Z0 S43gk3
Z0
0 0 0 0 0 0 S41 C41 S42 C42 0 0
0 0 0 0 0 0 C41c1gk1 S41c1gk1 C42c2gk2 S42c2gk2 S43lZ0 C43lZ0
0:
A:6
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where
c1 a2
; c2a2
; c3a 1
2 ; c4
1 a2
;
Si1 sincik1Ci1 coscik1
;
Sij sinhcikjCij coshcikj
; for i 1; 2; 3 and 4;
j 2; 3:
Z0 sinhc1k3; Z1 c1 g2k21 ml2
; Z2 c2 g2k22 ml2
;
Z3 g1 mlk3; Z4 2c1gk1l; Z5 2c2gk2l;
Z6 g2k23 l2; Z7 c1 1gk1; Z8 c2 1gk2;
Z9 c1gk1C21 6Habba
Z1S21; Z10 c1gk1S21 6Habba
Z1C21;
Z11 c2gk2C22 6Habba
Z2S22;
Z12 c2gk2S22 6Habba
Z2C22; Z13 lZ23 6Habba
Z3C23
Z0;
Z14 lC23 6Habba
Z3S23
Z0:
It should be pointed out that all elements, including cosh
(ci k3) and sinh (ci k3) (i= 1, 2, 3, 4), take very large values
Fig. 10. The variation ofbC/bUCwith respect to the crack location a for an SCSC rectangular plate (g= 0.5, d = 0.1).
Fig. 11. The percent of discrepancy for the first seven mode shapes of an SSSS
cracked rectangular plate (g= 2/3) whena = 0.5 andf = 0.5.
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in comparison with other elements excluding the aforementioned
parameters, whend< 0.1. In order to eliminate the ill-conditioning
problem in the matrices, given by Eqs. (A.1)(A.6), the Cartesian
coordinate systems should be considered as those shown in
Fig. 3 so that the elements with large values appear in the
fifth, sixth, eleventh and twelfth columns. Then, dividing all ele-
ments in those four columns by Z0 results in well-conditioning
matrices.
Fig. 12. Stress loading modes: (a) mode I (opening mode); (b) modeII (shearing mode); (c) modeIII(tearing mode).
Fig. 13. The nodal-point displacement for a crack.
Fig. 14. The effect ofKIIand KIIIfor first two shape modes of an SSSS cracked rectangular plate ( g= 2/3) witha = 0.5 andf = 0.5.
Table 8
The values ofDV, DUand DWfor the first two modes of an SSSS cracked rectangular
plate (g= 2/3, d = 0.1) when a = 0.5 and f = 0.5.
X1 First mode (1, 1) Second mode (1, 2)
DV 102 DU 102 DW 102 DV 102 DU 102 DW 102
0 0 0 0 0 0 1.109
0.25 0.884 0 0 0 0.443 0.912
0.5 1.310 0 0 0 0.658 0.148
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Table 9
Fundamental frequency parameters (b) of a rectangular plate (g = 2/3, d = 0.1) with multiple cracks when f = 0.5.
Crack location (a) Boundary conditions
SFSF SSSF SFSC SSSS SSSC SCSC
Uncracked plate 9.5157 10.4404 10.7099 13.9085 15.0884 16.6455
0.5 9.4740 10.3725 10.6559 13.3938 14.4601 15.8052
1/3 and 2/3 9.4472 10.3247 10.6131 13.2023 14.3147 15.8258
0.25, 0.5 and 0.75 9.4261 10.2907 10.5762 13.0518 14.1518 15.6941
1032 Sh. Hosseini-Hashemi et al. / Computers and Structures 88 (2010) 10151032