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Vibrations of Stepped Plate Strips with Cracks
Jaan Lellep, Esta K¨ago
Institute of Mathematics
University of Tartu
2 Liivi str., Tartu 50409
Estonia
Email: jaan.lellep@ut.ee, esta20@ut.ee
Abstract—An approximate method is suggested for the vibration
analysis of elastic plate strips. The strips supported at two opposite
edges and free at the other edges are of piece wise constant thickness.
It is assumed that stable part-through cracks are located at the re-
entrant corners of steps. Making use of the basic concepts of the
fracture mechanics a method for determination of eigenfrequencies
of stepped plates with cracks is developed. The influence of a crack on
the behavior of the strip is modeled as a change of the local flexibility
or as a distributed line spring. Numerical results are presented for
strips with cracks and without any crack subjected to the tension
applied at an edge of the strip.
Keywords—Crack, eigenfrequency, plate, strip, vibration
I. INTRODUCTION
The matter that the presence of surface flaws or intrinsic
cracks in a machine element is a source of local flexibility
which in turn influences the dynamic behavior of the whole
system was recognized long ago. The idea of an equivalent
elastic spring, a local compliance, was used first to quantify the
relation between the applied load and the strain concentration
in the vicinity of the crack tip by Irwin [1].
Later Rice and Levy [2] computed the local flexibility in
the case of a combined loading consisting of the bending and
tension.
Dimarogonas and Paipetis [3], Dimarogonas [4], Chondros
et al. [5], [6] combined this spring model in the case of
a vibrating beam with the methods of the elastic fracture
mechanics. As a result the frequency spectral method was
developed. This idea was exploited by Rizos et al. [7],
Chondros et al. [6] for the analysis of cracked beams. It was
extended by Lellep et al. [8], [9], [10] for axisymmetrical
vibrations of cylindrical shells.
In the present paper free vibrations of stepped plates and
strips are studied in the case of presence of cracks.
II. FORMULATION OF THE PROBLEM
Let us consider natural vibrations of a plate strip subjected
to the in-plane tension N(Fig. 1). Let the dimensions of the
strip in xand ydirection be land b, respectively. The plate
under consideration is clamped at the edge x= 0, the other
edges are free.
The thickness hof the plate us assumed to be piece wise
constant. Thus
h(x, y) = hj(1)
y
x
b
l
NN
(a) Dimensions of the strip
x
h0
h1
h2
a1a2l
0
.
(b) Piece wise constant thickness
Fig. 1. Plate strip
for x∈(aj, aj+1), where j= 0,...,n. The quantities aj
and hj(j= 0,...,n)are given constants whereas a0= 0,
an+1 =l.
It is assumed that at cross sections x=aj(j= 1,...,n)
where the thickness has jumps cracks of constant depth cjare
located. These flaws or cracks are treated as stable surface
cracks.
The aim of the paper is to elucidate the sensitivity of
natural frequencies on the crack parameters and geometrical
parameters of the plate.
III. BASIC EQUATIONS
In the present case of the plate in-plane forces have to be
taken into account. If, moreover, the inertia of the rotation
is not neglected as well the equilibrium equations of a plate
element can be presented as (Reddy [11])
Dj
∂4W
∂x4−N∂2W
∂x2=ρhj
∂2W
∂t2−Ij
∂4W
∂x2∂t2(2)
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for x∈(aj, aj+1)where j= 0,...,n. Here Wis the
transverse deflection,
Ij=ρh3
j
12 , Dj=Eh3
j
12(1 −ν2),(3)
Eand νare elastic moduli and ρ– density of the material.
In the case of a free edge of a strip the boundary conditions
are ∂2W
∂x2= 0,∂3W
∂x3= 0.(4)
However, in the case of the clamped edge one has
∂W
∂x = 0, W = 0.(5)
Let at the initial moment
∂W
∂t = 0, W =ϕ(x)(6)
where ϕis a given function.
IV. SOLUTION OF THE EQUATION OF MOTION
It is reasonable to look for the general solution of (2) in the
form
W(x, t) = wj(x)T(t)(7)
for x∈(aj, aj+1)where j= 0,...,n.
Differentiating (7) with respect to variables x,tand substi-
tuting in (2) one easily obtains
DjwIV
jT−Nw′′
jT=ρhjwj¨
T−Ijw′′
j¨
T(8)
for x∈(aj, aj+1)where j= 0,...,n. Here prims denote
the differentiation with respect to the coordinate xand dots –
with respect to time t.
Separating variables in (7) yields
DjwIV
j+ (Ijω2−N)w′′
j−ρhjω2wj= 0 (9)
for j= 0,...,n and
¨
T+ω2T= 0 (10)
where ωstands for the frequency of natural vibrations. Ev-
idently, the solution of (10) which satisfied according to (6)
initial conditions T(0) = d,˙
T(0) = 0 has the form
T=dcos ωt (11)
where dis a constant.
The equation (9) is a linear fourth order ordinary equation
with respect to the variable wj. The characteristic equation
corresponding to (9) is
Djr4
j+ (Ijω2−N)r2
j−ρhjω2= 0 (12)
Form (12) one easily obtains the roots
rj=±v
u
u
t−Ijω2+N
2Dj±s(Ijω2−N)2
4D2
j
+ρhj
Dj
.(13)
Introducing the notation
(r2
j)1=−λ2
j
(r2
j)2=µ2
j
(14)
one can present the general solution of (9) as
wj(x) = A1jcos λjx+A2jsin λjx+
+A3jsinh µjx+A4jcosh µjx(15)
which holds good for x∈(aj, aj+1 ),j= 0,...,n. Here
A1j,...,A4jstand for unknown constants of integration.
These will be determined from boundary conditions and re-
quirements on the continuity of displacements and generalized
stresses.
However, it appears that the quantity W′can not be con-
tinuous at x=ajaccording to the model of distributed line
springs developed by Rice and Levy [2]; Dimarogonas [4],
Chondros et al. [5].
V. LOCAL COMPLIANCE OF THE PLATE STRIP
Let us consider the influence of the crack located at the
cross section x=aon the stress-strain state of the sheet in
the vicinity of the crack. For the conciseness sake we shall
study the case when n= 1 and thus in the adjacent segments
to the crack the thickness equals to h0and h1, respectively.
Let h=min(h0, h1).
According to the distributed line spring method the slope
of the deflection has a jump
Θ = w′(a+ 0) −w′(a−0) (16)
at the cross section x=a. The angle Θcan be treated as
ageneralized displacement corresponding to the generalized
stress Mx. Thus
Θ = CMx(a)(17)
or
C=∂Θ
∂Mx(a)(18)
where Cis the local compliance due to the crack. It is known
in the linear elastic fracture mechanics that (see Anderson [12],
Broberg [13])
Θ = ∂UT
∂Mx(a)(19)
where UTis the extra strain energy caused by the crack.
Combining (17)–(19) one obtains
C=∂2UT
∂M 2
x(a).(20)
According to the concept of the distributed line spring 1/C =
K, where Kstands for the stress intensity coefficient. It is
known in the fracture mechanics that (see Anderson [12])
KM=σM√πcFMc
h.(21)
In (21) cis the crack depth and
σM=6Mx(a)
bh2,(22)
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provided the element involving the cross section x=ais
loaded by the bending moment Mxonly. Here the function
FMis to be approximated on the basis of experimental data
[14].
If the element is loaded by the axial tension Nthen the
stress intensity coefficient
KN=σN√πcFNc
h.(23)
where
σN=N
bh .(24)
In the case of a combined loading the stress intensity coeffi-
cient
KT=KM+KN.(25)
Note that (25) holds good under the condition that (21)–(24)
refer to the common mode of fracture (see Anderson [12] and
Broek [15]).
In the present case this requirement is fulfilled, KMand KN
regard to the first mode of the fracture. It was shown in the
previous studies (Lellep, Roots [10]; Lellep, Puman, Tungel,
Roots [8]; Lellep, Sakkov [16]) that in the case of loading by
the moment
KM=E′h2b
72πf (s)(26)
where E′=Efor plane stress state and E′=E/(1 −ν2)in
the case plane deformation state.
Here s=c/h and the compliance
C=72π
E′h2bZs
0
sF 2
M(s)ds (27)
whereas
f(s) = Zs
0
sF 2
M(s)ds. (28)
The function FMwas taken in the studies by Dimarogonas
[4]; Rizos, Aspragathos, Dimarogonas [7] as
FM= 1.93 −3.07s+ 14.53s2−25.11s3+ 25.8s4.(29)
According to the handbook by Tada, Paris, Irwin [14] the
function FNcan be approximated as
FN= 1.122 −0.23s+ 10.55s2−21.71s3+ 30.38s4.(30)
VI. DETERMINATION OF NATURAL FREQUENCIES
In the case when the plate has a unique step the deflected
shape of the plate can be presented according to (15) as
w(x) = A1sin λ0x+A2cos λ0x+A3sinh µ0x+A4cosh µ0x
(31)
for x∈[0, a]and
w(x) = B1sin λ1x+B2cos λ1x+B3sinh µ1x+B4cosh µ1x
(32)
for x∈[a, l].
Arbitrary constants Ai,Bi(i= 1,...,4) have to meet
boundary requirements and intermediate conditions at x=a.
The latter can be presented as (Lellep, Roots [10])
w(a−0) = w(a+ 0)
w′(a−0) = w′(a+ 0) −pw′′ (a+ 0)
h3
0w′′(a−0) = h3
1w′′(a+ 0)
h3
0w′′′(a−0) = h3
1w′′′(a+ 0)
(33)
where according to (25), (26)
p=Eh3
12(1 −ν2)KT
.(34)
It is worthwhile to mention that the third and the fourth
equality in (33) express the continuity of the bending moment
and the shear force, respectively, when passing the step at x=
a. It is known from the solid mechanics that these quantities
must be continuous (Soedel [17]).
Boundary conditions (5) at x= 0 admit to eliminate from
(31) the unknown constants
A4=−A2,
A3=−A1
λ0
µ0
.(35)
The intermediate conditions (33) with boundary require-
ments (5) at x=llead to the system of six equations which
will be presented in the matrix form. The continuity of the
deflection leads to the equation
A1
A2
B1
B2
B3
B4
⊤
×
sin λ0a−λ0
µ0sinh µ0a
cos λ0a−cosh µ0a
−sin λ1a
−cos λ1a
−sinh µ1a
−cosh µ1a
= 0.(36)
According to the second relation in (33) one has
A1
A2
B1
B2
B3
B4
⊤
×
λ0(cos λ0a−cosh µ0a)
−λ0sin λ0a−µ0sinh µ0a)
λ1(pλ1sin λ1α−cos λ1α)
λ1(sin λ1α−pλ1cos λ1α)
−µ1(cosh µ1α+pµ1sinh µ1α)
−µ1(sinh µ1α+pµ1cosh µ1α)
= 0.(37)
The continuity requirements imposed on the bending moment
and the sheare force, respectively, lead to the equations
A1
A2
B1
B2
B3
B4
⊤
×
−h3
0λ0(λ0sin λ0a−µ0sinh µ0a)
−h3
0(λ2
0cos λ0a−µ2
0cosh µ0a)
h3
1λ2
1sin λ1a
h3
1λ2
1cos λ1a
−h3
1µ2
1sinh µ1a
−h3
1µ2
1cosh µ1a
= 0
(38)
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150 200 250 300 350 400 450
alpha
omega
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
s=0.0
s=0.2
s=0.4
s=0.6
s=0.8
Fig. 2. Natural frequency vrs a/l,N= 0,γ= 1
and
A1
A2
B1
B2
B3
B4
⊤
×
−h3
0λ0(λ2
0cos λ0a−µ2
0cosh µ0a)
h3
0(λ3
0sin λ0a−µ3
0sinh µ0a)
h3
1λ3
1cos λ1a
−h3
1λ3
1sin λ1a
−h3
1µ3
1cosh µ1a
−h3
1µ3
1sinh µ1a
= 0.
(39)
The boundary conditions (4) can be expressed as
A1
A2
B1
B2
B3
B4
⊤
×
0
0
−λ2
1sin λ1l
−λ2
1cos λ1l
µ2sinh µ1l
µ2cosh µ1l
= 0 (40)
and
A1
A2
B1
B2
B3
B4
⊤
×
0
0
−λ3
1sin λ1l
−λ3
1cos λ1l
µ3sinh µ1l
µ3cosh µ1l
= 0.(41)
The system (36)–(41) is a linear homogeneous system of
algebraic equations. It has a non-trivial solution only in the
case, if its determinant ∆equals to zero. The equation ∆ = 0
is solved up to the end numerically.
VII. NUMERICAL RESULTS
The results of calculations are presented in Fig. 2–8. In
calculations the plate with dimensions l= 0.5m, h0= 0.02 m
was considered. The material parameters are ρ= 7860 kg/m3,
E= 2.1·1011 N/m2,ν= 0.3.
0 100 200 300 400 500
gamma
omega
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
s=0.0
s=0.2
s=0.4
s=0.6
s=0.8
Fig. 3. Natural frequency vrs h1/h0,N= 0,α= 0.5
100 200 300 400 500
alpha
omega
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
s=0.0
s=0.2
s=0.4
s=0.6
s=0.8
Fig. 4. Eigenfrequency vrs a/l,N= 0,γ= 0.5
In following the notation
α=a
l, γ =h1
h0
is used.
In Fig. 2 the frequency of natural vibrations is shown versus
awhere ais the location of the crack. Different curves in Fig.
2 correspond to different values of the crack depth.
The curves depicted in Fig. 2 are associated with the case
when no tension is applied to the plate, e.g. N= 0. It can be
seen from Fig. 2 that when the crack depth increases then the
natural frequency decreases, as might be expected.
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100 200 300 400 500
alpha
omega
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
s=0.0
s=0.2
s=0.4
s=0.6
s=0.8
Fig. 5. Stretched strip, N=106,γ= 0.5
The natural frequency versus αand γin the case of a plate
of piece wise constant thickness is presented in Fig. 3, 4. Here
N= 0 whereas Fig. 3 and 4 are associated with α= 0.5and
γ= 0.5, respectively.
Similarly to the case of a plate of constant thickness it
reveals from Fig. 3 and 4 that the natural frequency is maximal
for the intact plate in comparison to that for a cracked sheet. It
is interesting to note that when increasing the ratio of thickness
in the case of fixed position of the step then the natural
frequency ωincreases until a certain value and in the course of
subsequent increase of γthe quantity ωslowly decreases (Fig.
3). For instance, if c= 0, the point of maximum is achieved
for γ= 0.5and if c= 0.6h1then γ= 0.7.
If the ratio of thickness γis fixed then there exists also
a maximum of ωwith respect to the step location α. If, for
instance, N= 0,γ= 0.5(Fig. 4) then the maximum points
are α= 0.65 (for uncracked plate, s= 0), α= 0.7(for
s= 0.4) and α= 0.85 (for s= 0.9).
Calculations carried out showed that the natural frequency
depends quite weakly on the edge tension Nif N < N∗where
N∗is a critical value of the edge tension. The dependence of
the frequency ωon the ratio of thickness and on the crack
length is depicted in Fig. 5. Here N= 106and h1= 0.5h0.
Comparing the results presented in Fig. 3 and Fig. 5 one
can see that the corresponding curves are relatively close each
other.
Similar results are presented in Fig. 6 and 7 for N= 5·106.
The curves depicted in Fig. 6 correspond to the case a= 0.5l
and these shown in Fig. 7 are associated with the fixed value
of the ratio of thicknesses h1= 0.5h0.
Comparing Figures 6 and 7 with Fig. 3 and 4, respectively,
one can see that in the case of larger edge tension values of
the natural frequency are smaller than those corresponding to
0 50 100 150 200 250 300 350
gamma
omega
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
s=0.0
s=0.2
s=0.4
s=0.6
s=0.8
Fig. 6. Eigenfrequency vrs h1/h0,N= 5 ·106,α= 0.5
0 100 200 300
alpha
omega
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
s=0.0
s=0.2
s=0.4
s=0.6
s=0.8
Fig. 7. Eigenfrequency vrs a/l,N= 5 ·106,γ= 0.5
the plate without edge tension. Naturally, it is assumed herin
that the strips with the same geometrical parameters and with
the same crack parameters are compared. It can be seen from
Fig. 6, 7 as well, that the increase of the crack length entails
reduced values of the natural frequency.
The influence of the edge loading Non the natural fre-
quency ωis presented in Fig. 8. Here α= 0.5and γ= 0.5
whereas the units in the horizontal axis are taken in millions.
It can be seen from Fig. 8 that in this scale the natural
frequency ωdoes depend on the crack length and on the edge
tension. The larger is the edge load the smaller is the natural
frequency of the plate.
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100 200 300 400
N*10^6
omega
12345678910
s=0.0
s=0.2
s=0.4
s=0.6
s=0.8
Fig. 8. Eigenfrequency vrs edge tension, α=0.5,γ= 0.5
VIII. CONLUDING REMARKS
Combing the methods of the theory of elastic plates and of
the linear elastic fracture mechanics an approximate technique
for determination of natural frequencies of plate strips is
developed. This technique admits to account for the tension
applied at the edge of the plate weakened by part-through
surface cracks occuring at the re-entrant corners of steps.
Calculations carried out showed that the crack location and
the crack depth have an essential influence in the frequency of
free vibrations. It was established numerically that when the
crack extends then the natural frequency decreases, provided
the other geometrical parameters remain unchanged. It is
interesting to remark that the reduce of the frequency takes
place independently of the ratio of thicknesses, of the location
of the step and of other geometrical parameters.
ACKNOWLEDGMENT
The partial support from the Estonian Ministry of Edu-
cation and Research through the target financed project SF
0180081S08 ”Models of applied mathematics and mechanics”
and the Estonian Science Foundation trough the Grant ESF
7461 ”Optimization of elastic and inelastic shells” is gratefully
acknowledged.
REFERENCES
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Pergamon Press, 1960.
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[10] ——, “Vibrations of cylindrical shells with circumferential cracks,”
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