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ELASTIC BEAM FINITE ELEMENT WITH
TRANSVERSE ELASTO-PLASTIC CRACK
ARINDAM DEY
Y3103010
M.Tech.Geotechnical Engineering
IIT Kanpur
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INTRODUCTION
Cracks occurring in structural elements of machines lead to local changes in thestiffness of these elements and consequently their static, dynamic and stability
behaviour is altered
Cracked structures have been modelled by various methods such as one-dimensional,
discrete-continuous model in which the crack was modelled by a flexibility matrix
(Dimarogonas),where the elements of this matrix have been calculated from relations oflinear fracture mechanics; the method of sub-domains (Lee) for modelling cracked
annular plates; & singular finite elements for the dynamic analysis of a cracked beam
(Shen & Pierre)
The above citations assumed that material around the crack tip behaved in a purely
elastic manner.
In real structures, for many materials, a plastic zone appears around the crack tip & the
flexibility of such structures increases, more than is observed in the case of purely elastic
materials.
The main objective of this work is to develop a model of a cracked beam finite element,
which will take into account the effect of plasticity around the crack tip
A beam finite element model with single-edge crack is developed,which takes into
account the influence of the plastic zone ahead of the crack tip on flexibility of the
element.
The inertia and stiffness matrices of the element are presented in closed forms. Theforms of these matrices are modified by the presence of the crack
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ELASTO-PLASTIC LOCAL FLEXIBILITY OF A
CRACKED BEAM The radius of the plastic zone around the crack tipr
p, for I fracture mode of the
crack evaluation (opening mode) can be calculated from the following
relationship
where, KI
is the elastic stress intensity factor, Y the material yield strength,andd
pdenotes the diameter of the plastic zone around the crack tip
2
112
==
y
pp Krd ..(1)
Fig. 1 Plastic zone around a cracked rip
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Contd
For a crack with the plastic zone around its tip, the stress intensity factor (SIF)
can be expressed as follows
where,a is the crack length,F is a correction function which takes into account
the body and crack geometry, while is the applied nominal stress Substituting the relationship (1) into (2) gives
which allows calculation of the stress intensity factor for an elastoplastic crack
Additional local flexibilities of the body due to the crack cij
are given by the
relation
where Jc
denotes the strain energy density function (SEDF), P is a
corresponding force at the cracked cross-section, whileA is the crack area
pIp raFK += (2)
....(3)
= Ac
ji
ij dAJPPc
2
...(4)
22
1
2
1
21
2
1
+=
+=
YY
Ip
FK
KaFK
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Contd
For I fracture mode of the crack tip evaluation SEDF is given by
where, E'=E for the plane state of stress, E'=E/(1- 2) for the plane state ofstrain,E denotes Young's modulus and is Poisson's ratio.
In the present case the beam is subjected only to the bending momentM. Thus,
SIF may be calculated as
Taking into account the relation (4) the flexibility at the cracked cross-section
of the beam under consideration can be written as
where,B andHdenote the width and height of the beam, respectively
= =
=
6
1'
1 i
i
Ipic KE
J ..(5)
22
2
2
16
+=
Y
Ipi
F
BH
MK
..(6)
+=AYA
adAFHBEadAFHEBc4
2
42
2
42
21672
..(7)
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Contd
The correction functionF is given as follows
Eqn (7) is expressed in a more convenient manner as
where,
( )[ ]( )Ha
Ha
Ha
HaF
2/cos
2/sin1199.0933.0
2/
)2/tan(4
+=
(8)
( ) ( ) adaaFBHE
adaaF
EBH
ckk a
Y
a
+=
0
4
2
2
0
2
2
21672
(9)
H
aa
H
aa kk == ,
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A FINITE ELEMENT MODEL
A cracked beam finite element model has been chosen
Three different parts can be distinguished in the element: left segment, crack
and right segment
The left and right segments are represented by non-cracked beams of length
L /2, whereas the crack segment is represented by a massless spring of length
zero
The element has two nodes with two degrees of freedom (transverse
displacement and rotation) in each node
Fig. 2 A cracked beam finiteelement model
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Contd
The field of displacements for the left and right segments of the element can be written
in the following form
The constants a1-a8 can be expressed by nodal displacements using the elementboundary conditions
where K=EJc, and J denotes the geometrical moment of inertia of the beam cross-section
( )
( )
( )( ) 2
8762
3
8
2
7652
2
4321
3
4
2
3211
32
32
xaxaaxxaxaxaaxw
xaxaax
xaxaxaaxw
++=+++=
++=
+++=
....(10)
( ) ( )( ) ( )
( ) ( )
( ) ( ) ( )( ) ( )
( ) ( )22
22222
22
,0,0
'''
2
'''
1
"
2
"
1
"
221
21
4221
321
LwLw
LwLwLKwLL
LwLw
qLqqLwqw l
==
=
====
......(11)
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Contd
Taking into account the formulas (10) and (11) the shape function matrix for
left and right segments of the element can be determined, as follows
Uusing the matrices of the shape functions and applying the standard finite
element definitions of the characteristic matrices, the mass matrix of the
element can be evaluated as
+
+
+
+
++
+
+
=
+
+
+
+
=
2323
22
2
32
2
2323
222
32
1
1212
)(2
)32(3
)(2
)34(3
00
)(20
)(21
3210
1
1212)(2
)32(3
)(2
)34(30010
0001
3210
1
LLLL
KLL
KL
LKLL
KL
L
KL
K
KL
LK
KL
LK
KL
LK
xx
xxxN
LLLL
KLL
KL
LKLL
KL
Lxx
xxxN
.....(12)
21
420ccc JLMMALM += .(13)
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Contd
The parts of the inertia matrix corresponding to the transverse and rotation
motion, respectively are as follows
( )( )( )
( )
( )
( )
( )
( )
( )( )
( )
( )
+
+++
++
++
+
+
+
+++
+
+
+
=
2
223
2
2
2232
2
223
22
1
16
14819164
4
12388156
)(16
4453163
4
8752
16
14819164
4
875254
4
)12388(156
KL
KLKLLsym
KL
KLLKL
KLKLL
KL
KLL
KL
KLKLL
KL
KLL
KL
LLL
Mc
( )( ) ( )( )
( )( )
+
++
+
+++
++
=
2
22
2
2
22
2
22
22
2
60
1821810
1
5
6
60
18218
10
1
60
1821810
1
5
6
10
1
5
6
KL
KLKLLsym
LL
KLKLKLL
LKLKLKLL
LLLL
Mc
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Contd
The above inertia matrix contain element which are functions of coefficient of
flexibilityKat the crack location. By this, only the terms corresponding to therotational degrees of freedom and those which couple rotational and
translational degrees of freedom are only affected
WhenK=0, the matrix takes the form of the matrix representing a non-cracked
element The stiffness matrix an be represented as
Here alsoK=0 represents the stiffness matrix of a non-cracked element
( ) ( )
( )
+
++
+
+++
++
=
KLL
KLKLsym
LL
KLL
KLKL
LKLL
KLKLLLLL
EJK
22
23
22
2
22
2222
364
612
3626364
612612
..(14)
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NUMERICAL EXAMPLES TO CHECK THE
VALIDITY OF THE PROPOSED MODEL
Investigates influence of the grid density on the first three natural frequencies
of a cantilever beam with a crack
Beam data: length 1 m, height 0.01 m, width 0.01 m, Young's modulus 2.11011
Pa, density 7860 kg/m
3
The crack was located in the middle of the beam length. The depth of the crack
was varying in the range 0.00.6 and no plastic zone around the crack tip was
assumed here
For the non-cracked beam the first free natural frequencies analytically
calculated were: 52.463, 328.741, 920.641 rad/s
In this example several grids: 5 elements, 9 elements, 13 elements and 17
elements were tested
EXAMPLE I
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Table 1. Changes in the first natural frequency (rad/s)
as a function of the element grid density and the crack
depth
Table 2. Changes in the second natural frequency
(rad/s) as a function of the elements grid density and
the crack depth
Table 3. Changes in the third natural frequency (rad/s)
as a function of the elements grid density and the crackdepth
These results show that even for
relatively deep cracks changes in
natural frequencies of the beam
investigated, do not depend on the
grid density
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EXAMPLE II
Shows how the inertia matrix form influences natural frequencies of cracked
beam-like structures Numerical calculation in this example was carried out for a cantilever beam
with dimensions the same as in the example 1
The crack was located 0.05 m from the fixed end of the beam and the Y/
ratio was 0.8 Inertia matrix form does not influence natural frequencies of the cracked beam
under investigation. Thus it is possible and convenient to use in dynamic
analysis of beam-like cracked structures the inertia matrix in the same form as
for non-cracked structures
Table 4.
Relative natural
frequencies of the
cracked beam for
two forms of theinertia matrix
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EXAMPLE III
Illustrates the validity of the proposed model of the cracked beam element
Changes in natural frequencies obtained by applying the elaborated element
are compared to results of experimental investigation and analytical calculation
It is assumed here that the plastic zone around the crack tip does not exist
The calculation in this example was carried out for the following beam data:
length 0.4 m, height 0.02 m, width 0.01 m, Young's modulus 2.11011 Pa,
density 7860 kg/m3
The crack was located 0.08 m from the fixed end of the beam. The depth of the
crack was varying in the range 0.00.6
The beam was modelled by 20 finite elements
Fig. 3. The comparison of changes in the
first bending natural frequency as a
function of the crack depth betweenanalytical, experimental and FEM models
The compatibility obtained between the
results of the experimental investigation
and the results of FEM analysis is much
better than in the case of the analyticalmodel
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EXAMPLE IV
Shows the influence of the plastic zone around the crack tip on the intensity of
changes in natural frequencies The calculations were done with the following beam data: length 1.0 m, depth
0.05 m, width 0.01 m. The material data were: Young's modulus 2.11011 pa,
density 7860 kg/m3
The beam was modelled by 20 finite elements
Fig. 4, Fig. 5 & Fig. 6 shows that the fall in natural frequencies depends not
only on the crack location but also on the size of the plastic zone around the
crack tip
Fig. 7, Fig. 8 & Fig 9 clearly denotes that the fall in natural frequencies depend
on from the crack depth and the size of the plastic zone around the crack tip
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Fig. 4 Fig. 5 Fig. 6
Fig. 7 Fig. 8 Fig. 9
The relative changes of the first bending natural frequency as a function of the crack location and the ratio Y/
The relative changes of the first bending natural frequency as a function of the crack depth and the ratio Y/
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CONCLUSION
A beam finite element with a transverse, one-edge, open, elasto-plastic crack
for static, dynamic and stability analysis of cracked beam-like structures has
been developed in this paper. The model has been successfully employed to
investigate the influence of cracking on changes in natural frequencies of acantilever beam. Numerical examples presented in this paper allow the
following conclusions to be drawn:
1. Crack tip plasticity increases the predictive capabilities of the model.2. From a numerical point of view the form of the inertia matrix does not affect
natural frequencies of the beams analysed. Because of this the inertia matrix in
the same form as for a non-cracked structure can be applied.
Using the method described in the paper it is possible to calculate the full form
of the local flexibility matrix, which also takes into account: axial, torsion and
shear deformations of the element