Engineering, 2013, 5, 56-60
http://dx.doi.org/10.4236/eng.2013.51009 Published Online January 2013 (http://www.scirp.org/journal/eng)
Analyzing Some Behavior of a Beam with Different Crack
Positions Transversely inside It
Behrooz Yazdizadeh
Department of Mechanics, Kerman Branch, Islamic Azad University, Kerman, Iran
Received October 4, 2012; revised November 6, 2012; accepted November 25, 2012
ABSTRACT
Vertical displacement, critical Euler buckling load and vibration behavior of a cracked beam are considered in this re-
search. The crack inside the beam is placed in different positions and results compared for each crack position. On first
Eigenvalue of free vibration results, there is a border that first Eigenvalue of free vibration does not change if center of
crack is located on that border, and after that border, the first Eigenvalue of free vibration is increased that is a counter-
example relation of critical Euler buckling load and first Eigenvalue of free vibration.
Keywords: Crack; Mesh; Vibration; Euler Buckling; Displacement
1. Introduction
Fracture mechanics, was originated by Wieghardt and
Inglis [1]. Both independently showed that cavities and
flaws in continuum materials act as stress concentrators
which, in the limit of sharp edges (cracks), produce infi-
nite stress at the tip [2]. A fairly thorough description of
the approaches for solving the crack problems is made by
many researchers [3-6]. These were the first attempts to
bring closer the theories of fracture mechanics (FM) and
continuum mechanics (CM). About the same time, the
Finite Element Method (FEM) and digital computers
dashed into the engineering community as a gifted means
for quantifying solutions in structural and solid mechan-
ics. Naturally, fracture mechanic researchers implement-
ed their FE methods, while continuum mechanic re-
searchers implemented theirs [7].
Over the years, the finite element technique has been
so well established that today it is considered one of the
best methods for solving a wide variety of practical
problems efficiently and is more and more feasible, pro-
vided that finite element (FE) models can be shown to be
correct [8]. It can be used for the solution of any problem
simply by changing the input data [9].
A lot of effort has been spent in the last 30 years to
investigate and treatment the observed drawbacks of (FE)
method [10] and Finite element analysis is one of the
usual ways to solve crack problems.
In this research, some beam characteristic behavior
comparing with each other in different crack Positions
inside the beam. These comparing consist of maximum
vertical displacement under simple bending, first mode
first Eigenvalue of free vibration and critical Euler buck-
ling load.
Critical Euler buckling load for single beam-columns
can be evaluated from analytical expressions. These so-
lutions, as given for various boundary conditions, follow
the first analytical method given by Euler [11] for pre-
dicting the reduced strengths of slender columns. The
finite elements method can be easily implemented for
beam elements without cracks since the stiffness and
generalized geometrical stiffness matrices of a non-
cracked beam are already commonly known (for example
in [12]). However, the situation essentially changes if the
structural elements are transversely cracked. Two dimen-
sion (2D) or three dimension (3D) finite element ana-
lyzes must be implemented in order to achieve a com-
plete model of the structure. When studying the elastic
Euler buckling load of a column, it is necessary to deter-
mine the maximum load at which the structure remains in
equilibrium at the deformed position.
Stress intensity factor, normal, and shear stresses (at the
tip of the crack) are calculated for simple cracked beam
under vertical pressure before [13].
Here we examine changes of some parameters of a
beam with different crack position inside of it for con-
clusion that whether crack position has an important role
to change the vertical displacement, first Eigenvalue of
free vibration and critical Euler buckling load or not.
2. Material and Methods
Consider a beam as shown in Figure 1. This beam
loaded with a vertical load at end of it then calculating
C
opyright © 2013 SciRes. ENG

B. YAZDIZADEH 57
P
Figure 1. Beam under loading.
the Maximum vertical displacement at the end of it.
These procedures do again with applying a horizontal
crack inside the beam as shown in Figure 2. Also again
when crack moves along the beam vertically and hori-
zontally. They are fifteen Positions that consider for
crack. For each position, vertical displacement, first Ei-
genvalue of free vibration and critical Euler buckling
load are determined and compared with each other.
The beam has 2 meter (m) length and 0.12 m width
with thickness of 0.12 m. (Young’s modulus) E = 2 ×
1012 Pa, (Poisson’s ratio ν = 0, (density) D = 7800 kg/m3
and (load at the end of the beam) P = 10 kN.
Crack parameters are as follows:
Length = 0.38 meter;
Depth = through the beam;
Space between surface = 0.002 meter;
Radius of crack at tip = 0.002 meter.
Center of crack locations are as follows:
Horizontally (x dimensions—fixed end of the beam is
the origin of axes):
1) 0.2 m; 2) 0.6 m; 3) 1 m; 4) 1.4 m; 5) 1.8 m.
Vertically (y dimensions—lower surface of the beam is
the origin of axes):
1) 0.02 m; 2) 0.06 m; 3) 0.1 m.
The displacement formula for a beam [14]:
4
40
y
EI x

 (1)
where E is module of elasticity; I is moment of inertia; y
is vertical displacement of a point in x position.
The solution of above differential equation with
boundary conditions is as follows (See Figure 1):

323
23
6
p
y
LLxx
EI

The maximum displacement (at x = 0) is
3
3
PL
EI



.
There are several ways to calculate the beam fre-
quency. Relation (2) is obtained from Rayleigh approxi-
mation method and the error of this approximated solu-
tion is less than 0.5% [15].
4
1
2π
EIg
fk
L 

P
Figure 2. Crack positions inside the beam.
where f igt per
nit lenhooses
is first mode lateral frequency; ω is we
gth; g is the volume coefficient that is c
h
u
here to be equal to 1 and k is the constant that is equal to
3.53.
The critical first Eigenvalue-Euler buckling load (Pcr)
of a beam is determined from Equation (3) [14]:
2
2
π
cr
EI
PL  (3)
where E is module of elasticity; I is moment of inertia
and L is the length of a beam.
Tables 1-3 show maximum vertical displacement; first
If there is no crack inside the beam then from relations
(1)-(3), maximum displacement vertical displacement,
vibration and first Eigenvalue-Euler buckling load of the
beam can obtained that they are 7.72 × 10–4 m, 77.91 s–1,
21.30 × 103 kN, respectively.
For every location maximum vertical displacement,
first mode vibration and first critical Euler Euler buck-
ling load are obtained.
To reduce the error of meshing (because for each new
crack location we must mesh the beam again, which is
different from previous one), whole shape of the beam
does not mesh at once. The beam divided into 15 equi-
valent parts that are shown in Figure 3. One part is for
crack element and another are normal beam parts. Hence,
a part of the beam, which includes crack, can be moved
to different location. Other divided parts of the beam
could be moved to different location too. In this condi-
tion, meshing of the whole area of the beam is constant
for every crack position.
Note that in ANSYS software we cannot model the
crack with two crack tips. Only one tip is allowable. To
solve this problem, half of the crack is modeling and
meshing (as shown in Figure 4) then with mirror option
in ANSYS software, copy it to another part of the crack
(see Figures 5 and 6).
After each modeling, all nodes and elements on the
borders (edge) of each part must be merging to each
other. Then construction of the whole beam is complete.
After modeling of each crack position and apply boun-
dary conditions, maximum vertical displacement, fre-
quency and Euler Euler buckling of the beam are calcu-
lated and analyzed.
Now we consider the results of this problem.
3. Results and Discussion
(2)

B. YAZDIZADEH 58
Figure 3. Beam divided to 15 parts (each part meshes se-
parately). Figure 5. Total crack modeling in ANSYS software.
Figure 4. Half of crack modeling in ANSYS software (crack
part of the beam is specified in right).
Figure 6. Total crack modeling in ANSYS software (crack
part of the beam is bounded).
Table 1. Vertical maximum displacement data (m).
Center of crack positions horizontally
Simple beam
(without crack)
Center of crack
positions vertic1.4 1.8 ally 0.2 0.6 1
0.1 7.79 × 10−3 7.77 × 10−3 7.75 × 10−3 10−3 7.76 × 10−3 7.75 ×
0.06 7.78 × 10−3 7.78 × 10−3 7.78 × 10−3 7.78 × 10−3 7.79 × 10−3 7.73 × 10−3
0.02 7.79 × 10−3 7.77 × 10−3 7.76 × 10−3 7.75 × 10−3 7.75 × 10−3
le 2. FirstEigenee vibration data (1/s).
Center of crack positions horizontally
Tab mode first value of fr
Simple beam
(without crack)
Center of crack positions
verti 1.4 1.8 cally 0.2 0.6 1
0.1 77.013 77.1577.583 3 77.29 77.435
0.06 7 7 77.5 7 7 6.8336.86017.2797.564
77.418
0.02 77.013 77.153 77.29 77.435 77.583
Table 3. Fler bucklical load data (N).
Center of crack positions horizontally
irst Euing crit
Simple beam
(without crack)
Center of crack
positions vertica1.4 1.8 lly 0.2 0.6 1
0.1 2.12 × 107 2.12 × 107 2.12 × 107 107 2.12 × 107 2.12 ×
0.06 2.13 × 107 2.12 × 107 2.12 × 107 2.11 × 107 2.10 × 107 2.13 × 107
0.02 2.12 × 107 2.12 × 107 2.12 × 107 2.12 × 103 2.12 × 107
mode vibration and first critical Euler load
spectively and Figures 7-10 are showing these data on
nt tnd om firstue
of free vibration increased up to near the first Eigenvalue
fixed poioward the ef the bea Eigenvalbuckling
re
diagrams (graphs).
Note that Figures 7 and 8 are equivalent. For having
the best view of diagrams, (2D) view is chosen instead of
(3D) view.
In Figure 8, when crack is not placed at the center and
moved horizontally from fixed point toward the end of
the beam displacement decreased. However, when crack
is placed at the center, there is some little increasing of
displacement.
In Figure 9, when crack moved horizontally from
of free vibration of the beam without crack. When crack
moved vertically first Eigenvalue of free vibration in-
creeasd. It means that when there is a crack near the
edges of a beam, first Eigenvalue of free vibration is
more than when crack is located in the middle of the
beam.
In Figure 10, when crack position, is at the center and
moved horizontally from fixed point toward the end of
the beam critical load reduced, meaning that the system
is more unstable.

B. YAZDIZADEH 59
0.1
0.77
0.775
0.78
No crack
0.2 0.6
1
1.4 1.8
0.1
0.06
0.02
Figure 7. Vertical maximum displacement diagram (3D
view) (×10−3 m).
0.77
0.772
0.774
0.776
0.778
0.78
0.1
0.06
0.02
Figure 8. Vertical maximum displacement diagram (2D
view) (×10−3 m).
0.722
0.724
0.726
0.728
0.73
0.732
0.734
0.736
0.1
0.06
0.02
Figure 9. First mode first Eigenvalue of free vibration dia-
gram (2D view).
2080
2090
2100
2110
2120
2130
buckling
0.1
0.06
0.02
Figure 10. First buckling critical load diagram (2D view)
(×104 N).
There is more stability for beam when crack not placed
at the center of the beam except near the fixed point (left
of the beam). In this chart, we show only (2D) display
ck is placed at the center and moved hori-
ontally from fixed point toward the end of the beam first
Eu
instead of (3D) chart for better comparing. In addition,
when cra
z
ler buckling critical load is decreased. However, when
crack is not placed at the center, there is some little
increasing of first Euler buckling critical load from fixed
point toward the end of the beam.
In general discussion; different shape of characteristic
behavior of the beam when crack positioned at the center
may be for that the center is located in the neutral line of
Figure 11. First Eigenvalue of free vibration border for the
beam.
the beam that there is no any tensile stress on it or
displacement.
Percentage different of Maximum value of crack posi-
t means insignificant distinct for all condition that
iscussed:
free vibration different: 0.754%;
ment increases when
cr o
y-n
th in the center). How-
alue of free vibration when the crack
beam end, first Eigenvalue of free
de
) more than that one from the beam without
cr
and first Eigenvalue of free vibration.
[1] C. E. Inglis, “Stresses in a Plate Due to the Presence of
tion with a non-crack beam shows the different below
5% tha
d
Euler buckling different: 1.14%;
First Eigenvalue of
Displacement different: 0.712%.
4. Conclusions
Results show that vertical displace
ack is located in the center of the beam (respect t
kling is decreasing iaxes) and critical load Euler buc
at position (when crack is located
ever, in first Eigenv
is moved near the
vibration increased more than the beam first Eigenvalue
of free vibration when there is no crack inside it. It can
be conclude that on first Eigenvalue of free vibration
results, there is a border that first Eigenvalue of free
vibration does not change if one crack is located on that
border, and after that border, the first Eigenvalue of free
vibration is increased. The border is shown in Figure 11.
Vertical displacement is increased or constant when
crack is moved from the fixed pint to free end of the
beam (left to right) when crack is located near the edge
of the beam but when crack is located in the center of the
beam height and again crack is moved from the fixed
pint to free end of the beam vertical displacement is
crease.
First Euler buckling critical load diagram shows a
same result as first Eigenvalue of free vibration, except
that for all crack location critical loads are less that the
beam without cracks, however first Eigenvalue of free
vibration is increased (after the border that shows in
Figure 11
acks.
Finally the border in the beam shows that after that the
first Eigenvalue of free vibration is increased more than a
simple beam without crack but critical Euler buckling
load is decreased lower that a simple beam without crack
that is a counterexample relation of critical Euler buck-
ling load
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