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ACKNOWLEDGEMENT
Thanks go to Bahir Dar University Engineering Faculty for sponsoring my education. Also
a special thanks to the head, Solomon T/Mariam, and all the staffs of the Mechanical
Engineering Department for their kind and unforgettable collaboration during study period.
I really give thanks to my advisor, Dr. Alem Bazezew, for the inspiration and
encouragement to work on this project. I also appreciate not only for his professional,
timely and valuable advices, but also for his continuous scheduled follow up and valuable
comments during my research work. I can say that without his guidance I may not be the
one finalize this project soon enough.
It is really hard to skip many thanks to friends and family who were always with me in bliss
and despair. A special thanks goes to all my family members and friends: Korbaga Fantu,
Birhane Hagos, Seifu Admasu, Yoseph Alemu, Melkam Tegegn, Dereje Engda and all
members of Applied Mechanics stream. Also I would like to thank Nebil Mohammed,
Fikrea and Tamrat for giving me valuable reference materials specially at the beginning of
my research.
Generally, I would like to extend my gratitude for all the above people and those who are
not mentioned here but contributed their part a lot towards the success of this research.
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TABLE OF CONTENTS
ACKNOWLEDGEMENT .................................................................................................... i
TABLE OF CONTENTS .....................................................................................................ii
LIST OF FIGURES..............................................................................................................v
LIST OF TABLES..............................................................................................................vii
NOTATION....................................................................................................................... viii
ABSTRACT.........................................................................................................................xii
ABSTRACT............................................................................................................................i
1. INTRODUCTION ........................................................................................................ 1
1.1 Overview and Objective of the thesis.....................................................................1
1.2 Literature Review ...................................................................................................3
1.3 Organization of the Thesis......................................................................................7
2 FORMULATION OF CRACK MODELING............................................................9
2.1 Introduction.............................................................................................................9
2.2 Modeling of Crack................................................................................................10
2.2.1 Modes of Fracture.........................................................................................10
2.2.2 The Stress Intensity Factor ...........................................................................11
2.2.3 The J-Contour Integral..................................................................................13
2.2.4 Castiglianos Theorem..................................................................................16
2.2.5 Crack Modeling ............................................................................................16
3 EULER- BERNOULLI BEAM .................................................................................21
3.1 Euler- Bernoulli Beam formulation......................................................................21
3.2 Finite Element Method .........................................................................................22
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3.3 Critical load selection ...........................................................................................30
3.4 Establishment of Element Stiffness Matrix for Cracked Element ........................31
4 TIMOSHEKNO BEAM............................................................................................. 35
4.1 Timoshenko Beam Formulation ...........................................................................35
4.2 Isoparametric Element ..........................................................................................37
4.3 Establishment of matrix for cracked beam element..............................................45
4.4 Assembly of Element Matrices and Derivation of System Equation ...................52
4.5 Algorithm of assembly procedure.........................................................................56
5 THE COMPUTER PROGRAMMING....................................................................57
5.1 Program Algorithm...............................................................................................58
5.1.1 For Euler-Bernoulli.......................................................................................58
5.1.2 For Timoshenko Beam..................................................................................59
5.1.3 Program Algorithm for Graphical User Interface (GUI)..............................60
5.2 The Graphic User Interface Program....................................................................61
6 RESULT DISCUSSIONS...........................................................................................65
6.1 Comparison of Timoshenko and Euler-Bernoulli beams. ....................................65
6.2 Effect of crack position as a function of crack depth ratio (for Tim. Beam)........68
6.3 Effects of mass on beam.......................................................................................69
6.4 Effects of crack and mass on mode shape ............................................................72
6.5 Result comparison for Timoshenko Beam............................................................75
7 CONCLUSION ........................................................................................................... 78
8 FUTURE OUTLOOK ................................................................................................ 81
REFERENCES ...................................................................................................................83
APPENDICES.....................................................................................................................93
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Appendix I. Program For Euler-Bernoulli Beam..............................................................93
Appendix II. Program For Timoshenko Beam ..............................................................103
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LIST OF FIGURES
Fig 2-1 The three modes of fracture .....................................................................................11
Fig 2-2 Arbitrary contour around the tip of crack ................................................................13
Fig 2-3 Loaded beam element with transverse crack ...........................................................17
Fig 3-1 Euler-Bernoulli beam element .................................................................................21
Fig 3-2 A two node beam element........................................................................................24
Fig 3-3 Deformation of an Euler Bernoulli Beam ................................................................24
Fig 3-4 A cantilever beamwith one end clamped and a concentrated mass attached at the
other. .............................................................................................................................29
Fig 3-5 Shear force and bending moment diagram...............................................................31
Fig 3-6 Schematic representation of an element with a crack. .............................................34
Fig 4-1 Deformation of a Timoshenko Beam.......................................................................35
Fig 4-2 Two Node Linear Element. ......................................................................................38
Fig 4-3 Linear Shape Functions............................................................................................39
Fig 4-4 Linear Element in the natural Coordinate system....................................................41
Fig 4-5 Deformation of beam including shear......................................................................46
Fig 4-6 Cross-section of a beam ...........................................................................................47
Fig 4-7 Beam with two elements. .........................................................................................53
Fig 5-1 The program algorithm for Euler-Bernoulli beam...................................................58
Fig 5-2 The program algorithm for Timoshenko beam........................................................59
Fig 5-3 The program algorithm for the GUI program ..........................................................60
Fig 5-4 Front page of GUI....................................................................................................61
Fig 5-5 The input Window for Euler-Bernoulli beam..........................................................62
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Fig 5-6 The Output window for Euler-Bernoulli beam. .......................................................63
Fig 5-7 The input window for Timoshenko beam. ...............................................................63
Fig 5-8 The output window for Timoshenko beam..............................................................64
Fig 6-1 Error analysis for comparison of Timoshenko and Euler-Bernoulli beams.............67
Fig 6-2 Fundamental (first) frequency ratios for different crack positions. .........................69
Fig 6-3 The changes of the first natural frequencies as a function of the crack depth at
element seven, a) for Timoshenko beam, b) for the Euler-Bernoulli beam..................71
Fig 6-4 Mode shape graphs without mass: a) for Euler-Bernoulli beam with crack
(continuous line) and without crack (dash line), b) for Timoshenko beam with crack
(continuous line) and without crack (dash line)............................................................72
Fig 6-5 Mode shape graphs with mass: a) for Euler-Bernoulli beam with crack (continuous
line) and without crack (dash line), b) for Timoshenko beam with crack (continuous
line) and without crack (dash line). ..............................................................................73
Fig 6-6 Mode shape graphs without mass for second mode shape: a) for Euler-Bernoulli
beam with crack (continuous line) and without crack (dash line), b) for Timoshenko
beam with crack (continuous line) and without crack (dash line). ...............................73
Fig 6-7Mode shape graphs with mass for second mode shape: a) for Euler-Bernoulli beam
with crack (continuous line) and without crack (dash line), b) for Timoshenko beam
with crack (continuous line) and without crack (dash line)..........................................74
Fig 6-8 Deviation of first mode shape due to crack for Timoshenko beam without mass. ..74
Fig 6-9 Deviation of first mode shape due to crack for Euler-Bernoulli without mass........75
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LIST OF TABLES
Table 2-1...............................................................................................................................12
Table 6-1Geometry and Property of Timoshenko beam and Euler-Bernoulli beams ..........66
Table 6-2 Comparison of the first three natural frequencies of Timoshenko beam and Euler-
Bernoulli for various L/h ratios. ...................................................................................66
Table 6-3 Geometry and Property of Timoshenko beam .....................................................68
Table 6-5 Determination of Natural Frequencies with different crack depth ratio at
element 7 for Timoshenko ........................................................................................70
Table 6-6 Determination of Natural Frequencies With different crack depth at
element 7 for Euler-Bernoulli beam. .......................................................................70
Table 6-7 For the First Natural Frequency at e/L=0.4..........................................................76
Table 6-8 For the Second Natural Frequency at e/L=0.4 .....................................................76
Table 6-9 For Third Natural Frequency at e/L=0.4 ..............................................................76
Table 6-10 For the First Natural Frequency at e/L=0.6........................................................76
Table 6-11 For the Second Natural Frequency at e/L=0.6 ...................................................77
Table 6-12 For the Third Natural Frequency at e/L=0.6 ......................................................77
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NOTATION
A Cross sectional area
z , z1 Distance from the neutral axis to the centroid of an area
b Width of beam
C0, C1, C2, and C3Arbitrary constants
E Modulus of elasticity for plane stress
e nth
element
E Modulus of elasticity for plane strain
F Form factor
G Crack driving force.
G Shear modulus
h Height of beam
H1,H2Linear shape functions for Timoshenko beam
I Moment of inertia
J Strain energy density function (SEDF).
KE Kinetic energy
Ki Stress intensity factor for different modes of fracture, for i=I, II, and III
L Total length of the beam
l Element length of beam
M Bending moment
m Mass per unit of beam length
Ml Lamped mass
n Number of elements for the beam
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nel Total number of elements
Ni Shape function for Euler Bernoulli beam
P1 Axial load
P2 Shear force along z-axis
P3 Shear force along y-axis
P4 Bending Moment about y-axis
P5 Bending Moment about z-axis
Pi Applied load (force or bending moment)
Q First moment
q(x, t) Externally applied pressure loading.
r Gyration radius of the cross section
R Weigh residual
s Arc length
sdof Total nodal degree of freedoms
U The total strain energy.
V Shear forces
v(x, t) Transverse displacement
wi Test function
Characteristic stress
An arbitrary counter-clockwise path around the crack tip of a crack
Characteristic crack dimension
eF Element force vector
eM Element mass matrix for Euler Bernoulli beam
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eK Element Stiffens matrix for Euler-Bernoulli beam
Shear stress
Slop (Angular displacement)
Correction factor for shear energy
e Element domain
{ }d Modal shape for Euler-Bernoulli beam
{ } Modal shape for Timoshenko beam
Rotational angle of cross-section
Shear angle
w Strain energy
( )oTU Total strain energy for Timoshenko beam
Transverse shear strain
q Vector of sdofnodal degrees of freedom
[ ]T Transfer matrix
iu Displacement component
Mass density per length
21 , Natural coordinate for Isoparametric element
eRM Rotary Inertia
ij Strain tensor
i The angular frequency for Euler-Bernoulli beam
iT Angular frequency in radians per second for Timoshenko beam
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( )oTijc Compliance of Timoshenko beam without crack
[ ]ekk Element expanded characteristic matrix
ij
c Local flexibility
Poisson ratio
[ ]cK Stiffness matrix of the cracked element for Euler Bernoulli beam
[ ]cTK Stiffness matrix of the cracked element for Timoshenko beam
)0(
ijc Total flexibility coefficient matrix for an element without crack
( )1ijc Total flexibility coefficient matrix for cracked beam
ip Traction load
eTM Translating inertia
bU Bending strain energy
sU Shear strain energy
e
bK Stiffness matrix for bending strain energy
e
sK Stiffness matrix for shear strain energy
Mass density
ij Stress tensor
l,mThe global degree of freedom
[kk] Assembled stiffness matrix
[mm] Assembled mass matrix
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ABSTRACT
Beams are widely used as machine elements and structural elements in civil, mechanical,
naval and aeronautical engineering with quite complex design features. These machine and
structural elements are designed with more care for different load conditions, with good
range of safety factors, and are inspected regularly. Still there are unexpected sudden
failures.
In order to attain the maximum reliability of machinery and structures, there is no way
except monitoring the health of susceptible critical components. This leads to continuous
gathering of information of changes in their static and/or dynamic behavior.
The main objective of this thesis is to develop a method for the investigation of cracked
beam behavior of a Timoshenko beam under different conditions such as orientation of
crack, size of crack and inclusion of additional mass. Moreover, the results have been
compared with Euler-Bernoulli beam. The methods, formulation and results obtained can
be used to understanding the behavior of a cracked beam structure.
The results obtained are compared with other published results. The comparison shows that
the method used in the thesis is eligible to investigate the behavior of cracked Timoshenko
beams under different loading conditions.
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ABSTRACT
Beams are widely used as machine elements and structural elements in civil, mechanical,
naval and aeronautical engineering with quite complex design features. These machine and
structural elements are designed with more care for different load conditions, with good
range of safety factors, and are inspected regularly. Still there are unexpected sudden
failures.
In order to attain the maximum reliability of machinery and structures, there is no way
except monitoring the health of susceptible critical components. This leads to continuous
gathering of information of changes in their static and/or dynamic behavior.
The main objective of this thesis is to develop a method for the investigation of cracked
beam behavior of a Timoshenko beam under different conditions such as orientation of
crack, size of crack and inclusion of additional mass. Moreover, the results have been
compared with Euler-Bernoulli beam. The methods, formulation and results obtained can
be used to understanding the behavior of a cracked beam structure.
The results obtained are compared with other published results. The comparison shows that
the method used in the thesis is eligible to investigate the behavior of cracked Timoshenko
beams under different loading conditions.
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1. INTRODUCTION
1.1 Overview and Objective of the thesis
Now a days sophisticated structures and machinery parts are constructed by using
metallic beams. Beams are widely used as structural element in civil, mechanical, naval,
aeronautical engineering. During the time leading to World War, every structure and part of
machinery were designed based on the tensile strength of a material. However, unforeseen
failure had been frequently observed. One of the major disasters of structural failure was
the sinking of Liberty Ships. These ships were participating in the war. Though they were
designed well, they collapsed without any external force. After careful investigations, the
cause of failure was determined to be fracture of components. And that was the main reason
for an introduction of fracture mechanics. Due to this new design concept, substantial
improvement the life of machinery and saving was observed.
In structures and machinery, one undesirable phenomenon is crack initiation in which the
impact cannot be seen overnight. Cracks develop gradually through time that lead finally to
catastrophic failure. Therefore, crack should be monitored regularly with more care. This
will lead to more effective preventive measure and ensure continuous operation of the
structure and machine.
In order to investigate the behavior of cracks in structures and machinery, there are
different methods like ultrasonic inspection, X-ray inspection, experimental method, Eddy
current inspection, etc. However the above methods require high cost and time even if they
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are easy to apply them. Moreover, most of them are limited to detection of cracks. So it is
better to establish a new method for simple geometric structures that helps to see the
behavior of cracked beam element using Finite Element Method, FEM, based on vibration
analysis. When cracks are predicted using this method, time and money will be saved.
Using FEM method based on vibration analysis we can observe the effects of inclusion and
orientation of crack on the natural frequency of the beam, since the presence of crack
reduce the system natural frequency of the beam. Most of the beams in the structures and
machinery have mass so that the effects of additional mass attachment on the cracked beam
will be investigated.
Therefore the main objective of this thesis is to develop a method for investigation of
cracked beam behavior for Timoshenko beam under different conditions such as orientation
of crack, size of crack and inclusion of additional mass. Moreover, the results have been
compared to results obtained for Euler-Bernoulli beams. The results obtained can be used
for determining behavior of cracked beams which can eventually be used for prediction of
cracks in beams
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1.2 Literature Review
The tendency to monitor a structure and detect damage at the earlier stage is pervasive
throughout the civil, mechanical and aerospace engineering fields. Most currently used
damage investigation methods are included in one of the following categories: visual or
localization experimental methods such as ultrasonic method, magnetic field methods,
radiography, eddy-current method and etc. All of these experimental techniques require
that the vicinity of damage be known a priori and that the portion of the structure being
inspected be readily accessibly.
The need for quantitative global damage investigation and detection method that can be
applied to complex structure has led to the development and continued research of
methods, which examine change in static and dynamic characteristic of the structure. In this
literature review, different ways of investigation of crack behavior will be discussed.
To study the behavior of cracked beam, in the past decade researchers have used open and
closed (breathing) crack model in their studies. In 1970s, Dimarogonas and Chondros [26]
used local flexibility matrix to simulate the stiffness of the shaft system with opening crack.
Also Maiti[76], Tsai et. Al.[80] and Ostachowitwz et. Al.[84] assumed in their work that
the crack in a structural element is open and remains open during vibration. Such an
assumption give an advantage to avoid the complexities that results from the non-linear
characteristics presented by introduced a breathing crack.
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On the other hand, different researchers have implemented closed crack model in their
work for investigation of crack behavior. Among them, Chondros and Dimarogonas [29],
Rivola and White [15], Dimarogonas and Paipetis [36], and Shen[62] dealt with closed
crack model. Dimarogonas and Paipetis [36] devoted almost one chapter to the
discussion of the dynamic response of structural members with variable elasticity
including for closing cracks. Also, Rivola and White analyzed the behavior of crack based
on closed crack model and they have done experimental test to show the effectiveness of
their method. Even if all the above researchers did their work on closed crack model, they
didnt show the effectiveness of their method with respect to open crack model. However,
the application of open and closed crack models depend on different condition such as
static and dynamic load conditions.
To study the behavior of crack in the structures, vibration parameter like compliance,
mechanical impedance and damping factors have played great roll. The presence of crack
in the structure affects directly or indirectly these vibration characteristics. Specifically, the
eigen frequency and mode shape of structures are changed from their original value due to
an inclusion of a crack. That is why many researchers focus on these parameters to
investigate the behavior of crack. Pandey [13] investigated the behavior of crack related
with curvature mode shape of structure. He has shown that the absolute change in the
curvature made shapes are localized in the region of damage and hence can be used to
analyzed the damage in the structure. He proposed an experimental method to verify his
work.
Y. Bamnios, E. Douka and A. Trochidis [89] used mechanical impedance model in order
to investigate the crack behavior and to predict the damaged zone. They investigated the
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effect of a transverse surface crack on the mechanical impedance both analytically and
experimentally. However, their method lacks accuracy for smaller damage.
Qian, Du and Jiang [43] have derived an element stiffens matrix of a beam with crack from
an integration of stress intensity factors and then established a finite element model of
cracked beam even though they didnt consider additional mass on their model. The
results that they have obtained analytically agree quite well with the experimental data.
Several methods were used to deal with the behavior of crack in the structure. Zheng et.
al. [22] used modified Fourier series to investigate the response of natural frequencies of
cracked beam. However, their method is applied only for standard linear eigen value
equation. T.G. Chondors and Dimarogonas studied the dynamic sensitivity of structure to
cracks using Rayleight principle. As per their conclusion, the method reduced the
computational effort needed for the full eigen solution of cracked structures and gave
acceptable accuracy. Also different researchers have used Finite Element Method (FEM)
for solving the problem related with crack behavior. Among them, Pandey et. al.[13],
Sekhar et. al. [17], Qain et. al. [43], Sinha et. al.[48], Chinchalkes[68], Maiti et. al. [73],
Ostachowitcz et. al.[85], Matijaz[5], G.D. Gouanaris and C.A. Papadoulso and also A.D.
Dimarogonas [8], P. G. Nikolakopouloz, D. E. Katsreas and C.A. Papadopouloss[9]. All
of them confirmed that their results are very close to the experimental methods. Matijaz
presented a generalization of a simple mathematical model based on FEM for transverse
motion of a beam with crack. However, he didnt show the effectiveness of his method by
comparing with other methods. Moreover, mass wasnt considered on his model.
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Many investigators have studied the problem of crack detection in rotating shaft in the last
three decades. A.D Dimarogonas and C.A. Papadopoulos [10], [11], [91], [92], [93], have
investigated the behavior of crack on rotating shaft. In [98], they considered the system to
be bi-linear. A de Laval rotor with an open crack was investigated by way of application of
the theory of shafts with dissimilar moment of inertia. Furthermore, they found analytical
solution for the closing crack under the assumption of large static deflection, which is a
situation common in turbomachinery. In [11], they investigate the coupling of longitudinal
and bending vibration of rotating shaft, due to an open transverse surface crack. Also in
another next work [93], the coupling of vibration modes of vibration of a clamped-free
circular cross section of Timoshenko beam with a transverse crack was investigated.
On the Timoshenko beam, different investigators have used various approaches to
investigate the crack behavior. Among them are S.P. Lene and S.K. Maiti [73], Zhou, and
Y.K. Cheung [6], and M. Kisa, J Brandon and M Topeu [48]. S.P. and S.K. Maiti, [73],
used both method forward (determination of frequency of beam knowing the crack
parameter) and inverse (determination of crack knowing the natural frequency). They
give numerical and experimental demonstration in order to illustrate the effectiveness of
their methods of accuracy and it is quite encouraging. Also M Kisa, J Brandon and M
Topeu analyzed the vibration characteristic of a cracked Timoshenko beam by applying
finite element method and component mode synthesis integrating together. To illustrate the
effectiveness of their approach, results had been compared with experimental data and
previous published literature. However, they didnt include additional mass to observe
effect of masseson the cracked beam behavior.
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In this thesis investigation of crack behavior will be dealt by using the Finite Element
Method. In this method, the beam will be divided in to several elements and then by
taking boundary conditions the eigen frequency of the beam will be found. The great
advantage of finding natural frequency isits measurability from the machine and structure
at any single point and easily without dismantling much access requirement. In this thesis
the local flexibility of beam element model follows the approach of Dimarogonas [26, 11].
To avoid the non-linearity of the system, in this thesis work, crack will be modeled based
on open crack model and additional mass will be included. For the sake of verification the
beam model used is the cantilever beam, since many authors have analyzed the cantilever
beam and have got experimental results.
1.3 Organization of the Thesis
This thesis is organized in to eight chapters. In the first chapter, the objective and overview
of the thesis are discussed. Also a literature review is given detailing information about
investigations and methods of analysis of cracked beams and their behavior, which have
been investigated by different researchers.
In chapter two, mathematical model is developed for cracked beams. Also related
concepts like stress intensity factors, mode of cracks, J-Integral and Castiglianos theorem
are discussed briefly.
In Chapter Three, the equation of motion for an Euler-Bernoulli beam is developed. Using
the governing equation of motion, Finite Element method is implemented for cracked and
uncracked beam elements.
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In chapter four, Timoshenko beam is discussed. In this chapter a mathematical model is
developed based on FEM for cracked and uncracked element. Also related topics, like
isoparametric element and strain energy formulations will be discussed.
In chapter five, a computer programming is developed with the help of algorithms to study
the behavior of cracked beams for two cases: Euler-Bernoulli and Timoshenko beams. In
chapter six, detail discussion of results is presented. Finally, chapter seven gives conclusion
and future outlook.
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2 FORMULATION OF CRACK MODELING
2.1 Introduction
Crack is a problem that society has faced for as long as there have been man-made
structures. The occurrence of crack problems may actually be worse today than in the
previous century, because more can go wrong in our complex technological society.
The cause of crack initiation in structures generally falls in to one of the following major
groups: First, negligence during design, construction or operation of the structure: and
second, application of new design or material, which produce an unexpected results. In the
first case, existing procedures are sufficient to avoid failure, but are not followed by one
or more of the parties involved, due to human error, ignorance, or willful misconduct.
Unskillful workmanship, substandard or inappropriate materials, error in stress analysis,
and operator error are example of where the appropriate technology and experience are
available, but not applied well.
In the second case, the initiation of crack is much more difficult to prevent. For instant,
when an improved design is introduced, there are invariably factors that the designer
may not anticipate. New materials can offer tremendous advantage, but also potential
problems. Consequently, a new design or material should be placed in to service only after
extensive testing and analysis. Such an approach will reduce the frequency of failure due to
crack, but not eliminate them entirely.
To avoid or minimize the structural failure due to the above cases, there are two design
approaches. Those are the strength of material approach and the fracture mechanics
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approach. In the first approach, the anticipated design stress is compared to the flow
properties of a candidate material; a material is assumed to be adequate if its strength is
greater than the expected applied stress. This approach may attempt to guard against brittle
fracture by imposing a safety factor on stress, combined with minimum tensile elongation
requirements on material.
In the second approach, that of fracture mechanics has three important variables: applied
stress, flow size and fracture toughness. In fracture analysis there are two approaches:
energy criterion and the stress intensity approach. In this thesis the stress intensity approach
will be discussed in detail and will be employed to investigate the behavior of cracks in
vibration analysis.
2.2 Modeling of Crack
2.2.1 Modes of Fracture
In cracked structure, the stress field near crack-tips may be one of the three modes of
fracture, Fig 2.1. The opening mode, Mode I, is associated with local displacement in
which the crack surfaces move directly apart, symmetric with respect to the x-y and x-z
plane. The edge-sliding mode, Mode II, is characterized by displacement in which the
cracked surfaces slide over one another perpendicular to the leading edge of the crack,
symmetric with respect to the x-y plane and skew-symmetric with respect to the x-z plane.
Mode III, the tearing mode, finds the crack surfaces sliding with respect to one another
parallel to the leading edge, skew-symmetric with respect to the x-y and x-z planes.
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y x
y x y x
Mode I Mode II Mode III
Fig 2-1 The three modes of fracture
Even if these are the basic fracture modes, most of the time the crack growth usually takes
place in Mode I or close to it [2], especially for member like slender beams [57]. If there is
a load on the structure, due to shear force, Mode II will be considered combined with Mode
I to study the crack behavior. Therefore, Mode I and Mode II will be applied to investigate
the behavior of crack if there is a load on the beam.
2.2.2 The Stress Intensity Factor
The stress intensity factor defines the amplitude of the crack tip singularity. That is stresses
near the crack tip increase proportional to the stress intensity factor. Physically, stress
intensity factor may be regarded as the intensity of load transmittal through the crack-tip
region caused by the introduction of a crack into the body of interest.
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Generally the stress intensity factor is given by
FKI = 2.1
where a characteristic stress
a characteristic crack dimension, and
F is a form factor, which is dimensionless constant that depend on geometry and
mode of loading.
Different researchers have got formulas for stress intensity factor experimentally,
numerically or analytically for various cases, such as for the Center Cracked Test
Specimen, the Double Edge Notch Test Specimen, the Single Edge Notch Test Specimen
and the Pure Bending etc.
Different authors have given different empirical relations and value for ( )h
F , where his
the height of the beam,and some of them are given in Table 2.1
Table 2-1
Person ( )h
F Accuracy
Brown 43
0.148.0.1333.740.1122.1
+
+
=hhhh
0.2% for 6.0
h
Tada
h
h
h
h
2
cos
2sin1199.0923.0
2tan
2
4
+
=
Better than
hanyfor
%5.0
Anderson 1.12
Papadopoulos
( )( )21
32
/1
18.0085.056.0122.1
h
hhh
+
=
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2.2.3 The J-Contour Integral
The J contour integral is the strain energy density function (SEDF). It has enjoyed great
success as a fracture characterizing parameter for nonlinear and linear materials.
Consider an arbitrary counter-clockwise path ( ) around the crack tip of a crack, Fig 2.2.
The J integral is given by [2].
= ds
x
upwdyJ
i
i
i 2.2
Where ijijdwii
= 0 , is the strain energy, i,j =1, 2, 3
ijij and are the stress and strain tensors, respectively.
sis the arc length.
ip is the traction exerted on the boundary and the crack surface.
iu is displacement component
s
y
x
Fig 2-2 Arbitrary contour around the tip of crack
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To implement J-integral in modeling of crack, the following argument plays a great roll.
Let represent the area enclosed by the curve in Fig 2.2 and assume that the curve is
shrunk toward the crack tip ( 0 ). Within this area the gradients are so large (toward
singularities at the crack tip) that they dominate all local derivatives with respect to the
crack length. Thus, the field within 0 will be stationary in the sense that they
mainly translate with the crack tip during a differential crack motion. Give the external
action, when the crack tip moves a small step forward, the changes observed at a fixed
location in will therefore be the same as when the observer moves the same length back
toward the stationary crack. [57]
=
x 2.4
applying to some function of x and with x measured form a fixed origin. Then the
second right-hand term of Eq. 2.2 equals
dsu
pdsx
up ii
i
i
=
2.5
which can be interpreted as the rate of work exerted per unit thickness by the outside
material on the material inside as the crack moves.
Similarly, from Eq. 2.2,
wdy
can be seen as total strain energy carried by particles in to per unit thickness and crack
advance when that region move the crack tip.. The sum J will therefore represent a net
expenditure of mechanical energy per unit crack area during virtual growth, which again
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equals to the crack driving force. We have thus arrived at a simple relation and an
important physical interpretation of theJintegral
J=G 2.6
where Gis crack driving force.
For linear elastic material Gwill be
G'
2
E
Ki= Where i=I, II, III 2.7
hence'
2
E
KJ i= 2.8
Where Kiis the stress intensity factor
( )21,' =EorEE for plane stress and plane strain respectively
E is the modulus of elasticity
is the Poisson ratio
Eq. 2.8 gives a relation betweenJ-integral and the stress intensity factor for linear elastic.
Generally Eq. 2.8 can be given in the following form
+
+
=
===
2
1
2
1
2
1
1 n
i
IIi
n
i
IIi
n
i
Ii KmKKE
J 2.9
where j=1, 2, 3., n the load index which are applied on the structure.
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2.2.4 Castiglianos Theorem
Due to the presence of crack in the structure additional displacement will be created. This
additional displacement will introduce strain energy. Castiglianos theorem says, When
forces act on elastic system subjected to small displacement, the displacement
corresponding to any force, collinear with the force, is equal to the partial derivative of the
total strain energy with respect to the force. Mathematically that is
i
iP
Uu
= 2.10
where iu is the displacement of the point of the application of thePi
Uis the total strain energy.
Piis the applied load (force or bending moment)
2.2.5 Crack Modeling
In order to study the behavior of crack in the beam we have to take some assumption. The
crack has been considered as open with transverse crack depth, and its depth is uniform.
Also the material has the sameEI.
According to the principle of Saint-Venant, the stress field is affected only in the region
adjacent to crack. The element stiffens matrix, except for the cracked element may be
regarded under a certain limitation of element size. It is very difficult to find an appropriate
shape function to express the kinetic energy and elastic potential energy approximately,
because of the discontinuity of deformation in the cracked element. Finding of the
additional stress energy of crack, however, has been studied deeply in fracture mechanics
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and the flexibility coefficient expressed by a stress intensity factor can be easily derived by
means of Castilianos theorem, in linear range.
Consider a beam with a given stiffness properties, dimension b h, and a transverse crack
depth of , see Fig. 2.3.
a
y
PP
P
xP
zP
Fig 2-3 Loaded beam element with transverse crack
Where P1 Axial load
P2& P3 Shear forces
P4& P5 - bending Moments
Paris [36] give the additional displacement iu due to a crack of depth , in the i direction
as
( )
=
0
dJP
ui
i 2.11
where ( )J is strain energy density function [SEDF] or J-Integral, which is found in Eq. 2.9
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Pi is corresponding load and is the crack depth
The local flexibility due to the crack can be given as
( )
=
=
0
2
dJPPP
uc
jii
i
ij 2.12
Integrating the local flexibility along the width, b,of the crack,
( )
=
=
00
21dzdJ
PPbP
uc
b
jii
i
ij 2.13
Since the energy density is a scalar quantity, it is permissible to integrate along tip of the
crack it being assumed that the crack depth is variable and that the stress intensity factor is
given for the element strip.
bh
Pwhere
hFKI
11111 =
=
( ) 24
344144
6
12)( hb
Pz
bh
Pwhere
hFKI ==
=
( )0
6
12
632
2
5
3
5
5155
===
==
=
III
I
KKK
bh
Py
bh
Pwhere
hFK
2.14
II Fbh
LPK
2
33
3= , the stress intensity due to shear force for mode I
0421
3
333
===
=
=
IIIIII
IIII
KKK
bh
Pwhere
hFK
And we can find for the rest of stress intensity factors.
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In this thesis only bending moment about z-axis, P5,and shear force in the direction of y,
P3, are considered.
Now we can find the local flexibility of c33, c35, c55by combining Eq. 2.9, 2.13, and 2.14,
then we will make the non-dimensional term.
( )
( ) ( )[ ]
+
+
+
=
++=
=
=
2
3
2
2
5
2
5
2
3
2
2
3
2
3
2
53
0033
2
3
33
6
632
3
'
1
'
1
1
bh
FP
bh
FP
bh
FP
bh
LFP
bh
LFP
E
KKKE
Jwhere
dzdJPPbP
uc
III
III
IIII
bi
Up on substitution Eq. 2.14 to the above equation the following result is obtain
2
218
'
2
22
2
42
22
33
+=
hb
F
hb
LF
Ec III 2.15
For coupled load the compliance will be
( )[ ]
( )[ ]2
32
53
0
2
3
2
530
53
2
35
'1
'
1
IIII
IIII
b
KKKE
Jwhere
dzdKKKEPP
c
++=
++
=
Over integration
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2
18
'
2
42
2
35
=
hb
LF
Ec I 2.16
And finally
( )[ ]
( )[ ]2 32
53
0
2
3
2
530
55
2
5
55
'
1
'
1
IIII
IIII
bi
KKKE
Jwhere
dxdKKKEPPP
uc
++=
++
=
=
2
72
'
2
42
2
55
=
hb
F
Ec I 2.17
In the case of this thesis I assume that the only available loads are P3and P5, where P3is
bending load and P5is shear load due to mass.
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3 EULER- BERNOULLI BEAM
3.1 Euler- Bernoulli Beam formulation
In this thesis the beam is first modeled based on the Euler- Bernoulli beam theory.
The Euler-Bernoulli assumption of elementary beam theory will be employed, namely:
a) There is an axis of the beam, which undergoes no extension or contraction. The x-
axis is located along this neutral axis.
b) Cross sections perpendicular to the neutral axis in the undeformed beam remain
plane and remain perpendicular to the deformed neutral axis, that is, transverse
shear deformation is neglected.
c) The material is linearly elastic and the beam is homogenous at any crass section.
d) zy and are negligible compared to x
M(x , t)
V
y
xM(x + dx, t)
q(x, t)
Fig 3-1 Euler-Bernoulli beam element
The Euler Bernoulli equation for beam bending can be written as follow
( )txqx
vEI
xt
v,
2
2
2
2
2
2
=
+
3.1
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where v(x, t)is the transverse displacement;
is mass density per length;
EIis the beam rigidity;
q(x, t) is the externally applied pressure loading.
3.2 Finite Element Method
We apply one of the methods of weighted residual, Galerkins method, to the beam
equation to develop the finite element formulation and the corresponding matrix equation.
The weigh residual of Eq. 3.1 gives
00 2
2
2
2
2
2
=
+
= dxwqx
vEI
xt
vR i
l
3.2
Where l- is the length of the beam element
wi is a test function
The weak formulation of Eq. 3.2 is obtained from integration by parts for the second term
of the equation as follow.
( ) 0|1
3
3
03
3
2
2
=
+
=
=
n
i
i
il
iie e e
dxxqwdxdx
vdEI
dx
dw
x
vdEIwdxw
tR
( ) 0|1
3
3
2
2
02
2
3
3
2
2
=
+
+
= =
n
i
iili
iie e e
dxxqwdxdx
vdEIdx
wddx
vdEIdx
dwx
vdEIwdxwt
R 3.3
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01 0
2
2
2
2
2
2
=
+
+
=
=
n
i
l
i
ii
i
ix
dwMVwdxqw
x
w
x
vEIdxw
t
vR
eee
3.4
where3
3
x
vEIV
= is shear forces
2
2
x
vEIM
= is bending moment
e is the element domain
n in number of elements for the beam
For the time being we consider shape function for special interpolation of transverse static
deflection, v, in terms of nodal variable. Interpolation in terms of time domain will be
discussed latter on. Also in Galarkins method, the shape functions are the same as the
weight function, thus
wi=Ni
where Niis shape function which is supposed to be found in the in the for going
discussion
Then
( ) 00
2
2
2
2
=
+
l
i
ii
i
ee
dx
dNMVNdxxqNdx
dx
vd
dx
NdEI 3.5
To formulate the shape function now we consider an element, which has two nodes on each
end, Fig 3.2
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y
v1 v2
Fig 3-2 A two node beam element.
The deformation of a beam must have continuous slope as well as continuous deflection at
any two neighboring beam elements.
The Euler-Bernoulli beam equation is based on the assumption that the plane normal to the
natural axis before deformation remains normal to the natural axis after deformation (see
Fig. 3.3).
Fig 3-3 Deformation of an Euler Bernoulli Beam
12
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This assumption denoteddx
dv= (i.e. slop is the first derivative of deflection in terms ofx).
Because there are four nodal variables for the beam element, we assume a cubic polynomial
function for v (x).
The elastic curve of a beam can be approximated by.
( ) 332
210 CCCC xxxxv +++= 3.6
( ) ( ) 2
321 C3C2C xxdx
xdvx ++== 3.7
Atx=0, v(0)=C0=v1, C0=v1
(0)=C0=1, C0=1
Atx=l, v (l)=v1+ 1l + C2l2+ C3l
3=v2
(l)= 1+ 2C2l + 3C3l2 =2
From the above relations, C2and C3can be obtained by simplification:
( ) ( )
( )ll
vvl
C
lvv
lC
211222
2132133
23
12
=
++=
By substituting the C0, C1, C2, and C3and rearranging then we found the following results.
( ) 22
3
2
2
23
3
2
2
12
32
12
3
2
2 3232231
+
+
+
++
+=
l
x
l
xv
l
x
l
x
l
x
l
xxv
l
x
l
xxv 3.8
Thus, the shape functions are:
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+
=
=
+=
+=
2
3
2
2
4
3
3
2
2
3
2
32
2
2
3
2
2
1
3
23
2
231
l
x
l
xN
l
x
l
xN
l
x
l
xxN
l
x
l
xN
3.9
It is important to note two shape functions corresponding to v and are used for each.
Such types of shape function are called Hermitian shape function.
Let { } { } [ ]''
4
''
3
''
2
''
12
2
4
3
2
1
NNNNdx
NdB
N
N
N
N
N ==
=
From interpolation rule finite element
( ) { } { }iT
T
vNv
v
N
N
NN
xv =
=
4
3
2
1
4
3
2
1
{ } { } { }
==2
2
2
2
dx
NdBwherevB
dx
vdi
T 3.10
Substitute Eq. 3.10 in and 3.5 when concentrated moment or shear forces are absent
{ } { } { } { } ( ) 0=+ dxxqNdxvBEIB ee iT
[ ]{ } { }eie FvK =
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[ ] { } { } dxBEIBK Tee= 3.11
The third term in Eq 3.4, results in the element force vector. For a generally distributed
pressure loading, we need to compute
{ } { } ( ) dx
N
N
N
N
dxxqNFe
e
==
4
3
2
1
3.12
In the case concentrated shear forces and moments act on a node they have to be added
after.
Integrating Eq. 3.11 we can find the stiffens matrix
[ ]
=
2
22
3
4626
612612
2646
612612
llll
ll
llll
ll
l
EIK
l
e 3.13
If we have a uniform pressure load 0q within the element force vector become
{ }
=
= 2
2
0
4
3
2
1
00
6
6
12
l
l
l
l
qdx
N
N
N
N
qFl
e 3.14
In case of concentrated forces at nodes
{ }
+
=
2
2
1
2
2
0
6
6
12
M
V
M
V
l
l
l
l
qF
e 3.15
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The last term in Eq 3.4) represents the boundary condition of shear forces and bending
moment at the two boundary points, x=0 and x=l, of the beam. If these boundary
condition are known, the known shear forces and/or bending moment are included in the
system forces vector at the two boundary nodes. Otherwise they remain as unknowns.
However, deflection and /or slope are known as geometric boundary conditions for this
case.
For dynamic analysis of beams the inertia forces must be included. In this case the
transverse deflection is a function of xand t.The deflection is interpolated within a beam
element as given below.
)()()()()()()()(),( 24231211 txNtvxNtxNtvxNtxv +++= 3.16
As we see Eq. 3.16 states that the shape functions are used to interpolate the deflection in
terms of the spatial domain and the nodal variation are function of time. Now the first terms
in Eq. 3.4 becomes
[ ] [ ] { }eTl
ddxNN &&0 3.17
where [ ] [ ]4321 NNNNN =
{ }ed is nodal degree of freedom vector
And the superimposed dot denote temporal derivative for Eq. 3.17 and A= , the
element mass matrix becomes
[ ] [ ] [ ]dxNNAMTl
e
= 0 3.18
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=
22
22
422313
221561354
313422
135422156
420
lll
ll
llll
ll
A 3.19
If we have a mass Mlat the end of cantilever, see Fig. 3.4
Ml
Fig 3-4 A cantilever beamwith one end clamped and a concentrated mass attached at the
other.
The mass matrix of an element which contain mass will be changed to
[ ] [ ] [ ] ( )[ ] ( )[ ]lNlNMdxxNxNAM TlTl
e += )()(0 3.20
Up on substitution, l in shape function we have got the following result
+
=
0000
0100
00000000
422313
221561354
313422135422156
42022
22
lM
lll
ll
llllll
A
[ ]
+
=
22
22
422313
22420
1561354
313422
135422156
420
lll
lMAl
l
llll
ll
AlM
l
e
3.21
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The element stiffness matrix does not change for the dynamics analysis because the shape
function are the same for both static and dynamics analysis. However the force term may
vary as a function of time. The force vector is for the dynamic analysis
( ){ } ( )[ ]dxNtxqtFl
e
= 0 , 3.22
The mass matrix equation for a dynamic beam analysis is, after assembly of element
matrix and vectors,
[ ]{ } [ ]{ } ( ){ }tFdkdM =+&& 3.23
where { }d is displacement vector
For free vibration of a beam, the eigen value problem
[ ] [ ]( ){ } 02 = dMK 3.24
Where is the angular frequency in radians per second.
{ }d is the mode shape.
3.3
Critical load selection
Comparison for section critical load (mass) application between distributed load (mass)
and concentrated load (mass) here as follow, see Fig. 3.5.
When we make comparison between the concentrated load and distributed load for cracked
beam, the shear force in the case of distributed load the shear force is varied with
respect to length.
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Shear Force Diagram
x
bending moment Diagram
Fig 3-5 Shear force and bending moment diagram
Since shear force has its impact on the behavior of cracked beam it is advisable to take
the shear forces which has uniform value through the length in order to take the critical
condition in every part of the beam, by assuming lqq o= .
3.4 Establishment of Element Stiffness Matrix for Cracked Element
In order to develop an element stiffness matrix for a cracked beam element, there are two
parts to the strain energy: The strain energy for the uncracked beam element and the
additional strain energy due to the crack. The strain energy of an element without crack is
obtained from the existing moment and load (mass). The additional strain energy due to
the crack has been studied in chapter two, which is the cause for creation of additional
compliance in the beam.
With shear action neglected, the strain energy of an element without a crack is
dxbhE
dVUx
v == 021
2
1 3.25
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Where E= 3.26
bhAAdxdV == , , A is cross-sectional area of a beam 3.27
Up on substitution Eq. 3.26 and 3.27 in Eq. 3.25 the strain energy can be given as follow
dxE
bhU
x
= 02
2 3.28
The stress, , in Eq.3.28 refers to the stress due to bending and the stress due to shear
force, which is
PM += 3.29
where2
h
I
My
I
MM == 3.30
2
)()( h
I
xlPy
I
xlPP
=
= 3.31
where P is found due to concentrated load at the end
By substituting Eq. 3.30 and 3.31 in Eq. 3.28 the following equation is found.
dxI
xhlP
I
Mh
E
bhU
l2
0 2
)(
22
+=
dxI
hxlP
I
hxlP
I
Mh
I
Mh
E
bhU
l
+
+
=
0
22
2
)(
2
)(
22
22 3.32
Over integration the above equation, the strain energy will be
++=
323
32
22 lPMPllMEI
U 3.33
Now, the flexibility coefficient for an element without a crack, in different load condition is
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( )( )
( ) UU
jiMPPPWherePP
Uc
ji
o
ij
=
===
=
0
53
0
5,3,,, 3.34
( )
++
=
323
32
22
2
3
2
033
lPMPllMEIP
c
( )
=
EI
lc
33
30
33 3.35
( )
++
=
32
3 3222
53
20
35
lPMPllM
EIPPc
( ) ( )053
20
352
3 cEI
lc =
= 3.36
( )
++
=
32
3 32222
5
20
55
lPMPllM
EIPc
( )
EI
lc
3055 =
The total flexibility coefficient matrix for an element without a crack will be
( ) ( )
( ) ( )
=
0
55
0
53
0
35
0
33)0(
cc
cccij 3.37
The total flexibility coefficient is
( ) ( )10ijijij ccc += 3.38
Where ( )1ijc is the compliance for cracked beam, which was derived in Eq. 2.15-2.17.
( )( ) ( )
( ) ( )
=
1
55
1
53
1
35
1
331
cc
cccij
From the beam load condition we have the following diagram
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P3=P
P5
=M
Fig 3-6 Schematic representation of an element with a crack.
From equilibrium condition of the element, transfer of moment and shear from one node to
the other is obtained by,
[ ] [ ] matrixTransferl
TWhereM
PT
M
P
M
PT
i
i
i
i
i
i
,
10
01
1
01
1
1
1
1
=
=
+
+
+
+
So, the stiffness matrix of the cracked element can be written as [43]
[ ] [ ][ ] [ ]Tc
TcTK1= 3.39
[ ]
=
10
01
1
01
lKc [ ]
1c
1010
011 l 3.40
where [ ] 1c is the inverse matrix of compliance.
Once we have got the stiffness matrix for the cracked beam we can assemble it and find the
global matrix, which will be discussed in chapter four. In this case the number of elements
in the beam can be varied based on desired accuracy.
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4 TIMOSHEKNO BEAM
4.1 Timoshenko Beam Formulation
In the case of Timoshenko beam, a plane normal to the beam axis before deformation does
not remain normal to the axis after deformation. Thus the effects of rotary inertia and
transverse shear deformation have to be included in the analysis of a Timoshenko beam.
y , v
x , u
j v j x g
j v j x
g
Fig 4-1 Deformation of a Timoshenko Beam
Let uand vbe the axial and transverse displacement of a beam, respectively. Because of
transverse shear deformation, the slope of the beam is different fromdx
dv. Instead, the
slope equals dx
dv where is the transverse shear strain. As result, the displacement
fields in the Timoshenko beam can be written as
( )xyyxu =),( 4.1
vxv =)( 4.2
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Where the x-axis is located along the neutral axis of the beam and the beam is not subjected
to an axial load such that the neutral axis does not have the axial strain. From Eq. 4.1 and
4.2, the axial and shear strain are
dx
dy
= 4.3
dx
dv+= 4.4
The element stiffness matrix can be obtained from the strain energy expression for an
element. The strain energy for an element of length , l,is
+=l h
h
Tl h
h
T dxdyGb
dxdyEb
U0
2
20
2
222
4.5
The first term in Eq. 4.5 is the bending strain energy and the second term is the shear strain
energy. b and hare the width and height of the beams respectively, and is the correction
factor for shear energy where value is normally6
5. [1]
Substitute Eq. 4.3 and Eq. 4.4 into Eq. 4.5 and taking integration with respect to y gives
dxdx
dvGA
dx
dvdx
dx
dEI
dx
dU
Tll T
+
++
=
0022
1 4.6
whereIandAare the moment of inertia and area of the beam cross-section.
To derive the element stiffness matrix for the Timoshenko beam, the variables v and
need to be interpolated within each element. As it has been observed form Eq. 4.6, v and
are independent variables. That is, we can interpolate them independently using proper
shape functions. This results in the satisfaction of inter-element compatibility, i.e continuity
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of both the transverse displacement vand slope between two neighboring elements. As a
result, any kind of C0shape function can be used for the present elements. Shape function
of order C0are much easier to construct than shape functions of order C1. It is especially
very difficult to construct shape function of order C1 for two-dimensional and three-
dimensional analysis such as the classical plate theory. C1means both vandx
v
continuous
between two neighboring elements. In general, Cntype continuity means the shape function
have continuity up to the nth
order derivative between two neighboring element elements
To derive the stiffness matrix we use the simple linear shape function for both variables.
That is,
[ ]
=v
vHHv
1
21 4.7
[ ]
=2
1
21
HH 4.8
whereH1andH2are linear shape functions for Timoshenko beam. The linear element looks
like that in Fig 3.2, but the shape functions used are totally different from those for the
Hermitian beam element in Euler Bernoulli beam. To develop the stiffness matrices using
linear shape function for Timoshenko beam, the concept of isoparametric mapping will be
applied.
4.2
Isoparametric Element
Isoparametric elements use mathematical mapping from one coordinate system to another
coordinate system. The former coordinate system is called the natural coordinate system
while the latter is called thephysicalcoordinate system.
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To derive the isoparametric element shape functions, the shape functions with respect to
physical coordinate should be derived, first. Consider a subdomain or a finite element
shown in Fig. 4.2. The element has two nodes, one at each end. At each node, the
coordinate value (x1orx2) and the nodal variable (u1or u2) are assigned. Let us assume the
unknown trial function to be
21 cxcu += 4.9
where u is unknown trial function
c1and c2are constants
x1
u1
x
u2
x2
Fig 4-2 Two Node Linear Element.
Eq. 4.9 will be express in terms of nodal variables. In other word, c1 and c2 need to be
replaced by u1and u2. To this end, u will be evaluated atx=x1andx=x2. Then
( ) 12111 ucxcxu =+= 4.10
( ) 22212 ucxcxu =+= 4.11
Now solving Eq. 4.10 and 4.11 simultaneously for c1andc2gives
12
121
xx
uuc
+= 4.12
12
12211
xx
xuxuc
= 4.13
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Substitution of Eq. 4.12 and 4.13 into Eq. 4.9 and rearrangement of the resultant expression
result in
( ) ( ) 2211 uxHuxHu += 4.14
where
( )l
xxxH
= 21 4.15
( )l
xxxH 12
= 4.16
12 xxl = 4.17
Equation 4.14 gives an expression for the variable uin terms of nodal variables, and Eq.15
and Eq. 16 are called linear shape functions. The shape functions are plotted in Fig. 4.3.
1x
1H (x)
x2
H (x)2
Fig 4-3 Linear Shape Functions
These functions have the following properties:
1. The shape function associated with node 1 has a unit value at node 1 and vanishes
at other nodes. That is,
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( ) ( ) ( ) ( ) ( ) 1,0,0,0,1 2222122111 ===== xHxHxHxHxH 4.18
2. The sum of all shape functions is unity.
( ) 12
=
ii
xH 4.19
These are important properties for shape functions. The first property, Eq. 4.18, states that
the variable umust be equal to the corresponding nodal variable at each node (i.e. u(x1)=u1
andu(x2)=u2as enforced in Eq. 4.10 and 4.11. The second property, Eq. 4.19, says that the
variable u can represent a uniform solution within the element.
Once the shape function for physical coordinate system is developed, the shape function for
isoparametric element will be given in terms of the natural coordinate system as seen in Fig
4.4. The two nodes are located at 0.10.1 21 == and , originally, which werex1andx2in
physical coordinate system. These nodal positions are arbitrary but the proposed selection
is very useful for numerical integration because the element in the natural coordinate
system is normalized between 1 and 1. The shape function can be written as [1]
( ) ( ) = 12
11H 4.20
( ) ( ) += 12
12H 4.21
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Fig 4-4 Linear Element in the natural Coordinate system
Any point between 11 21 == and in the natural coordinate system can be mapped onto
a point between 21 xandx in the physical coordinate system using the shape function
defined in Eqs. 4.20 and 4.21.
( ) ( ) 2211 xHxHx += 4.22
The same shape functions are also used to interpolate the variables u and v with in the
element
( ) ( ) 2211 uHuHu += 4.23
( ) ( ) 2211 vHvHv += 4.24
If the same shape functions are used for the geometric mapping as well as nodal variable
interpolation, such as Eq. 4.22, 4.23 and 4.24, the element is called the isoparametric
element.
In order to computedx
dv, which is necessary in Eq. 4.6 to compute element matrix for
Timoshenko beam, we use the chain rule such that
node 1 node 2
x 1 x 1x
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( ) ( )2
21
1 vdx
dHv
dx
dH
dx
dv +=
( ) ( )2
21
1 vdx
d
d
dHv
dx
d
d
dH
+= 4.25
where the expression requiresdx
d, which is the inverse of
d
dx. The latter can be computed
from Eq. 4.22.
( ) ( )( )122
21
1
2
1xxx
d
dHx
d
dH
d
dx=+=
4.26
Substituting Eq 4.26 into Eq. 4.25 yields
2
12
1
12
11v
xxv
xxdx
dv
+
= ,
21
11v
lv
ldx
dv+= 4.27
where 12 xxl = is the element size
In matrix form, Eq. 4.27 can be written as follow
=
2
111
v
v
lldx
dv 4.28
With the same idea we can have an equation fordx
das follow
=
2
111
v
v
lldx
dv 4.29
Also Eq. 4.8 can be expressed in terms of isoparametric element by substituting Eq. 4.20
and 4.21.
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[ ]
=2
1
21
HH = ( ) ( )
+
2
11
2
11
2
1
4.30
Now using Eq. 4.7-4.30 along with the strain energy expression Eq. 4.6 yields the
following stiffness matrix for the Timoshenko beam.
Stiffness matrix for bending strain energy:
From Eq. 4.6 the bending strain is taken as follow
[ ] dxdx
dEI
dx
dK
l T
e
b
=
02
1 4.31
Derivate with respect to x and substitute in to Eq. 4.31 yields the following result.
[ ]
=
1010
0000
1010
0000
l
EIKeb 4.32
Stiffness matrix for shear strain energy:
Also for shear strain energy an equation will be taken from Eq. 4.6.
[ ] dxdx
dvGA
dx
dvK
Tl
e
s
+
+=
02
4.33
Using the concept of isoparametric mapping discussed previously the stiffness matrix for
shear will be derived as follow.
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Substituting Eq. 4.29 and 4.30 in to Eq.4.33 also changing the limit of integration of
physical coordinate 12 xandx to natural coordinate system, 1 and 1, then
[ ] ( )
( )
d
l
lll
l
GAKes
22
11
2
11
21
1
21
1
2
1
1
+
+
=
4.34
[ ]
=
22
22
22
2424
22
2424
4
llll
ll
llll
ll
l
GAKe
s
4.35
where dl
dx2
= , see Eq. 4.26
At this point one thing to be noted is that the bending stiffness term, Eq. 4.32, is obtained
using the exact integration of the bending strain energy but the shear stiffness term, Eq.
4.35, is obtained using one point Gauss quadrature rule. The major reason is if the beam
thickness becomes so small compared to its length, the shear energy dominates over the
bending energy. As we have seen Eq. 4.32 and Eq. 4.35, the bending stiffness is
proportional to lh3 while the transverse shear stiffness is proportional to hl, where hand l
are the thickness and length of beam element, respectively. Hence, as lh becomes smaller
for a very thin beam,the bending term become negligible compared to the shear term. This
is not correct in the physical sense. As the beam becomes thinner, the bending strain energy
is more significant than the shear energy. This phenomenon is called shear locking. In
order to avoid shear locking, the shear strain energy is under-integration. Because of the
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under-integration the presence beam stiffness matrix is rank deficient. That is, it contains
some fictitious rigid body mode (i.e. zero energy modes).
4.3
Establishment of matrix for cracked beam element
In the case of Euler-Bernoulli beam, by neglecting the shear action, the strain energy
without crack is derived. But in the case of Timoshenko beam the shear action will be
included to model the crack entirely.
The strain energy of an element without a crack is given for two cases as follows.
For bending strain energy,
dxI
xhlP
I
hxlP
I
Mh
I
Mh
E
bhU
l
b
+
+
=
0
22
2
(
2
)(
22
22, from Eq. 3.32
++=
32
3 3222 lPMPllM
EI
Ub 4.36
The shear strain energy can be expressed [3]
=l
sAdxU
02
1 4.37
where the shear coefficient which is equal to =5/6 for rectangular beam.[1, 3]
A is cross-section of beam
is the shear angle, see Fig 4.5
is shear stress
x
v
= , where is the rotation of cross-section 4.38
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dv/dx
v
x
Fig 4-5 Deformation of beam including shear
Eq. 4.37 can be written as follow
=
l
s AdxG
U0
2
1
=l
s AdxG
U0
2
2
1 4.39
whereG
= , G is the shear modulus 4.40
Once the equation of shear strain energy is determiend, it can be evaluated by substituting
the shear stress value in to Eq. 4.39.
( )=l
s AdxG
IbPQU
0
2/
2
1 4.41
whereIb
PQ= 4.42
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Pis the shear force at the section.
Iis the moment of Inertia about the neutral axis
bis the width of the section
Qis the first moment with respect to the neutral axis of the area below the point
at which the shear stress is derived.
==A
zAzdAQ ' 4.43
whereA is the area of that part of the section below the point desired.
z is the distance from the neutral axis to the centroid ofA.
For beam of uniform cross section the maximum shear stress occurs at the section having
the greatest shear force, P. In the case of this thesis the shear force is uniform through the
length of the beam.
neutral axish
A'
b
Fig 4-6 Cross-section of a beam
If the shear stress is desired at level z1of the rectangular cross section, Fig 4.6, Q must be
calculated for the shaded area
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+
==
2
2
2' 111
zhzz
hbzAQ
= 21
2
42zhbQ 4.44
It follows from Eq. 4.42 that the shear stresses vary according to
= 21
2
42z
h
I
P 4.45
Eq. 4.45 shows that the shear stress varies parabolically with z1. For modeling of crack the
maximum value of shear will be taken for z1=0, at the natural axis.
A
P
bh
P
I
Ph
2
3
2
3
8
2
=== 4.46
In general, the shear strain energy can be express in the form of
dxbdzzh
I
P
GU
h
h
l
s
= 1
22
2
2
1
2
0422
1
However, for this thesis the maximum shear stress will be taken to get the shear strain
energy,
GA
lPUs
2
8
9= 4.46
The total strain energy will be the summation of strain energy due to bending and the strain
energy due to shear, by adding Eq. 4.36 Eq. 4.46.
( )sb
oT UUU +=
( )
GA
lPlPMPllM
EIU oT
23222
8
9
32
3 +
++= 4.47
We can find the flexibility coefficient for an element without a crack.
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( )( )
ji
oToT
ijPP
Uc
= where P3=P, P=M, i, j=3,5 4.48
( )
+
++
=
GA
lPlPMPllM
EIPc oT
23222
2
3
2
338
9
32
3
( )
GA
l
EI
lc oT
8
152
33 += , where A=bh 4.49
( )
+
++
=
GA
lPlPMPllM
EIPP
c oT232
22
53
2
35
8
9
32
3
( ) ( )oToT cEI
lc 53
2
35 3 =
= 4.50
( )
+
++
=
GA
lPlPMPllM
EIPc oT
23222
2
5
2
558
9
32
3
( )
EI
lc oT
2
3
55 = 4.51
The flexible coefficient matrix for a uniform beam will be
( ) ( )
( ) ( )
=
oToT
oToT
oT
ijcc
ccc
5553
3533)( 4.52
The total flexible coefficient will be
( ) ( )1ij
oT
ijij ccc += 4.53
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where( )oTijc is the compliance of beam without crack
( )1ij
c is the compliance of beam with crack , from Eq. 3.38
Now the stiffness matrix of the cracked element can be written as [43]
[ ] [ ][ ] [ ]TcT
TcTK1=
Once we get the stiffness matrix for the cracked Timoshenko beam element we can
assemble it and find the global matrix. In this case the number of elements in the beam can
be varied based on desired accuracy of results.
The consistent mass matrix for the Timoshenko beam is computed from the equation of
kinetic energy.
( )dAdxvuKEl
xA
+=0 )(
22
2
1&& 4.54
By substituting Eq. 4.1 the following equation will be found,
( )dAdxvyKEl
xA +=0 )(
222
21
&& 4.55
dxvAdxIKE
ll
+=0
2
0
2
2
1
2
1&& 4.56
where ( ) =xI A
dAy 2 , is the geometric moment of inertia of the cross section
Now, by defining
Am= the mass per unit of beam length
A
Ir =2 where ris the gyration radius of the cross section
we obtain the following expression for the kinetic energy.
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dxmrdxvmKE
ll
+=0
22
0
2
2
1
2
1&& 4.57
The first term on the right hand side of Eq. 4.57 is translating inertia and the second term is
the rotary inertia.
By taking the shape function from Eq. 4.7 and 4.8 substituting in Eq. 4.57, then the mass
matrix for translation will be
=l
T
eTHdxmHM
0 4.58
and the mass matrix for rotary inertia will be
dxHHmrM
l
T
eR =0
2
4.59
As we have seen from Eq. 4.58 and Eq. 4.59, the beam element mass matrix has two
components: Transverse and rotary component.
Eq. 4.58 and 4.59 result in
=
0000
0201
0000
0102
6
mlMeT
And 4.60
=
0000
0101
0000
01012
l
mrMeR
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Once the mass matrix and stiffness matrix are found, the system characteristic equation can
be found for free vibration as follow
[ ] [ ]( ) 02 = &&MK 4.61
Where [K] is the sum of component of bending and shear stiffness matrices
[M] is the sum of component of rotary and transverse mass matrices
is the angular frequency in radians per second.
{ } is the modal shape.
4.4 Assembly of Element Matrices and Derivation of System Equation
Once the element characteristics, namely, the element matrices stiffness and element mass
matrices are found in common global coordinate system, the next step is to construct the
overall or system equation. The procedure for constructing the system equation from the
element characteristic is the same regardless of the type of problem and the number and
type of elements used.
The procedure of assembling the element matrices is based on the requirement of
compatibility at the element nodes. This means that at the nodes where elements are
connected, the value(s) of the unknown nodal degree(s) of freedom or variable(s) is (are)
the same for all the elements joining at that node.
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Let nel and sdof denote the total number of elements and nodal degree of freedom
(including the boundary and restrained degrees of freedom), respectively. Let q denote the
vector of sdof nodal degrees of freedom and [kk] the assembled system characteristics
matrix of order sdof xsdof.Since the element characteristic matrix [Ke] is order of 4x4, it
can be expressed to order of sdof xsdofby including zeros in the remaining locations. Thus
the global characteristics matrix can be obtained by algebraic addition as
[ ]
=
=
nel
e
ekkkk1
4.62
where [ ]ekk is the expanded characteristic matrix of element e(of order sdof xsdof).
In actual computation, the expansion of the element matrix [Ke] to the size of the overall
[kk] is not necessary. [kk] can be generated by identifying the elements of [Ke] in [kk] and
adding them to the existing values as echanges from 1 to nel.
This procedure is shown withreference to the assemblage of beam elements as shown in
Fig. 4.7.
Fig 4-7 Beam with two elements.
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For assembling [Ke], we consider the elements one after another. For e=1, the element
stiffness matrix [K1] can be written as shown below.
[ ]
=
1
44
1
43
1
42
1
41
1
34
1
33
1
32
1
31
1
24
1
23
1
22
1
21
1
14
1
13
1
12
1
11
1
4
3
2
1
4
3
2
1
4321..
4321..
kkkk
kkkk
kkkk
kkkk
K
fodGlobal
fodLocal
4.63
The location of (raw lland column ml) of any component 1ijK in the global stiffness matrix
[kk] is identified by the global degree of freedom lland mlcorresponding to the local
degree of freedom ( )1i and( )2i respectively for i=1to 4andj=1to 4. The corresponding
between ll and( )1i , and ml and
( )1i is also shown already in Eq. 4.63. Thus the location
of the components 1ijK in [kk] will be show in Eq. 4.64.
[ ]
=
0
0
00
0
0
0
0
00
0
0
0
0
0
0
0
0
0
0
6
5
43
2
1
6
5
43
2
1
654321..
654321..
1
44
1
43
1
42
1
41
1
34
1
33
1
32
1
31
1
24
1
23
1
22
1
21
114
113
112
111
1
kkkk
kkkk
kkkk
kkkk
kk
fodGlobal
fodLocal
4.64
For, the element stiffness matrix [K(2)] can be written as shown in Eq. 4.65 below
( )[ ]
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
=
2
44
2
43
2
42
2
41
2
34
2
33
2
32
2
31
2
24
2
23
2
22
2
21
2
14
2
13
2
12
2
11
2
6