Ermias Gebrekidan

7/24/2019 Ermias Gebrekidan

1/123

i

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.


2/123

ii

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


3/123

iii

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


4/123

iv

Appendix I. Program For Euler-Bernoulli Beam..............................................................93

Appendix II. Program For Timoshenko Beam ..............................................................103


5/123

v

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


6/123

vi

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


7/123

vii

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


8/123

viii

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


9/123

ix

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


10/123

x

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


11/123

xi

( )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


12/123

xii

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.


13/123

i

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.


14/123

1

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


15/123

2

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


16/123

3

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.


17/123

4

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


18/123

5

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.


19/123

6

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.


20/123

7

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.


21/123

8

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.


22/123

9

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


23/123

10

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.


24/123

11

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.


25/123

12

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

+

=


26/123

13

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


27/123

14

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


28/123

15

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.


29/123

16

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


30/123

17

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


31/123

18

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.


32/123

19

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


33/123

20

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.


34/123

21

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


35/123

22

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


36/123

23

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


37/123

24

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


38/123

25

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:


39/123

26

+

=

=

+=

+=

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 =


40/123

27

[ ] { } { } 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


41/123

28

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


42/123

29

=

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


43/123

30

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.


44/123

31

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


45/123

32

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


46/123

33

( )( )

( ) 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


47/123

34

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.


48/123

35

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


49/123

36

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


50/123

37

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.


51/123

38

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


52/123

39

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,


53/123

40

( ) ( ) ( ) ( ) ( ) 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


54/123

41

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


55/123

42

( ) ( )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.


56/123

43

[ ]

=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.


57/123

44

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


58/123

45

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


59/123

46

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


60/123

47

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


61/123

48

+

==

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.


62/123

49

( )( )

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


63/123

50

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.


64/123

51

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


65/123

52

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.


66/123

53

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.


67/123

54

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

Documents

Ermias Gebrekidan