ELEMENT FREE GALERKIN METHOD OF COMPOSITE...

ELEMENT FREE GALERKIN METHOD OF COMPOSITE BEAMS

WITH PARTIAL INTERACTION

DZULKARNAIN BIN AHMAD

A thesis submitted in fulfilment of the

requirements for the award of the degree of

Doctor of Philosophy (Civil Engineering)

Faculty of Civil Engineering

Universiti Teknologi Malaysia

AUGUST 2013

iii

To my thoughtful father and mother, lovely wife and kids, and family.

iv

ACKNOWLEDGEMENT

Praise to Allah, the Almighty who sparks my intuition to pursue my post

graduate studies and provides me with invaluable guidance throughout my life.

Thanks to Universiti Selangor, for providing the financial support throughout my

post-graduate studies.

Thanks to my supervisor, Dr. Airil Yasreen for his supervision and support in seeing

this work through to completion; and also for being my friend.

My father and mother, your second son loves you both and thanks for taught me

about ‘kerja bersungguh-sungguh’.

For my lovely wife and cheerful kids, thanks for cheering up my life. To my family,

my deepest appreciation goes out to all of you.

Finally, my gratefulness to all authors who wrote the books and papers that I had

read.

v

ABSTRACT

Composite beam with partial interaction behaviour has ignited many studies,

not just on its mechanics but also on solutions of its one-dimensional partial

differential equation. Inadequate solution by available analytical methods for this

high order differential equation has demanded for numerical approach and therefore

Element Free Galerkin (EFG) method is applied for the first time in this present

work. The work consists of three parts; first is the formulation of Galerkin weak

form and assemblage of the EFG discrete equilibrium equation. One-dimensional

formulation of the weak form is performed by adopting the variational approach and

the discrete equation, which is in matrix form and written using the Matlab

programming code. Subsequently in second part, the EFG formulation is developed

for both the slip and uplift models, where the former adopted equal curvature

deflection assumption while the latter considered the unequal curvature. The

proposed EFG formulation gives comparable results in both models, after been

validated by established analytical solutions, thus signify its application in partial

interaction problems. The third part provides numerical tests result on EFG

numerical parameters such as size of support domain, polynomial basis and

quadrature points with seven different types of weight functions for this composite

beams behaviour. Conclusively, Cubic Spline and Quartic Spline weight functions

yield better accuracy for the EFG formulation results, compares to other weight

functions. The capability of the EFG formulation was also studied in terms of its

application on free vibration problem and various composite beam cross-sections.

Results from the numerical tests deduced the demand for optimised parameters value

as the parameters are highly reliant on user-defined value. Additionally, the research

supports the need for more efficient EFG code’s algorithm, stiffness matrix, shape

function formulation and background integration methods, in approximating the

higher order differential equation which refers to dynamics analysis.

vi

ABSTRAK

Kelakuan rasuk komposit dalam keadaan interaksi separa telah mencetuskan pelbagai kajian, dan tidak hanya melibatkan perihal mekanik tetapi juga untuk penyelesaian satu matra bagi persamaan pembezaan separanya. Ketidakupayaan yang wujud pada kaedah analitikal sediada di dalam menyelesaikan persamaan pembezaan tertib tinggi telah menuntut kepada penggunaan kaedah berangka, yang dengan itu kaedah Element Free Galerkin (EFG) diaplikasi, buat pertama kalinya. Kajian ini terbahagi kepada tiga bahagian, bahagian pertama menyentuh tentang formulasi bentuk lemah Galerkin dan persamaan keseimbangan diskrit EFG. Bentuk lemah satu matra diterbitkan mengikut kaedah perubahan manakala persamaan diskrit di dalam bentuk matrik ditulis menggunakan kod program Matlab. Di bahagian kedua, formulasi EFG dibangunkan untuk pertamanya bagi keadaan gelincir dengan anggapan pesongan adalah sama bagi kedua-dua komponen rasuk dan model kedua melibatkan bersama kesan angkat-naik yang pesongannya adalah berasingan. Formulasi EFG telah memberikan keputusan yang setara apabila dibandingkan dengan penyelesaian analitikal sediada, dengan itu memungkinkan aplikasinya di masalah interaksi separa. Bahagian ketiga memberikan keputusan ujian berangka terhadap beberapa parameter EFG seperti saiz sokong domain, asas polinomial dan titik kamiran yang melibatkan tujuh pemberat fungsi yang berbeza bagi kelakuan rasuk komposit tersebut. Kesimpulannya, pemberat fungsi Cubic Spline dan Quartic Spline memberikan keputusan ketepatan yang lebih baik berbanding yang lain. Kemampuan formulasi EFG juga diaplikasi terhadap frekuensi tabii untuk masalah getaran bebas dan pelbagai keratan-rentas rasuk komposit. Bersandar kepada keputusan ujian berangka yang dijalankan, didapati nilai parameter yang optimum adalah perlu memandangkan parameter yang diuji amat bergantung kepada nilai cadangan daripada penganalisis. Lanjutan itu, kajian ini menyokong kepada perlunya algoritma kod EFG, matrik kekukuhan, formulasi rangkap bentuk dan kaedah kamiran latarbelakang yang lebih efisen bagi penghampiran persamaan pembezaan tertib yang lebih tinggi, yang merujuk kepada analisis dinamik.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION ii

DEDICATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF TABLES xiii

LIST OF FIGURES xvi

LIST OF SYMBOLS xxiv

LIST OF APPENDICES xxviii

1 INTRODUCTION 1

1.1 Introduction 1

1.2 Research Background and Problem Statement 2

1.3 Purpose and Objectives of the Study 5

1.4 Scope and Limitation of the Study 6

1.5 Outline of the Thesis 6

viii

2 LITERATURE REVIEW 9

2.1 Introduction 9

2.2 Behaviour of Partial Interaction Composite Beam 10

2.2.1 Analytical Solution of Partial Interaction 11

Composite Beam-Shear Connectors


Composite Beam-Adhesive Bonded Steel

Plate


Composite Beam-Adhesive Bonded Fiber

Reinforced Materials

2.2.4 Finite Element Formulation of 17

Partial Interaction Composite Beam

2.3 Brief Developement of Meshless Method 19

2.4 Element Free Galerkin Method 20

2.5 Concluding Remarks 22

3 FUNDAMENTALS OF MOVING LEAST SQUARE 24

METHOD

3.1 Introduction 24

3.2 Basis Functions 25

3.3 Non-constant Coefficients 27

3.4 Weight Functions 30

3.5 Support Domain 32

3.6 Dimensions of Support Domain 34

3.7 Derivatives of Shape Functions 35

3.8 Element Free Galerkin for Euler-Bernoulli Beam 37

3.8.1 Galerkin Weak Form of 37

Euler-Bernoulli Beam

3.8.1.1 Weighted Integral Approach 38

ix

3.8.1.2 Principle of Minimum 39

Potential Energy

3.9 New EFG Solution of Simply Supported 41

Euler-Bernoulli Beam


4 ELEMENT FREE GALERKIN FORMULATION 44

OF COMPOSITE BEAM WITH

LONGITUDINAL SLIP

4.1 Introduction 44

4.2 Deflection Equations of Composite Beams 45

4.2.1 Deflection Equation of Full Interaction 45

4.2.2 Deflection Equation of Partial Interaction 46

4.2.2.1 Newmark’s Solution 47

4.2.2.2 Girhammar’s Solution 47

4.3 Principle of Minimum Potential Energy 48

4.3.1 The Variational Approach 50

4.3.2 The Galerkin Weak Form 53

4.4 Moving Least Square Shape Function 53

4.4.1 Selected Polynomial Basis 54

4.4.2 Selected Weight Function 55

4.4.3 Computation of MLS Shape Function 57

4.5 EFG Linear Algebraic Equations 59

4.5.1 Differential Operators 59

4.5.2 Global Nodal Stiffness Matrix 61

4.5.3 Global Nodal Displacement Matrix 63

4.5.4 Global Force Vector Matrix 64

4.5.5 Langrage Multiplier 66

4.6 General Matlab EFG Procedure for 71

Longitudinal Slip of Composite Beams

4.7 Numerical Solution and Validation 73

4.7.1 Domain Discretisation 74

x

4.7.2 Full Interaction Composite Beam 76

4.7.3 Newmark Model 77

4.7.4 Girhammar Model 79


5 NUMERICAL TESTS ON EFG FORMULATION 82

OF LONGITUDINAL SLIP COMPOSITE BEAM

5.1 Introduction 82

5.2 Influence of the Weight Functions 83



5.3 Influence of the Support Domain 92

5.4 Polynomial Basis 96

5.5 Quadrature Points 99

5.6 Weight Function Effect on Interfacial Shear 101

Force


6 ELEMENT FREE GALERKIN FORMULATION 106

OF COMPOSITE BEAM WITH

PARTIAL INTERACTION

6.1 Introduction 106

6.2 Element Free Galerkin Weak Form 107

6.2.1 Variational Approach for Partial 109

Interaction of Composite Beam

6.3 EFG Algebraic Equation System 111

6.4 Numerical Solution 115

6.5 Effect of Vertical Stiffness on EFG Formulation 118

6.6 Influence of Weight Functions 120


xi


6.7 Influence of Support Domain Size 129

6.8 Influence of Polynomial Basis 132

6.9 Influence of Quadrature Points 135

6.10 Interfacial Normal Stress Distribution 138


7 ELEMENT FREE GALERKIN APPLICATION 143

ON PARTIAL INTERACTION COMPOSITE

WITH VARIOUS SECTIONS AND BOUNDARY

CONDITIONS


7.2 Composite Sections and Boundary Conditions 144

7.3 EFG Formulation of Longitudinal Slip Interaction 145

7.3.1 Simply Supported Composite Beam 146

7.3.2 Fixed-end Composite Beam 153

7.3.3 Cantilever Composite Beam 160

7.4 EFG Formulation for Partial Interaction with 166

Uplift

7.4.1 Simply Supported Composite Beam 167

7.4.2 Fixed-end Composite Beam 170

7.4.3 Cantilever Composite Beam 173


8 FORMULATION AND SOLUTION OF 178

COMPOSITE BEAM FOR FREE VIBRATION

PROBLEM BY ELEMENT FREE GALERKIN

METHOD


8.2 Weak Form of Free Vibration Equation 179

xii

8.3 EFG Linear Algebraic Equation 181

8.4 Numerical Solution 183

8.5 Influence of Weight Functions 185

8.6 Influence of Support Domain Size 186

8.7 Influence of Polynomial Basis 187


9 SUMMARY, CONCLUSIONS AND 190

SUGGESTIONS FOR FUTURE WORK

9.1 Summary 190

9.2 Conclusions 194

9.3 Suggestions for Future Work 197

REFERENCES 199

Appendix A 208-222

xiii

LIST OF TABLES

TABLE NO. TITLE PAGE

1.1 Differences between FEM and Meshless

method (Liu and Gu, 2005)

3

3.1 Commonly used polynomial basis in the

Meshless method

26

3.2 Numerical parameters for Euler-

Bernoulli beam

42

4.1 Imposition of essential boundary

condition techniques (Fries and

Matthies, 2003)

66

4.2 Input parameters adopted in the analysis 75

4.3 Comparison of vertical displacement

(m) at mid-span of the composite beam

with longitudinal slip (Newmark model)

78

4.4 Comparison of vertical displacement

(m) at mid-span of the composite beam

with longitudinal slip (Girhammar

model)

79

5.1 Displacement norm and relative

discrepancy obtained by using different

EFG weight functions for Newmark

model

86

xiv



EFG weight functions for Girhammar

model

90

5.3 Displacement norm obtained by using

different dimensionless size

93


different polynomial basis order

97

5.5 Comparisons of displacement norms for

different quadrature point

99




model

121



EFG weight functions for Girhammar

model.

125



129

6.4 Displacement norm from different

polynomial basis order

132

6.5 Comparisons of displacement norms for

different quadrature point

135

6.6 Geometric and material properties

(Roberts and Haji-Kazemi, 1989)

139

7.1 Properties of composite beam cross-

sections

145

7.2 Comparison of maximum deflection for

simply supported composite beam

exerted by uniform load

147

xv


simply supported composite beam

exerted by point load

150


fixed support composite beam exerted

by uniform load

154


fixed support composite beam exerted

by point load

157


cantilever composite beam exerted by

uniform load

160


cantilever composite beam exerted by

point load

164

8.1 Geometric and material properties (after

Xu and Wu, 2007)

184

8.2 Comparison of EFG natural frequency

result

184




model

185



186

8.5 Displacement norm from different

polynomial basis order

187

xvi

LIST OF FIGURES

FIGURE NO. TITLE PAGE

1.1 Partial interaction in composite beams 1

1.2 Flowchart for FEM and Meshless

method (Liu and Gu, 2005)

3

1.3 Flow of the thesis contents 8

2.1 Differential element of partial

interaction in composite beam

13

3.1 The approximate function uh (x) and the

nodal parameter in the MLS

approximation (Liu, 2002)

28

3.2 Different sizes and shapes of support

domain at different quadrature points

33

3.3 Influence domain(s) of nodes. Only

node 1 and 3 are included for

quadrature point domain; X although

node 2 is nearer than node 1

34

3.4 Comparison of (a) deflection and (b)

rotation of EFG with exact solution

42

4.1 Boundary conditions of (a) concentrated

load (after Newmark, Siess and Viest,

1951) and (b) uniformly distributed load

(after Girhammar and Gopu, 1993).

46

xvii

4.2 Quartic spline weight function at mid-

span of the beam using 101 nodes

56

4.3 (a) First and (b) second derivatives of

quartic spline weight function using 101

nodes at mid-span of the beam.

56

4.4 MLS Shape function at mid-span of the

beam using 101 nodes

58

4.5 (a) First and (b) second derivatives of

MLS Shape function using 101 nodes at

mid-span of the beam

58

4.6 EFG algorithm flow chart of the

longitudinal slip model

73

4.7 Nodal arrangement (21 nodes) and

background cell of one-dimensional

beam

75

4.8 Convergence of deflection in full

interaction model subjected to mid-

point load

76

4.9 Convergence of deflection in full

interaction model subjected to uniform

load

77

4.10 Comparison of deflection between EFG

and Newmark solution

78

4.11 Comparison of deflection between EFG

and Girhammar solution

80

5.1 Selection of weight function at mid-

span of the beam using 67 nodes

83

5.2 Various shape function at mid-span of

the beam using 67 nodes

84

5.3 Various shape function at quarter-span

of the beam using 67 nodes

84

xviii

5.4 Comparison of deflection using

different weight functions for the

Newmark model

87

5.5 Convergence of displacement in (a)

displacement norm and (b) relative

discrepancy using different weight

functions for the Newmark model

88


different weight functions for the

Girhammar model

90




functions for the Girhammar model

91

5.8 Convergence of displacement norm

using different dimensionless size for

Newmark model

94



Girhammar model

95


using different polynomial orders for

the Newmark model

98


using different polynomial orders for

the Girhammar model

98

5.12 Displacement norm using different

quadrature points for Newmark model

100


quadrature points for Girhammar model

100

5.14 Comparison of interfacial slip between

EFG and Newmark solution

102

xix

5.15 Comparison of interfacial shear force

between EFG (QS weight function) and

Newmark solution

102

5.16 Comparison of interfacial shear force

between EFG (Qu, Hy, Ex, Co weight

function) and Newmark solution

103

6.1 Comparison of deflection in the

Newmark model between EFG and

analytical solution (55 nodes)

116

6.2 Comparison of deflection in the

Girhammar model between EFG and

analytical solution (63 nodes).

117

6.3 Vertical stiffness effect on top(T) and

bottom(B) parts of composite beam

with partial interaction

119


different weight functions for Newmark

model (53 nodes)

122




functions for Newmark model

123


different weight functions for

Girhammar model

126




functions for Girhammar model

127



Newmark model

130

xx



Girhammar model

131


using different polynomial order for

Newmark model

133


using different polynomial order for

Girhammar model

134


quadrature points for Newmark model

137


quadrature points for Girhammar model

137

6.14 Comparison of interfacial normal stress

distribution for RC beam with a steel

plate subjected to uniform load

140

7.1 Composite beam sections (Girhammar,

2009)

144

7.2 Deflection of EFG formula and exact

solution of simply supported composite

beam (Steel-concrete section-uniform

load)

148

7.3 Deflection of EFG code and exact


beam (Wood-wood section-uniform

load)

148



beam (Fully cracked section-uniform

load)

149

xxi



beam (Non-cracked section-uniform

load)

149



beam (Steel-concrete section-point

load)

151



beam (Wood-wood section-point load)

152



beam (Fully cracked section-point load)

152



beam (Non-cracked section-point load)

153


solution of fixed supported composite

beam (Steel-concrete section-uniform

load)

155

7.11

Deflection of EFG code and exact


beam (Wood-wood section-uniform

load)

155



beam (Fully cracked section-uniform

load)

156



beam (Non-cracked section-uniform

load)

156

xxii



beam (Steel-concrete section-point

load)

158



beam (Wood-wood section-point load)

158



beam (Fully cracked section-point load)

159



beam (Non-cracked section-point load)

159


solution of cantilever composite beam

(Steel-concrete section-uniform load)

162



(Wood-wood section-uniform load)

162



(Fully cracked section-uniform load)

162



(Non-cracked section-uniform load)

163



(Steel-concrete section-point load)

164



(Wood-wood section-point load)

165

xxiii



(Fully cracked section-point load)

165



(Non-cracked section-point load)

166

7.26 Simply supported’s mid-span deflection

of top and bottom components; a) Steel-

conrete, b) Wood-wood sections

168

7.27 Simply supported’s mid-span deflection

of top and bottom components; a) Fully

cracked concrete, b) Non-cracked

concrete sections

169

7.28 Fixed support’s mid-span deflection of

top and bottom components, (a) Steel-

concrete, (b) Wood-wood sections

171

7.29 Fixed support’s mid-span deflection of

top and bottom components; a) Fully


concrete sections

172

7.30 Cantilever free end’s deflection of top

and bottom components; (a) Steel-

concrete, (b) Wood-wood sections

174

7.31 Cantilever free end’s deflection of top

and bottom components; a) Fully


concrete sections

175

xxiv

LIST OF SYMBOLS

A(x) - weighted moment matrix

A, - x first derivative of the weighted moment

A, - xx second derivative of the weighted moment

A - b cross-sectional areas (of bottom component)

ai - (x) non-constant coefficients

A - t cross-sectional areas (of top component)

b - matrix of distributed load

B(x) - non-symmetry matrix

C - constant value of shear connector

c - order of consistency

c - stiffness matrix

C - b distance between the contact surface of the lower

components from the centroidal axis

C - t distance between the contact surface of the top

components from the centroidal axis

d - dimension of the problem

d - c nodal spacing

d - s support domain

E - a adhesive’s Modulus

E - b modulus elasticity (of bottom component)

Er - u displacement norm

E - t modulus elasticity (of top component)

F - applied forces acting at the centroid of top and

lower components

)(xf - distributed load

F - global nodal force vector

xxv

F - b global force vector of body force

f - I nodal force vector of body force

F - Q global force vector of traction force

I - b moment inertia (of bottom component)

i - c connectors spacing

I - t moment inertia (of top component)

J - weighted least-square discrete error norm

K - global nodal stiffness matrix

kc - slip modulus of connectors

K - IJ nodal stiffness matrix

k - s shear connector modulus

k - t foundation modulus

k - y vertical connection stiffness

L - Length

L, H - matrix of differential operators

m - number of terms in basis function

M - b resistant moments acting at the surface (of bottom

component)

M - t resistant moments acting at the surface (of top

component)

n - number of nodes in support domain

tnλ - total number of nodes on the essential boundary

N - I Langrage interpolants

n - r number of rows of connectors

P - concentrated load

pi - (x) basis functions

Q - matrix of concentrated forces

q - c shear flow at interface

q - y vertical force per unit length at interface

r - normalised distance

r - u displacement relative discrepancy

sgn - signum function

T - uplift force per unit length

xxvi

t - a adhesive thickness

t - b nodal force vector of traction force

U - strain energy

u - column vector of approximate displacement

U - global displacement parameter vector

ū - boundary condition matrix

u(x) - unknown scalar function of a field variable

u - B longitudinal slip of bottom component

uh - (x) approximate unknown scalar function of a field

variable

u - I nodal parameter

u - T longitudinal slip of top component

v - vertical displacement

v - B bottom component deflection

v - full vertical displacement due correspond tofull

interaction

v - slip vertical displacement correspond to slip

v - T top component deflection

W - potential load energy

w(x-xI - ) moving weight function

x - quadrature point/ spatial coordinates

y - distance of mass centre of components from

barycentre

z - distance between the centroidal axis of the top and

lower components

α - s dimensionless size of support domain

β - shape control parameter

∆ - s interface slip

∆ - u vertical uplift

δ - variational term

δu - test function

ε - Strain

λ - Langrange multiplier

xxvii

σ - Stress

σ - normal interfacial normal stress

∏ - total potential energy

µ - coefficient of friction

Φ - shape function vector

Φ, - x first derivative of the shape function

Φ, - xx second derivative of the shape function

ω - Frequency

Ω - problem domain

xxviii

LIST OF APPENDICES

APPENDIX TITLE PAGE

A EFG code for one dimensional

composite beam with longitudinal slip

208

CHAPTER 1

INTRODUCTION

1.1 Introduction

Composite members consisting of reinforced concrete and steel sections as

well as retrofitted members are widely used in modern building constructions as well

as strengthening and rehabilitation purposes. Since this type of member provides the

beneficial effect of higher bending strength due to its composite behaviour, the

interaction between two materials in the composite conditions should be assumed as

partially interacting, if actual behaviour is required to be considered and analysed.

This behaviour is known as partial interaction at the interface surfaces. Typical

behaviour of partial interaction composite beam is as shown in Figure 1.1.

Figure 1.1 Partial interaction in composite beams

2

Composite member behaviour is mathematically represented by partial

differential equation and can result in higher order when longitudinal slip, uplift and

dynamic effects on the beam are considered. Generally, an analytical solution for

such effect is difficult to obtain due to the high order. Therefore, numerical

techniques such as the Finite Element method (FEM) are widely used, in solving this

type of difficulty.

The Finite Element method (FEM) has been used as one of the methods in

solving differential equations, numerically, for almost five decades circa the 1960’s.

This approximate method is well established and among the most popular choices of

analysis tools, either by engineers or scientists. Due to its vast and deep researches,

the FEM is well accepted throughout the engineering committees, due to its

flexibility in analysing complex geometry and capability to simulate nonlinear

behaviour. Nevertheless the FEM still suffers from two shortcomings; discontinuity

of meshed elements and computing cost of re-meshing.

In recent years, several developments in providing solutions for those

shortcomings have unfolded, hence the initiation of an alternative to the FEM have

been initiated. New methods have been introduced with promising procedures,

where the approximate solution is constructed solely by a set of nodes instead of

element. These new types of methods are usually known as the Meshfree or

Meshless methods. In this study the ‘Meshless’ term is used, as it is favourably used

in most academic papers as compared to the former term.

1.2 Research Background and Problem Statement

The meshless method obtained its name from its mathematical algorithm

ability to discretise the problem domain by simply adding or deleting nodes where

desired. No element mesh is needed to connect those nodes during the discretisation

process which is contrary to the FEM’s procedure. This is possible due to

3

formulation of the shape functions that is based on nodes in local support domains

thus eliminating the needs for pre-defined elements.

The major difference between the meshless method and the FEM is that the

problem domain is discretised only by nodes and the ability to use higher order

continuous shape functions. Liu and Gu (2005) has provided a good overview on the

procedure and differences between both methods as shown in Figure 1.2 and

tabulated in Table 1.1, respectively. Those advantages and the infant status of the

meshless method are factors that motivate this study, as they provide the first

insights for the application of the present method onto partial interaction of

composite beams.

Figure 1.2 Flowchart for the FEM and the meshless methods (Liu and Gu, 2005).

Geometry creation

Mesh generation

Shape functions based on a pre-defined element

Discretised system equations

Solution for field variables

Post-processing

Node generation

Shape functions based on nodes in a local support domain

FEM Meshless

4

Table 1.1: Differences between FEM and meshless method (Liu and Gu, 2005)

Items FEM Meshless method Mesh Yes No Shape function creation Based on pre-defined elements Based on local support domains Discretised system stiffness matrix

Banded, symmetric Banded, may or may not be symmetric depending on the method used

Imposition of essential boundary conditions

Easy and standard Special treatments may be required, depending on the method used

Computation speed Fast Slower compared to the FEM depending on the method used

Accuracy Accurate compared to FDM More accurate than FEM Adaptive analysis Difficulty for 3D cases Easier Stage of development Well developed Infant, with many challenging

problems Commercial software packages availability

Many Few

In regards to the partial interaction analysis of composite beam, as an

alternative method to the Finite Element Method, the meshless method, with

abovementioned advantages can give a new perspective on the research subject.

Proposing a new element formulation for FEM in partial interaction problem is

literally cumbersome, due to problems of shear locking, element remeshing and

computing cost. Thus, new FEM’s element with partial interaction feature will not

be possible in future development of FEM software. Regards to that, meshless

method formulation which apparently formulated to overcome the last two

shortcomings, has also open-up an opportunity for the first shortcoming solution,

unintentionally. However, before the feature can be realised in meshless software, a

one-dimensional formulation of the partial interaction behaviour has to be

formulated. This will be the important step in meshless formulation of partial

interaction behaviour, before it can be extended to higher dimensional problems.

Element Free Galerkin (EFG) is one of the meshless methods that have been

used widely among researchers to model solid mechanic problems. Its formulation is

closer to FEM compares to others meshless method and easily to be extended to

others meshless formulation, such as; meshless local Petrov-Galerkin (MLPG),

Reproducing kernel particle method (RKPM) and hp-clouds. The Element EFG

method was developed by Belytschko et al. (1994). An extensive review (as in

http://en.wikipedia.org/w/index.php?title=Hp-clouds&action=edit&redlink=1

5

Chapter 2) by the study, found that, there are no attempts yet to be found on the

application of EFG method on composite beam with slip and uplift effects. It is

therefore the main interest of this study to establish the EFG formulation for such an

engineering problem.

1.3 Purpose and Objectives of the Study

The purpose of this study is to establish new formulation, namely the

Element Free Galerkin (EFG) method, for the analysis of the partial interaction

behaviour of composite beams. The objectives of this study are as follows:-

a) To derive one-dimensional formulation and algorithm of EFG method for

composite beams with longitudinal slip effect, and validate the result with

established analytical solutions.

b) To conduct numerical tests on various numerical parameters of the developed

EFG formulation in assessing the accuracy of its result.

c) To extend derivation for an additional effect; the vertical uplift at interfacial

faces and subsequently assess the effect of the numerical parameters on its

convergence result.

d) To verify the viability of the EFG formulation by conducting analysis on

various typical composite beam cross-sections with various boundary

conditions.

e) To derive and evaluate the capability of developed EFG formulation in

finding natural frequency of free vibration problem and conduct its numerical

parameters study.

6

1.4 Scope and Limitation of the Study

Formulation of Element Free Galerkin method becomes the main purpose of

this study. The formulation is applied on partial interaction analysis of composite

beams that follows Euler-Bernoulli’s assumptions. Verification of the formulation is

conducted by comparing its result with available analytical solution. The study is

limited to static analysis. Even so initial effort is taken to formulate the method on

free vibration behaviour.

1.5 Outline of Thesis

This present chapter gives a brief introduction on the study which consists of

meshless development and its differences with a well known numerical method, the

finite element. The Element Free Galerkin (EFG) method is highlighted for this

research work and objectives of the study are presented at the end of the chapter. The

flow of the thesis contents is depicted as in Figure 1.3, for ease review.

In Chapter 2, an extensive review on the development of partial interaction

study in composite beam behavior is reviewed. The review starts with an early

development of longitudinal slip effect derivation of composite beam in early 50’s.

Till then, the study area has been extended due to the various application of

composite beam technology in construction industry. This development includes the

derivation of analytical solution and numerical solution, particularly finite element

method. The second part of the review touches on the development of EFG method

in various attempts in solving mechanics problems.

The shape function of numerical solution is the main perimeter that gives

FEM and EFG methods their own numerical characteristic. Therefore, in Chapter 3,

the fundamental of EFG shape function known as Moving Least Square (MLS)

7

method is derived in step-by-step manner. The effects of MLS numerical parameters

are reviewed based on previous researches and general formulation of the method is

reviewed from the weak form perspective. The numerical EFG solution for Euler-

Bernoulli beam problem is presented for introductory reason.

Chapter 4 provides the new developed EFG formulation with its essential

formulation procedure for the application of composite beams with longitudinal slip

at interface surfaces. The code is developed using Matlab programming language

and the results were verified against established analytical solutions.

Selected numerical parameter values are studied in Chapter 5 through various

numerical tests. Several weight functions are used in each numerical test and their

effects on the result are discussed. The convergence rates are plotted for comparison

purposes.

The developed EFG formulation is extended to consider the vertical uplift

effects and it is the main discussion in Chapter 6. Results are compared with

analytical solution in terms of deflection, for validation purposes. EFG formulation

is presented and similar numerical parameters from preceding chapter are

reconsidered in present numerical tests. Suggestions on suitable numerical

parameters for the EFG formulation are made available.

In Chapter 7, the capabilities of developed EFG formulation code are studied

and discussed. Four different cross-sections of composite beam are used in this study

and three types of boundary conditions for each cross-section have been chosen to

verify the code application. Further application on these various section and

boundary condition is extended to partial interaction EFG formulation, where

previously, longitudinal slip EFG formulation was involved.

Chapter 8 comprises the formulation, of the linear dynamic algebraic

equation and numerical study for free vibration problem. This study involves the

finding of the natural frequency of the partial interaction composite beam.

8

Chapter 9 summarises the development of EFG formulation and its capability

in terms of applications. Effects of numerical parameters on the formulation

accuracy are concluded and several recommendations for future works are

suggested.

Chapter 1Problem

Statement and Objectives

of the Study

Chapter 2Review on

Partial Interaction

Study, Analytical Solution and

EFG

Chapter 3Basic

Formulation of EFG’s Shape

Functions and EFG of Euler-

Bernoulli’s Beam

Chapter 4EFG Formulation

of Composite Beam with

Longitudinal Slip effect and its

Algorithm


of Free Vibration of

Partial Interaction Composite

Beams

Chapter 7Application of

EFG for Common

Composite Cross-sections and Boundary

Conditions


of Composite Beam with

Partial Interaction

effect and its Numerical Tests

Chapter 5Numerical Test

of the Longitudinal Slip

Algorithm

Chapter 9Summary,

Conclusion and Suggestion for Future Works

Figure 1.3 Flow of the thesis contents.

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