Luận án Vibration, buckling and static analysis of laminated composite beams with various cross - Sections

MINISTRY OF EDUCATION AND TRAINING 
HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY AND EDUCATION 
NGUYEN NGOC DUONG 
VIBRATION, BUCKLING AND STATIC ANALYSIS OF LAMINATED 
COMPOSITE BEAMS WITH VARIOUS CROSS-SECTIONS 
Ph.D THESIS 
MAJOR: ENGINEERING MECHANICS 
HCMC, December 2019 
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i 
Declaration 
I declare that this thesis is all my own work based on instruction of Associate 
Professor Dr. Trung-Kien Nguyen and Dr. Thuc P. Vo. 
The work contained in this thesis has not been submitted for any other award. 
Name: Ngoc-Duong Nguyen 
Signature: 
ii 
Acknowledgement 
Firstly, I wish to express my deep gratitude to my advisor, Associate Professor 
Dr. Trung-Kien Nguyen, for his warm guidance, suggestions and support during my 
study. He has influenced my career by coaching me the work ethics and 
responsibilities, along with research skills, which are required of a good researcher. 
The completion of this work would not have been possible without his detailed 
advice, constructive criticism and constant encouragement and patience. 
I am also extremely grateful to Dr. Thuc P. Vo at Northumbria University who 
generously spent a great deal of time providing me with alternative viewpoints to my 
ideas through many helpful discussions. His invaluable knowledge, experience and 
moral support proved to be of inestimable value to the revision and completion of 
this thesis. 
In addtion, I am grateful to Dr. Huu-Tai Thai for his comments in my publications, 
and Mr. Thien-Nhan Nguyen for sharing his Matlab code. 
My special thanks are extended to my colleagues at Department of Structural 
Engineering in Falcuty of Civil Engineering, HCMC University Technology and 
Education, who have offered me intellectual stimulation, friendship and provided a 
warm and inspiring environment. 
Finally, I wish to express my deep appreciation to my family and wife for their 
continued encouragement and support during my study. Without their presence, this 
work would have never been possible. 
Ngoc-Duong Nguyen 
iii 
Abstract 
Composite materials are widely used in many engineering fields owing to their 
high stiffness-to-weight, strength-to-weight ratios, low thermal expansion, enhanced 
fatigue life and good corrosive resistance. Among them, laminated composite beams 
are popular in application and attract a huge attention from reseacher to study the 
their structural behaviours. Many theories are proposed for the bending, buckling and 
vibration analysis. They can be divided into classical beam theory (CBT), first-order 
beam theory (FOBT), higher-order beam theory (HOBT) and quasi-three dimension 
(quasi-3D) beam theory. It should be noted that classical continuum mechanics 
theories are just suitable for macro beams. For analysing microbeams, researchers 
proposed many non-classical theories. Among them, the modified couple stress 
theory (MCST) is the most popular and commonly applied owing to its simplicity in 
formulation and programming. In order to accurately predict behaviours of beams, a 
large number of methods are developed. Numerical approaches are used increasingly, 
however, analytical methods are also used by researchers owing to their accuracy and 
efficiency. Among analytical approaches, Ritz method is the most general one, which 
accounts for various boundary conditions, however, it has seldom been used to 
analyse the bending, buckling and free vibration behaviours of beams. This is also 
the main motivation of this study. 
This dissertation focuses on propsing new approximation functions to analyse 
laminated composite beams with various cross-sections and boundary conditions. 
The displacement field is based on the FOBT, HOBT and quasi-3D theories. Size-
dependent effect for microbeams is investigated using the MCST. Poisson’s effect is 
considered by integrating in the constitutive equations. The governing equations of 
motion are derived from Lagrange’s equations. Numerical results for beam with 
various boundary conditions are presented and compared with existing ones available 
in the literature. The effects of fiber angle, length-to-height ratio, material anisotropy, 
shear and normal strains on the displacements, stresses, natural frequencies, mode 
shape and buckling loads of the composite beams are investigated. Some of numerical 
iv 
results are presented at the first time and can be used as the benchmark results for 
numerical methods. Besides, a study on efficacy of approximation functions for 
analysis of laminated composite beams with simply-supported boundary conditions 
is carried out. 
v 
List of Publications 
ISI papers with peer-reviews: 
1. N.-D. Nguyen, T.-K. Nguyen, T.P. Vo, T.-N. Nguyen, and S. Lee, Vibration 
and buckling behaviours of thin-walled composite and functionally graded sandwich 
I-beams, Composites Part B: Engineering. 166 (2019) 414-427. 
2. N.-D. Nguyen, T.-K. Nguyen, T.P. Vo, and H.-T. Thai, Ritz-based analytical 
solutions for bending, buckling and vibration behavior of laminated composite 
beams, International Journal of Structural Stability and Dynamics. 18(11) (2018) 
1850130. 
3. N.-D. Nguyen, T.-K. Nguyen, H.-T. Thai, and T.P. Vo, A Ritz type solution 
with exponential trial functions for laminated composite beams based on the modified 
couple stress theory, Composite Structures. 191 (2018) 154-167. 
4. N.-D. Nguyen, T.-K. Nguyen, T.-N. Nguyen, and H.-T. Thai, New Ritz-
solution shape functions for analysis of thermo-mechanical buckling and vibration of 
laminated composite beams, Composite Structures. 184 (2018) 452-460. 
5. T.-K. Nguyen, N.-D. Nguyen, T.P. Vo, and H.-T. Thai, Trigonometric-series 
solution for analysis of laminated composite beams, Composite Structures. 160 
(2017) 142-151. 
Domestic papers with peer-reviews: 
6. T.-K. Nguyen and N.-D. Nguyen, Effects of transverse normal strain on 
bending of laminated composite beams, Vietnam Journal of Mechanics. 40(3) (2018) 
217-232. 
7. X.-H. Dang, N.-D. Nguyen, T.-K. Nguyen, Dynamic analysis of composite 
beams resting on winkler foundation, Vietnam Journal of Construction (8-2017) 123-
129. 
8. N.-D. Nguyen, T.-K. Nguyen, T.-N. Nguyen, Ritz solution for buckling 
analysis of thin-walled composite channel beams based on a classical beam theory, 
Journal of Science and Technology in Civil Engineering (STCE)-NUCE. 13(3) 
(2019) 34-44. 
vi 
Conference papers: 
9. N.-D. Nguyen, T.-K. Nguyen, T.-N. Nguyen, and T.P. Vo, Bending Analysis 
of Laminated Composite Beams Using Hybrid Shape Functions, International 
Conference on Advances in Computational Mechanics. (2017), (503-517). 
10. N.-D. Nguyen, T.-K. Nguyen, Free vibration analysis of laminated composite 
beams based on higher – order shear deformation theory. Proceeding of National 
Confrence-Composite Material and Structure (2016) 157-164. 
11. N.-D. Nguyen, T.-K. Nguyen, and T.P. Vo, Hybrid-shape-functions for free 
vibration analysis of thin-walled laminated composite I-beams with different 
boundary conditions, Proceeding of National Mechanical Confrence (2017) 424-433 
vii 
Table of content 
Declaration .................................................................................................................. i 
Acknowledgement ...................................................................................................... ii 
Abstract ..................................................................................................................... iii 
List of Publications ..................................................................................................... v 
Table of content ........................................................................................................ vii 
List of Figures ........................................................................................................... xi 
List of Tables ............................................................................................................. xv 
Nomenclature .......................................................................................................... xix 
Abbreviations ......................................................................................................... xxii 
Chapter 1. INTRODUCTION ..................................................................................... 1 
1.1. Composite material ........................................................................................... 1 
1.1.1. Fiber and matrix ......................................................................................... 1 
1.1.2. Lamina and laminate .................................................................................. 1 
1.1.3. Applications................................................................................................ 2 
1.2. Review .............................................................................................................. 3 
1.2.1. Literature review ........................................................................................ 4 
1.2.2. Objectives of the thesis .............................................................................. 6 
1.2.3. Beam theory ............................................................................................... 7 
1.2.4. Constitutive relation ................................................................................. 10 
1.3. Organization ................................................................................................... 13 
Chapter 2. ANALYSIS OF LAMINATED COMPOSITE BEAMS BASED ON A 
HIGH-ORDER BEAM THEORY ........................................................................... 15 
2.1. Introduction ..................................................................................................... 15 
2.2. Beam model based on the HOBT ................................................................... 16 
2.2.1. Kinetic, strain and stress relations ............................................................ 16 
2.2.2. Variational formulation ............................................................................ 17 
2.3. Numerical examples ....................................................................................... 20 
2.3.1. Static analysis ........................................................................................... 22 
2.3.2. Vibration and buckling analysis ............................................................... 25 
viii 
2.4. Conclusion ...................................................................................................... 31 
Chapter 3. VIBRATION AND BUCKLING ANALYSIS OF LAMINATED 
COMPOSITE BEAMS UNDER THERMO-MECHANICAL LOAD ................... 32 
3.1. Introduction ..................................................................................................... 32 
3.2. Theoretical formulation .................................................................................. 33 
3.2.1. Beam model based on the HOBT ............................................................. 34 
3.2.2. Solution procedure ................................................................................... 34 
3.3. Numerical results ............................................................................................ 36 
3.3.1. Convergence study ................................................................................... 37 
3.3.2. Vibration analysis ..................................................................................... 38 
3.3.3. Buckling analysis ..................................................................................... 41 
3.4. Conclusions ..................................................................................................... 47 
Chapter 4. EFFECT OF TRANSVERSE NORMAL STRAIN ON BEHAVIOURS 
OF LAMINATED COMPOSITE BEAMS ............................................................. 48 
4.1. Introduction ..................................................................................................... 48 
4.2. Theoretical formulation .................................................................................. 49 
4.2.1. Kinetic, strain and stress relations ............................................................ 49 
4.2.2. Variational formulation ............................................................................ 50 
4.3. Numerical results ............................................................................................ 55 
4.3.1. Cross-ply beams ....................................................................................... 56 
4.3.2. Angle-ply beams....................................................................................... 62 
4.3.3. Arbitrary-ply beams ................................................................................. 70 
4.4. Conclusions ..................................................................................................... 74 
Chapter 5. SIZE DEPENDENT BEHAVIOURS OF MICRO GENERAL 
LAMINATED COMPOSITE BEAMS BASED ON MODIFIED COUPLE STRESS 
THEORY .................................................................................................................. 76 
5.1. Introduction ..................................................................................................... 76 
5.2. Theoretical formulation .................................................................................. 78 
5.2.1. Kinematics ................................................................................................ 78 
5.2.2. Constitutive relations ................................................................................ 80 
ix 
5.2.3. Variational formulation ............................................................................ 81 
5.2.4. Ritz solution ............................................................................................. 82 
5.3. Numerical results ............................................................................................ 83 
5.3.1. Convergence and accuracy studies ........................................................... 83 
5.3.2. Static analysis ........................................................................................... 87 
5.3.3. Vibration and buckling analysis ............................................................... 94 
5.4. Conclusions ..................................................................................................... 99 
Chapter 6. ANALYSIS OF THIN-WALLED LAMINATED COMPOSITE BEAMS 
BASED ON FIRST-ORDER BEAM THEORY ................................................... 101 
6.1. Introduction ................................................................................................... 101 
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